Many articles and videos also like to lead quantum mechanics in this direction, preaching that "one look" can determine the life or death of a cat, telling you how "terrifying" double-slit experiment is, and bringing consciousness and quantum mechanics together. etc. As a result, quantum mechanics has become more and more mysterious, weird, and terrifying in the eyes of the public.
In fact, quantum mechanics is not strange. You think it is strange, mainly because you always look at quantum mechanics from the perspective of classical mechanics, just like lightning is also strange in the eyes of the ancients.
We have been immersed in the classic world since childhood. Many classic concepts have become part of the subconscious. If you look at the quantum world like this, you will naturally find it strange. However, if you change your perspective and try to see the quantum world from a quantum perspective, you will find that everything is natural.
So, how to view the quantum world from a quantum perspective?
If we want to understand the way quantum mechanics looks at the world, we must first understand the way classical mechanics looks at the world. Only by knowing how classical mechanics views the world can we know which concepts are unique to classical mechanics and which concepts need to be modified after entering quantum mechanics. Only then can we know how to build a new quantum worldview .
So, what is the world of classical mechanics like?
1 The classic world
Everyone has learned Newtonian mechanics in middle school. I wrote in "What is high school physics?" "Also introduced in ".
In Newtonian mechanics , if you want to know how an object will move, you need to see what force F it receives, and then use Newton's second law F=ma to calculate its acceleration a. After calculating the acceleration, we can know how the object's motion state will change, and we can calculate its state at the next moment based on the object's current state (such as where the object is and what its speed is).
That is to say, in Newtonian mechanics, as long as we master the force condition of an object, we can know its state at any time based on the object's initial state. For example, we know that the fall of apple is due to the gravity of the earth. Knowing the gravity can know the acceleration of the falling apple, and then know the speed and position of the apple at any time.
This is a very typical example, and everyone is used to handling the movement of objects in this way. However, this very natural processing method implies an extremely important assumption: We know that Apple must be somewhere in space at a certain moment, and there must be a certain speed, whether we have it or not Go measure . What does
mean?
If you measure the position and speed of Apple, you will definitely get a numerical value. And, you know that no matter who does the measurement or how many times you measure it, it won’t change the result. It is impossible to say that Zhang San measured the apple on the tree, and Li Si went to measure it, and the apple ran to the ground. At most, the measuring instrument would bring a little error.
In other words, classical mechanics believes: The mechanical quantity of an apple has a definite value at any time, and its position and speed are determined. It has nothing to do with whether you measure it or how you measure it . No matter who does the measurement, no matter how it is measured, or how many times it is measured, the measurement results should be the same within the error range. Because, we are all convinced that Apple must have a determined position and speed. The measurement is just to know what the determined value is. This is common sense in our common sense.
If someone comes to you and says: No, Apple has no definite position and speed. If you want to know where Apple is, you have to measure it, and the measurement result will be there. Moreover, the measurement results of different people can be completely different. Zhang San can measure the apples on the tree, and Li Si can measure the apples on the ground. You must think this person is crazy.
Yes, any mechanical quantity has a definite value at any time, and it has nothing to do with measurement . This is the belief classical mechanics engraved deep in our souls.
But is this belief really absolutely reliable? Is it possible that it is not as natural as imagined?
With this question, let’s take a look at the famous Stern-Gerlach experiment .
2 Stern-Gerlach experiment
Since you think that mechanical quantities have definite values at any time, and they have nothing to do with measurement. Then let's do an experiment to test it. What should we test? Measure the spin of silver atoms .
Let's not worry about what spin is. Just know that this is an inherent property of particles, like mass and charge .
Then, everyone must know that the spin of a silver atom can only take two values and in any direction, which we record as up and down . In other words, if you measure the spin of a silver atom in any direction, there are only two possible results: either up or down, and there is no other value.
Now that we know the spin and its value, we can start to measure it. What should we use to measure it? Use magnetic field , to be precise, it is uneven magnetic field .
When we let the silver atoms pass through the inhomogeneous magnetic field , the silver atoms will deflect, and different spins will have different deflection directions. We agree that if the silver atom is deflected upward, it is said that its spin is upward ; if the silver atom is deflected downward, it is said that its spin is downward . Of course, this correspondence is not important, as long as we know that different spins will have different deflections. The reason why
chose spin is not because the spin is so special, but because it is simple enough. It is the same if the spin is replaced by position and momentum.
Then, we can start the experiment.
First, we add a magnetic field in the z direction (no special statement in the future, the magnetic field in the article refers to the uneven magnetic field ), and then let a beam of silver atoms pass through this magnetic field.
Since there are many silver atoms, some have spins up and some have spins down, silver atoms with different spins receive different forces in the magnetic field, so the deflection directions are also different. As a result, this beam of silver atoms split into two beams of in the z direction . There is nothing to say (the experimental picture is from "Modern Quantum Mechanics" by Teacher Zhuang Pengfei).
Next, there is the wonderful cascade Stern-Gerlach experiment .
3 Cascade Stern-Gerlach Experiment
The so-called cascade Stern-Gerlach experiment, as the name suggests, is to add a magnetic field after the original experiment and continue the experiment. The magnetic field added later may be the same as the original magnetic field, or may be different from the original magnetic field.
These cascaded Stern-Gerlach experiments have a total of three groups of and . Let’s look at them separately.
The first set of experiments : We first let the silver atoms pass through the magnetic field in the z direction, and the silver atoms split into two beams (original experiment). Then, we block the lower beam of silver atoms and let the upper beam pass through the z direction magnetic field again (Figure 1).
What do you think the result will be?
This result is easy to guess, because the silver atoms passed through the z direction magnetic field once and split into two beams. Then, the spins of the above bunch of silver atoms in the z direction should all be the same (all spin upward). If you let them pass through the magnetic field in the z direction again, they should all be deflected upward, so they will not split.
is right, and the experimental results are indeed like this: after allows one of the beams of silver atoms split in the z direction to pass through the magnetic field in the z direction again, they do not split again .
Next, let’s look at the second set of experiments.
The second set of experiments : Let the silver atoms pass through the z direction magnetic field first. After splitting into two beams, continue to let the upper beam of silver atoms pass through a magnetic field again. The difference is that what passes through this time is not the z-direction magnetic field, but the x-direction magnetic field.
As a result, we see that the silver atoms are split into and two bundles of (Figure 2).
In other words, the silver atoms that have been "screened" once by the z-direction magnetic field have the same spin in the z direction , but the spins of in the x direction seem to be different. Although the result of
is a bit unexpected, it is more or less acceptable. Because, your or will think that all silver atoms have certain values in the z direction and x direction. The first magnetic field screens out all the silver atoms with spin-up in the z direction, and the second magnetic field screens out all the silver atoms with spin-up in the x direction.
This is like a talent show, screening a group of people from different dimensions each time. In the first round, only those with good and good can pass, and in the second round, only those with good and good can pass. Then, those who pass the two rounds of screening are all the elites with good , good character and good .
In the same way, your current or will think that the silver atoms that have passed the two rounds of screening of z direction and x direction must all have silver atoms with spin up in the z direction and spin up in the x direction. These silver atoms are the elites who have gone through two rounds of screening. They are all pure. In the future, whether they pass through the z-direction magnetic field or the x-direction magnetic field, they will spin upward and will definitely not split again.
With this idea, we entered the third group of experiments .
The third set of experiments is to add a z direction magnetic field after the second set of experiments. That is to say, the silver atoms split into two beams after passing through the z direction magnetic field. We let one of the beams pass through the x direction magnetic field (the second set of experiments). After splitting again, we let one of the silver atoms pass through the z direction magnetic field again.
Originally, we thought that after two rounds of screening, silver atoms would spin upward in both the z direction and x direction, and would definitely not split again when passing through the z direction magnetic field again.
However, the experimental results shocked everyone: actually split again (as shown in Figure 3)!
This is a shocking split, this is a split that is puzzling, this is a split that completely draws a clear line with classical mechanics , this is a split that announces the coming of quantum mechanics .
You can think about the reasons why it split again, but as long as you are still thinking about the problem with the thinking of classic mechanics and , you will not find a way out. In other words, as long as you can realize the core reason for this split, you are already standing at the door of quantum mechanics .
Why?
4 Preliminary analysis of the experiment
If you think about the third group of experiments , you should use the analogy of a talent show. In the first round, we selected those with good conduct (spin upward in the z direction), and in the second round we selected those who were good at learning (spin upward in the x direction). Then, those who passed the two rounds of screening should all be excellent in both conduct and learning. people.
At this time, you test this group of people with both good character and academic performance. Logically speaking, whether it is a test of character (z direction) or learning (x direction), they should all be excellent (spin upward). But the test results showed that when we tested the character (z direction) of this group of people with both good character and academic performance, they were actually divided into two groups of people with excellent character and despicable character (split into two bunches in the z direction) , how is this not shocking?
But shock is still a shock. The experiment did indeed happen. Whether you want to believe it or not, the reality is right in front of you.
So, what is the problem? Which link is the problem? Why would a group of people who have passed two rounds of tests with excellent conduct and academic performance be divided into two groups, those with excellent conduct and those with despicable conduct, when their conduct was tested again?
Some people say that the standards for the first round of testing and the second round of testing are different for than for ? For example, the first round of character testing has a lower standard, and the second round of character testing has a higher standard. Therefore, those who passed the first round of testing may indeed fail to pass the second round of testing, resulting in lower performance in the second round of testing. Splitting occurs again (resplitting in the z direction).
sounds very reasonable, but in the experiment, is impossible to . The reason is very simple. In the experiment, we used the magnetic field to measure the spin of the silver atoms, and the magnetic fields are the same. You can suspect that the judges of the talent show are unfair, but you can’t say that the magnetic field is unfair, right?
So, if you plan to find problems with in the test session, then I'm sorry, this road is blocked! If there is no problem in the testing process, then we can only find the reason in the testee .
If the two rounds of test environments are exactly the same as , and a person has excellent conduct in the first round of testing but has despicable conduct in the second round of testing, it can only mean: This person did have good conduct in the first round of testing. Excellent, but turned into despicable by the second round of testing. The test standards have not changed, so the only thing that has changed is this person. It is him who has changed from a person with excellent conduct to a person with despicable conduct..
I know that many people find it difficult to accept this conclusion. The same people have just gone through two rounds of tests. How come they have changed? Of course, we can say that people's hearts are different, and it is not known that he has indeed changed in the two rounds of tests. However, people's hearts can change. The spin state of silver atoms is governed by the laws of physics. How can it change just by saying it?
also measures the spin of silver atoms in the z direction . In the first measurement, the spin was up . Why was the spin down in the second measurement?
If we replace spin with position , then this thing becomes: when the position of the silver atom is measured for the first time, it is in Beijing; when the position of the silver atom is measured for the second time, it becomes Wuhan, this is ridiculous!
In our subconscious mind, an object is where it is, and its position is determined. No matter who measures it, the results of the measurement should be the same several times. Within the error range, it is impossible for one person to measure it at position A and another person to measure it at position B.
However, friends who like to read detective novels must have heard Sherlock Holmes : When you eliminate all impossible situations, what is left, no matter how unbelievable, is the fact !
Because the external test environment is exactly the same, and the magnetic field in the z direction is also exactly the same, the reason why the two measurement results are different cannot come from the external environment, but must come from inside . It must be considered that the state of the person being tested has changed (from excellent conduct to despicable conduct), must be considered to be a change in the state of the silver atom (from spin up in the z direction to spin down), Only then can we explain the above experimental phenomena.
In other words, whether you want to believe it or not, you must accept the fact that " the spin state of the silver atom in the z direction has indeed changed ", so that the two measurement results will be different. And this is something that even classical mechanics would not believe even to death. Therefore, classical mechanics cannot explain the Stern-Gerlacher experiment .
5 New Mechanics
So, why does the spin state of silver atoms in the z direction change? The status has changed, of course, it is affected by other factors. What is affected?
Let’s look at the first set of cascade Stern-Gerlach experiments in : If silver atoms split after passing through the z-direction magnetic field, and we let one of the beams pass through the z-direction magnetic field again, it will not split in .
However, in the third set of experiments in , we just added an x-direction magnetic field between the two z-direction magnetic fields in the first set of experiments. Then, the silver atoms passing through the z-direction magnetic field for the second time split. . The first group of , , did not split. After adding an x-direction magnetic field in the middle (the third group of , ), it split. After such a comparison, it will be found that: can only affect the z-direction spin state of silver atoms. In the middle, the spin of silver atoms in the x direction is measured using .
In other words, measuring the spin of silver atoms in the x direction actually affects the spin state of the silver atoms in the z direction . The measurement will affect the system state , which is new.
In classic mechanics , once the system status is determined, the values of all mechanical quantities are determined. Measurement only reads these values and does not affect them. Where an apple is, its position and momentum are determined. No matter who measures it or how many times it is measured, the position and momentum of the apple will not change. If you measure the position of the apple, of course it will not affect the momentum of the apple.
However, the third set of cascade Stern-Gerlach experiments tells us: after passing through the first z-direction magnetic field, the above beam of silver atoms all spins upward . After passing through the second z-direction magnetic field, part of the silver atoms that originally had an upward spin turned into with a downward spin and (that's why they split). The intermediate operation of measuring the spin in the x direction indeed changes the spin state of the silver atom in the z direction, which is unimaginable in classical mechanics .
Now that we are here, I believe everyone can see it: if we want to describe the Stern-Gerlach experiment , we must develop a completely new mechanical system, because the characteristics displayed by this experiment are already the same as those of classical mechanics Fundamental concepts conflict. In this new mechanical system, "measurement" will have a completely different meaning from its meaning in classical mechanics. It is no longer simply reading out a certain value, but will change the state of the system. , will participate in the evolution of the system to .
This brand-new mechanics is naturally the famous quantum mechanics.
6 Measurement and Status
Realizing that " measurement will change the system status " is a key point, but just knowing this is not enough. You know that measurement can change the state of the system, but how does measurement change the state of the system? The system was originally in this state. What state will it become after the measurement? You have to figure all of this out. How to figure out
? Of course, go back to Stern-Gerlacher experiment .
Let’s go through the third set of experiments again. At the beginning, the silver atoms were chaotic and in various states. After passing through the first z-direction magnetic field, they split into two beams. At this time, we can conservatively draw a conclusion: the bunch of silver atoms deflected upward by all have spins upward, and the bunch of silver atoms deflected downwards spin down .
This conclusion seems reasonable, but is it right? We have just stepped into the door of quantum mechanics, and we must be extremely cautious in drawing any conclusions, because previous intuitions may not be valid yet. We want to judge whether all the silver atoms deflected upward have an upward spin. We cannot rely on feeling. We have to go to to measure . How to measure
? If you want to know the spin state of the silver atom in the z direction, just let it pass through the magnetic field in the z direction. If the upward-deflected bunch of silver atoms indeed all spin upward in the z-direction, they will not split when they pass through the z-direction magnetic field again. In fact, we have already done this experiment
. It is the first set of cascade Stern-Gerlach experiments in (making silver atoms that pass through the z-direction magnetic field pass through the z-direction magnetic field again).The experimental results are also very clear: does not split !
In this way, we can draw the conclusion: In the third set of experiments, after the silver atoms passed through the first z-direction magnetic field, the upward deflection beam did indeed spin upward .
However, after this beam of silver atoms passed through the x-direction magnetic field, and passed through the z-direction magnetic field, it split again (the last shocking split). In other words, after passing through the first z-direction magnetic field, the silver atoms all spin upward. However, before passing through the second z-direction magnetic field of , they become the state of spin up and spin down with . Why is this?
Obviously, there is only one x-direction magnetic field sandwiched between the two z-direction magnetic fields, so this change can only be caused by this x-direction magnetic field.
Therefore, the third set of cascade Stern-Gerlach experiments forces us to admit the fact: After silver atoms pass through the magnetic field in the x direction, they spin upward from the z direction to It becomes , which has both spin-up and spin-down states in the z direction.
7 Dead knot
Although this conclusion is a bit strange, it does not seem to be that difficult to accept. Because we have accepted that " measurement will change the system state ", it is not surprising that measuring the x-direction spin will slightly affect the spin state of some silver atoms in the z-direction.
But is it so simple? Let's keep digging.
Do you think that measuring the spin of in the x direction will affect the spin of a part of the silver atoms in the z direction of , so that some of the silver atoms that were originally spin-up will become part of the spin-up, part of the spin-down, and then With the rear split. But the question is: Which part of the silver atoms will change the state of ? ?
We are all equal silver atoms. Now some people say that you pick out a part and spin it down. So which part should I pick? No matter which part you choose, everyone will be unconvinced. Why? Everyone is the same, why choose it instead of me?
In order to expose this contradiction more sharply, let us make another assumption: Assume that the silver atoms passing through the magnetic field in the x direction are not a bunch, but and . What do you think the result will be? After passing through the magnetic field in the x direction, will its spin in the z direction be upward or downward?
Are you sure that must be spinning up to ? No, you don’t dare!
Because I picked a silver atom at random. If you are sure that the spin of the silver atom in the z direction must be upward after passing through the magnetic field in the x direction, can the same be true for other silver atoms? have to? If all the silver atoms pass through the x-direction magnetic field, their spins in the z-direction change upward, then there will not be that shocking split after the passes through the z-direction magnetic field for the second time.
In the same way, you are not sure that after the silver atom passes through the x-direction magnetic field, its spin in the z-direction must be downward .
However, after passing through the x-direction magnetic field, this beam of silver atoms has indeed become a state with both spin up and spin down in the z-direction. Otherwise, they will pass through the z-direction magnetic field for the second time. will no longer be divided.
In other words, faces the exact same bunch of silver atoms, and after passing through the same magnetic field, you can neither be sure that a certain silver atom must spin up, nor can you be sure that it must spin down. However, this beam of silver atoms must contain both spin-up and spin-down states, so that there will be subsequent split .
This seems to be a dead end of and , which is an unsolvable problem. Because the states of these silver atoms are all the same, but for each silver atom, it can neither be spin up , nor spin down .The experimental results require that this beam of silver atoms must contain both spin-up and spin-down states. Otherwise, there will not be that shocking split after passing through the magnetic field in the z direction for the second time. This is contradictory no matter how you look at it. contradiction! What to do with
?
does seem to be in a desperate situation, but there is a possibility in the crack. Although this possibility seems too earth-shattering and impossible, there seems to be no other way. This possibility is: We can only assume that each silver atom itself has a spin-up and spin-down state, and it is in a superposition state of spin-up and spin-down . What does
mean?
8 superposition state
means that we can no longer look at the spin of silver atoms in black and white. You can't think of a silver atom as either spin up or spin down, it can also be in both states at the same time, in their superposition state. If you measure the spin of a silver atom, the result will be either spin up or spin down. One person plays two roles.
Only in this way can we not only satisfy "the state of all silver atoms in is the same as " (all are superposition states of spin up and spin down), but also satisfy " contains two types: spin up and spin down. Status ", thereby untying the above knot.
Before, you thought a person was either an infantryman or an artilleryman. Now, you find that he can also be a special forces soldier, an infantryman and an artilleryman. A group of special forces that are exactly the same as can be "split" immediately into an infantry team and an artillery team according to battlefield needs, just like silver atoms split after passing through the z-direction magnetic field for the second time.
If the silver atom can be in a spin-up state or a spin-down state, or it can be in a superposition state of spin-up and spin-down, then we can think that after passing through the x-direction magnetic field Each silver atom of is in the superposition state with spin up and spin down in the z direction. Therefore, when passing through the z-direction magnetic field for the second time, each silver atom may be deflected upward or downward, thus splitting into two beams.
The core point here is: before passes through the magnetic field in the z direction for the second time, it does not mean that half of the silver atoms have spins upward and half of the silver atoms have spins downwards. After passing the magnetic field, the half with upward spins is biased upward. The half that spins downward is biased downward. Rather, each silver atom is in a superposition state of spin up and spin down (the states are the same). Each silver atom does not know whether it will deflect upward or downward before passing through the magnetic field in the z direction. Only after passing the magnetic field did I know .
Although these two situations will cause the silver atoms to split into two bundles, the essence is completely different: in the former, not every silver atom is in the same state, and the spin of each silver atom is determined. This is in classical mechanics. It can also appear in ; the latter is that the state of each silver atom is the same, and they are all in a superposition state, which is a situation only found in quantum mechanics and .
In this way, we solved the knot by introducing the superposition state , and explained the third set of and cascade Stern-Gerlach experiments in a more reasonable way.
is opposite to the superposition state . We call the state in which silver atoms are in a certain spin-up or spin-down state eigenstate . In other words, the current silver atom can be in a spin-up eigenstate, a spin-down eigenstate, and a superposition state of spin-up and spin-down.
9 rerun experiment
introduced superposition state and eigenstate . Let's go through the third set of cascade Stern-Gerlach experiment again.
Silver atoms split into two beams after passing through the z-direction magnetic field for the first time. The upper beam of silver atoms spins upward (because the first set of experiments in tells us that this beam of silver atoms will not spin after passing through the z-direction magnetic field again. split), that is, the eigenstate that is all in the z-direction spin upward.
I have repeatedly emphasized that " measuring " has a completely different meaning in quantum mechanics than in classical mechanics. It is no longer a simple display, but needs to participate in the evolution of the system.
We let the silver atoms pass through the magnetic field in the z direction. This is a measurement. What is being measured? Measure the spin of silver atoms in the z direction. We don’t know what state the silver atoms were in before passing through the first z-direction magnetic field, but after the magnetic field measurement, the upward-deflected silver atoms are in an eigenstate with an upward spin in the z-direction, and the downward-deflected silver atoms are in an eigenstate with an upward spin in the z-direction. The bunch of silver atoms is in the eigenstate with the spin down in the z direction.
So, we found that: measuring the spin of the silver atom in the z direction will change the silver atom from its original state to the spin eigenstate in the z direction. The measurement will change the state of the system in this way.
passed through the first z-direction magnetic field of , and the above beam of silver atoms will next pass through the x-direction magnetic field . Similarly, we have reason to believe that letting the silver atom pass through the x-direction magnetic field will also change it from its original state to the spin eigenstate of x-direction .
After passing through the x-direction magnetic field, the silver atoms split into two beams. Obviously, the upward deflection is in the x-direction spin-up eigenstate, and the downward deflection is in the x-direction spin-down eigenstate. And this beam of silver atoms can split , which means that before they pass through the x-direction magnetic field , must be in the superposition state with the x-direction spin up and down.
So, we have clarified the state of before and after the silver atom passes through the x-direction magnetic field : before passes through the x-direction magnetic field, the silver atom is in the spin superposition state in the x-direction, and is also in the spin-up state in the z-direction. eigenstate (because it has just passed the first z-direction magnetic field); after passing the x-direction magnetic field, the silver atom is in the x-direction spin eigenstate .
In other words, after passing through the magnetic field in the x direction, the spin of the silver atom in the x direction has indeed changed from a superposition state to an eigenstate. What about the spin of in the z direction? Before passing through the x-direction magnetic field, the silver atom is in the spin eigenstate in the z-direction. Then, after passing through the x-direction magnetic field, will its spin in the z-direction change?
0 Not interchangeable
At first glance, this problem is a bit strange: we let the silver atoms pass through the x-direction magnetic field, and what we measure is the spin of the silver atoms in the x-direction. It only affects the spin in the x-direction. Your spin in the z-direction. Why are you here to join in the fun? The spin in the z direction is still cool and idle. Before you pass through the x direction magnetic field, the spin eigenstate in the z direction is in the spin eigenstate. After passing through, you will continue to maintain the eigenstate . Okay, don't join in the fun.
However, if we think about it carefully, we find something is wrong: in the third set of experiments of , the silver atoms that passed through the x-direction magnetic field will then pass through the z-direction magnetic field for the second time and split (that is, the last shocking split) ). The silver atoms split after passing through the second z-direction magnetic field, which means that the silver atoms must be in the z-direction spin superposition state before passing through the second z-direction magnetic field.
and before passing the second z-direction magnetic field and after passing the x-direction magnetic field are at the same time. Therefore, before and after passing the x-direction magnetic field, the spin state of the silver atoms in the z-direction is also clear: before passing the x-direction magnetic field , the silver atom is in the z-direction spin-up eigenstate ; after passing the x-direction magnetic field (before the second z-direction magnetic field), the silver atom is in the z-direction spin superposition state .
In other words, measures the spin of the silver atom in the x direction (through the x-direction magnetic field), which not only changes the silver atom from a superposition state to an eigenstate in the x direction, but also changes the spin of the silver atom in the z direction. The upward eigenstate becomes the superposition state .
This is a conclusion that seems completely unreasonable in classical mechanics . If you measure the spin of the silver atom in the x direction, it only affects the spin of in the x direction. Why does it also affect the spin of in the z direction? Isn’t this a dog meddling in other people’s business?
Moreover, if measuring the spin in the x direction will affect the spin in the z direction, will it also affect other mechanical quantities? Will the spin in the y direction be affected? Will momentum, position, and energy be affected? If one mechanical quantity is measured, all mechanical quantities will be affected, wouldn't the world be in chaos?
Fortunately, things are not so messed up. Although measuring the spin in the x direction will affect the spin in the z direction, it does not offend everyone. It only offends the mechanical quantities that are not interchangeable with and . .
If two mechanical quantities are commutable to , they are independent of each other. Who is measured first and who is measured later does not affect the result. They can have a common eigenstate and can be measured accurately at the same time; if two mechanical quantities are not commutative , they are not independent. Generally speaking, whoever measures it first will have different results. They do not have a common eigenstate and cannot be measured accurately at the same time.
Obviously, the x-direction spin and the z-direction spin are not commutable with and , so measuring the x-direction spin will affect the z-direction spin. After measuring the x-direction spin, the silver atom is in the x-direction spin eigenstate and also in the z-direction spin superposition state. At this time, there is a definite value when measuring the spin in the x direction, but there is no definite value when measuring the spin in the z direction.
Therefore, if two mechanical quantities are not commutative (such as x- and z-direction spin, position and momentum), they cannot be in eigenstates at the same time. The system is in the eigenstate of one mechanical quantity. If this mechanical quantity can be measured accurately, the other mechanical quantity will be measured inaccurately because it is in a superposition state. Therefore, you cannot measure them accurately at the same time. This is the so-called uncertainty principle .
Of course, regarding the uncertainty principle, I will only mention it in passing. Now we only need to know that measuring the spin of in the x direction will not only put the silver atoms in the x-direction eigenstate, but also affect the z-direction spin, causing the silver atoms to change from the spin-up eigenstate to the superposition state in the z-direction. That's it.
In this way, the third set of Stern-Gerlach experiments can be completely solved: after passing through the first z-direction magnetic field of , the silver atoms become the z-direction spin eigenstate, and the upward deflection of the silver atoms After passing through the x-direction magnetic field, it becomes the x-direction spin eigenstate. At the same time, since the spins in the z direction and the x direction are not commutative, they cannot be in eigenstates at the same time. Therefore, when the silver atom is in the x-direction spin eigenstate, it will change from the spin-up eigenstate to the superposition state in the z direction.
Therefore, the silver atoms in the z-direction spin superposition state naturally split after passing through the second z-direction magnetic field of . This is the last shocking split, which is the split that makes classical mechanics puzzling.
At this point, the Stern-Gerlach experiment has passed on .
1 Quantum Mechanics
As you can see, in order to explain the Stern-Gerlach experiment, we have introduced many brand-new assumptions. We assume that silver atoms can be in a superposition state of spin up and spin down, assume that measurement will affect the state of the system, and assume that if two mechanical quantities are not commutative, measuring one mechanical quantity will affect the situation of the other...
These assumptions It is completely beyond the scope of classical mechanics , but following the Stern-Gerlach experiment, you will find that it is necessary. Physicists are actually very conservative. As long as classical physics can still be used after being tinkered with, people will not overturn the table. quantum mechanics was forced out.
With these new hypotheses, we can qualitatively analyze the Stern-Gerlach experiment. However, qualitative analysis alone is not enough. We must also use the mathematical language to quantitatively describe them .
For example, you said that silver atoms can be in a superposition state of spin up and spin down. How to describe this state? Whether the system is in a superposition state or an eigenstate, the results of measuring spin will be completely different. So how to describe the mechanical quantity of spin? The system status has changed, how to describe it? etc.
We know that when the system is in different states, measuring mechanical quantities will have different results: in the eigenstate, the measurement results are certain; in the superposition state, the measurement results are uncertain. If the system state changes, the measurement results of each mechanical quantity will also change accordingly.
In this context, system status is in a very core position. Therefore, we must first describe the system state. Then, how to describe the system state? The old way, if we want to know what is going on in quantum mechanics and , let's go to classical mechanics first. In Classic Mechanics , how do we describe the system state?
Suppose there are two apples, one in Beijing and one in Wuhan. We will feel that their status is different because their locations are different. Of course, even if their positions are the same, if one is stationary and the other is moving, we will still feel that their status is different, unless their position and speed are the same.
In other words, in classical mechanics, we can use mechanical quantities such as the position and velocity (or momentum) of an object to describe the state of the system .
If the position and momentum (velocity) of two particle points are the same, their state in space and time is uniquely determined. In Hamiltonian mechanics , which is equivalent to Newtonian mechanics, we will use position and momentum as the horizontal and vertical axes to construct something called phase space , a point in the phase space (with a definite position and momentum) It represents a state of movement.
At the same time, since position and momentum can be directly observed, we use these observables to describe the system state, so there is no difference between system state and observable . In addition, in classical mechanics, no matter what state the system is in, the measurement results are certain. Therefore, there is no difference between the measurement results and the observables .
Therefore, in classical mechanics, there is no difference between system state , observable quantity and observation result . They can all be described by position and momentum . If you want to determine the state of a particle, just determine its position and momentum; the observable quantities of the particle are also position and momentum; the final observation result is nothing more than reading the values of position and momentum.
However, the observation results in quantum mechanics are related to the state of the system. Whether the system is in an eigenstate or a superposition state, the observation results will be very different. Observable quantities such as spin and position are not the same thing as the state of the system. In this case, if you want to use position and momentum to send them three, it will be impossible.
So, when it comes to quantum mechanics, how do we describe the state of the system?
2 system state
Can we directly use observables to describe the system state like classical mechanics? For example, the spin of silver atoms can be oriented up or down, then we use S=0 to represent the state of spin up to , use S=1 to represent the state of spin down , and use the variable S like this Is it okay to describe the system status?
doesn’t work!
If the silver atom is only in the eigenstate , we can indeed use S=0 to describe the spin-up eigenstate and S=1 to describe the spin-down eigenstate. But what if the silver atoms are in the superposition state ?
Some people say, then I use S=0.5 to describe the silver atom being in a superposition state of spin up and down. S=0.7 means there is a greater probability of spin down during measurement, and S=0.3 means there is a greater probability of spin down. Can it be rotated upward?
works in this particular case, but it cannot be generalized.It happens that spin here can only take discrete values such as S=0 and S=1. What if what we are discussing now is not spin, but position ? The position x of the silver atom itself can take continuous values, and x=0.3 can only represent the eigenstate of a certain position. So how do you express the superposition state of the position?
Therefore, it is not possible to use a variable S to describe the spin state of silver atoms, because the variables are not enough. What should I do if there is not enough? It's simple. If one is not enough, just add another one. It doesn't waste electricity anyway.
For example, we can use S0 to represent the spin-up eigenstate and S1 to represent the spin-down eigenstate. If the silver atom is in a superposition state, we add them up and use S=S0+S1 to describe the superposition state. Wouldn’t be enough?
If you want to change the weight of the overlay, just adjust the coefficients in front of S0 and S1. For example, we can use S=0.6S0+0.8S1 to indicate that during measurement, there is a probability of (0.6)²=0.36 that the spin is upward, and there is a probability of (0.8)²=0.64 that the spin is downward (why is square we will see later clear).
In this way, no matter whether the mechanical quantity takes a discrete value (spin) or a continuous value (position), we can describe the superposition state . How many values you take, I will get a few variables, what kind of superposition state you are in, I will adjust the coefficients in front of the variables accordingly, and then add them up and that's it.
Moreover, when you write the superposition state of silver atoms as S=S0+S1, if the coefficient in front of S0 is 0, then it is S=0×S0+S1=S1. Isn’t this the spin-down book? Zhengtai ? In the same way, letting the coefficient of S1 be 0 can also represent the spin-up eigenstate. In this way, both the superposition state and the eigenstate can be described in the form of S=S0+S1. Adjusting the coefficients of S0 and S1 can represent superposition states with different weights, and the eigenstate can be regarded as a special (other coefficients are 0) superposition state.
Therefore, using S=S0+S1 to describe the spin state of silver atoms is a good choice.
So, when we write the system status as S=S0+S1, what kind of thing are we doing? Does it look familiar to you? If it doesn’t look familiar to you, let me replace S0 with x and S1 with y, so that S can be written as S=x+y. Doesn’t it look familiar?
Yes, this is a vector !
You see, if we regard S0 and S1 as the abscissa and the ordinate , then they form a plane, and S=S0+S1 represents a vector in this two-dimensional plane. Because the coefficients of S0 and S1 are both 1, S=S0+S1 represents a vector from the coordinate origin (0,0) to (1,1), which is recorded as S=(1,1).
In other words, if we want to describe the state of the system in quantum mechanics, we cannot use a number, we have to use an vector . This vector used to describe the system state is called the state vector .
state vector is determined, and the coefficient (coordinates) of each basis vector is determined. We can know whether the silver atom is in the eigenstate or the superposition state, and know the probability of spin up and spin down during measurement. . Although we don’t know whether the result is spin up or down, but knowing the probability, we can also calculate its average .
In other words, the state vector is determined. Although the specific value of the spin is uncertain, its average value is determined . It is in this sense that we say that the state vector completely describes the state of the system, which is completely different from classical mechanics.
But everyone also knows that spin is an intrinsic property of particles, just like mass and charge, and has nothing to do with the particle's position and speed in space and time. Therefore, when we only consider spin, the spin state space of a particle is actually an internal space. If we do not consider spin, but consider the movement of a particle in external space-time, then we have to look at its position and momentum.
The spin of a silver atom can take two values. We use S=S0+S1 to represent its state. This is a two-dimensional state vector, and the corresponding spin state space is an two-dimensional space .The position can take on infinite values, so we have to use S=S0+S1+S2+... to represent its state. This is an infinite-dimensional state vector, and the corresponding state space is an infinite-dimensional space .
If you want to describe both the spin of a particle and its situation in external space-time, then you have to "add" these two state spaces, which is mathematically a tensor product for them.
It can be seen that common vectors are in two-dimensional and three-dimensional Euclidean space, but state vectors can be in infinite-dimensional space . In addition, the state vectors in quantum mechanics are no longer limited to real numbers, but have been expanded to complex numbers . I don't plan to go into more detail on this part of mathematics. Everyone just needs to know that the space where the state vector is located is not a European space, but a space in which the range of is larger than . This space is called Hilbert space , and the state vector is a vector in Hilbert space.
In other words, In quantum mechanics, we use vectors in Hilbert space to describe the system state . This is our first very important conclusion.
3 Mechanical Quantities
Knowing how to describe the state of a system is a huge improvement, but there is a problem: what describes the state of the system is a vector in Hilbert space , which cannot be directly observed by . Think about it, the state vector is a vector in two-dimensional, three-dimensional, N-dimensional, or even infinite-dimensional space. Can you directly observe it?
cannot!
In classical mechanics , we use position and momentum to describe the state of the system, and the position and momentum themselves can be directly observed. When it comes to quantum mechanics , the state of the system is described by the state vector in the Hilbert space, which cannot be directly observed. What can be directly observed are mechanical quantities such as spin, position, and momentum.
So, if your theory does not want to be out of touch with reality, you have to find a way to describe these mechanical quantities . We use the state vector to describe the system state. How to describe the mechanical quantities such as spin, position, and momentum?
We know that the result of measuring spin is related to the system state : the silver atom is in the eigenstate, and the measurement result is the corresponding eigenvalue; the silver atom is in the superposition state, and the measurement result may be that the spin is upward, or it may be that the spin is upward. Spin down. If the state vector is determined, the coefficients (coordinates) in front of each basis vector are determined. Once the coefficient is determined, the probability of each result when measured is also determined.
If the probability distribution is determined, the average of the mechanical quantity is also determined. The average value is that can directly observe , which is very important.
In other words, although the state vector cannot be directly observed by , the mechanical quantities do not have definite values in general. However, if the state vector is determined, the average value of the mechanical quantity is determined. The state vector cannot be directly observed, but the average value of the mechanical quantity can be directly observed. We can start from here.
Since there is no classic correspondence between and , it is inconvenient to understand. Let’s take a look at the more familiar position .
Assume that the electron can only be in two positions: x=1 and x=2. Similar to spin, if the electron is in the position superposition state , there is a certain probability that the electron is found to be at x=1 when measuring the position. The electron is at x=2. If both probabilities are 50%, then the average at the position is x=1×0.5+2×0.5=1.5; if the probability of being at x=1 is 70%, and the probability of being at x=2 is 30%, The average value of at that position is x=1×0.7+2×0.3=1.3.
It can be seen that after the state vector is determined, the probability distribution is also determined. Although the position of each electron is still uncertain (it may be at x=1 or x=2), the average value of the position is determined (two The state vectors correspond to x=1.5 and x=1.3 respectively).
Let me explain here, the method of measuring the average value in classic mechanics is usually to measure once and write down a number, measure again, write down another number, and finally find the average.But you can't do this in quantum mechanics , because the measurement in quantum mechanics will change the state of the system .
The electron is in a certain superposition state. If you measure the position, it will become a certain position eigenstate. If you then measure the electron in the position eigenstate, the measurement result will always be this eigenvalue. This Obviously something is wrong.
Therefore, if you want to measure the average position of electrons in a superposition state, you must prepare many and electrons in the same state in advance, and then and measure the position of each electron respectively. After measuring one electron, record a position (note that each electron is only measured once), then measure the next electron, and finally average all the positions, so that the average position in this state can be measured.
So, we are clear: If the system status is determined, although the mechanical quantity does not necessarily have a definite value, the average value of the mechanical quantity must be determined . The average value can be directly observed, so we build a bridge between the system state and the observable quantity.
In quantum mechanics, the system state is described by the vector in Hilbert space. Now that we want to find the average value of mechanical quantities in the state of and , we must perform some operations on this vector to make it produce an real number (average value) . So, what are the things that can operate and transform vectors?
is the operator of and ! The
operator can act on a vector and turn it into another vector. For example, if we translate a vector to another place, the operation that completes this operation is called the translation operator ; to rotate a vector into another vector is called the rotation operator ; to project a vector to a certain The coordinate axis is called the projection operator .
In other words, if we measure the average position of the electron in a certain state, now you have to use the operator to perform an operation on the state vector describing this state, so that the state vector "vomits" a real number (Of course, the operator directly acting on the vector can only get another vector. If you want to get a number, you have to use its dual vector, we will not go into details here), and let this real number be equal to the average position we measured.
In this case, it seems that there is an operator acting on the state vector , and after some operations, the average position is obtained. In this sense, we say that this operator describes the mechanical quantity of position. Isn’t it an exaggeration to call it position operator ?
In mathematics, the operator can be represented by the matrix . When a vector is multiplied by a matrix, the result can also be a vector, which is equivalent to a transformation of the vector. Among various transformations, there is a very special transformation: transforms a vector as if it lengthens or shortens the original vector by a certain multiple.
Of course, this transformation of the matrix only holds true for some special vectors. We call these special vectors the eigenvector (eigenvector) of this matrix . This stretched or shortened multiple is called the eigenvalue. (eigenvalue).
is named like this. I believe it is not difficult for everyone to see its relationship with quantum mechanics. In quantum mechanics , we use vectors to describe system states and operators to describe mechanical quantities. The operator can be described by a matrix. Therefore, for the operator A, it can also appear that when it acts on a certain state vector |Ψ, it is like stretching the state vector |Ψ by a times. . The equation written as
is: A|Ψ=a|Ψ, which is called the eigenequation of operator A, |Ψ is the eigenstate, and a is the corresponding eigenvalue.
It should be noted that the A on the left side of this equation is an operator, which is described by a matrix, and the a on the right side is a number. Therefore, you must not make an appointment with |Ψ on the left and right sides of the equation, and then get A=a (many beginners are prone to making such a joke).
So, mathematics and physics are right: we use vector to describe the system state, and use operator to describe the mechanical quantity. The operator can be written in the form of a matrix, and the matrix has corresponding eigenvectors and eigenvalues, which correspond to the eigenstates and the possible results when measures mechanical quantities.
In this case, if you want to know what values the mechanical quantities can take, just solve the intrinsic equation A|Ψ=a|Ψ corresponding to the operator A. If you want to know what the average value of a mechanical quantity is in a certain state, you can use operator A to act on the corresponding state vector, and you can calculate it after some operations.
Moreover, the order of different operators generally cannot be exchanged, which is what we mentioned earlier that is not commutative to . This is a very important feature of quantum mechanics.
In this way, as long as you know the situation of operator , you can know the situation of corresponding mechanical quantity . So, we got the second extremely important conclusion: In quantum mechanics, we use operators to describe mechanical quantities, and the order of different operators generally cannot be exchanged.
Since mechanical quantities are closely related to measurement, the third extremely important conclusion of is about measuring : When we measure a mechanical quantity, the measurement result can only be one of the eigenvalues of the corresponding mechanical quantity operator. .
This conclusion hardly needs too much explanation, because this is what we have always done. We have long known that measuring the spin of silver atoms will cause the system to change from a superposition state to an eigenstate, and the measurement result is the corresponding eigenvalue. Now, we just know that these eigenstates and eigenvalues correspond to an operator .
In the Stern-Gerlach experiment, the operator corresponding to spin is Pauli matrix . Solving the eigenequation of the Pauli matrix can obtain two eigenvectors and two eigenvectors. values, respectively corresponding to spin-up and spin-down. To measure the spin of silver atoms, the result can only be one of the two eigenvalues of the Pauli matrix.
Of course, since the measurement result must be real number , there will be certain requirements for the operator (it must be Hermitian operator ), and the specific probabilities can also be calculated, so I won’t go into details.
In this way, the problem of mechanics and is successfully solved.
4 static image
At this time, if there is an electron here, we can know how to describe the state of the electron, how to describe its mechanical quantities, and also know what values the mechanical quantities can take, what is the corresponding probability, and the average value How much do we know about e- and everything about at the moment.
If you are a painter, you can draw the physical image of the electron at the moment, but you can only draw the and image at the moment. Because you don’t know the state of the electron at the next moment, you don’t know the probability distribution at the next moment, and you don’t know the average mechanical quantity at the next moment, so you can’t draw the physical image of the next moment.
So, what we are describing now is an static quantum image, which cannot move. If we want to make static quantum images move and describe the changing quantum world, we have to know what state the system will be in at the next moment.
In other words, we must know how the system state changes over time, and know how to find the state of the system at the next moment based on the current state of the system. This is the problem of quantum dynamics .
So, how to find out how the system status changes over time? Can it be deduced from the above conclusion? No, because now we only know that we need to use vectors to describe the system state, and we don't know how it changes over time.
It’s still the old rules. If you want to know the situation in quantum mechanics , let’s go to classical mechanics first.
In Newtonian mechanics, if you know the position and speed of an object, you can know the state of the object.If you still want to know the status of object and at the next moment, that is, you want to know the position and speed of the object at the next moment, what should you do?
is very simple. Friends who have studied high school physics know it well (if you are not sure, you can read "What is High School Physics?" first): If you want to know the position and speed of an object at the next moment, you must first find the net external force F on the object. Then use Newton's second law F=ma to calculate the acceleration a of the object. With acceleration, we can calculate the speed of an object at the next moment based on its speed at this moment, and then find the position at the next moment. Therefore, we know the state of the object at the next moment.
In other words, the key to the reason why we can find the state of an object at the next moment lies in Newton's second law F=ma. It is precisely because of F=ma that we can calculate the position and speed of the object at the next moment based on its position and speed at this moment, know how the state of the system will change over time, and depict the movement of the object.
In the same way, if we want to make the quantum image also move, and want to know how the system state in quantum mechanics changes with time, we also need to find an equation similar to Newton's second law F=ma.
Then where did Newton's second law come from? Is it deduced from other conclusions of Newtonian mechanics?
Of course not! Every theory has some most basic assumptions. They are the lowest things in the system and cannot be deduced. (Of course, if a more profound theory is discovered in the future and more basic assumptions are made, these can be transferred from there. If they are deduced, that is another matter), and their correctness can only be guaranteed by experiments. Obviously, Newton's second law F=ma is a basic assumption of Newtonian mechanics.
Similarly, the equation describing the change of system state over time in quantum mechanics should also be a basic assumption. It cannot be deduced from other conclusions of quantum mechanics, and its correctness can only be guaranteed by experiments.
In 1925, in the snow-capped Alps, stimulated by various new ideas and accompanied by a mysterious woman, someone obtained this equation that describes the change of the system state over time, and obtained this equation equivalent to Newtonian mechanics. The equation of F=ma, this is the famous Schrödinger equation . The big guy who wrote this equation is naturally Schrödinger .
5 Schrödinger’s work
I believe everyone has heard of the Schrödinger equation, and various popular science books will also mention it. However, most people only know that the Schrödinger equation is important, but they don’t know why it is important or what it is about.
Now everyone knows: Schrödinger's equation is that describes the change of system state over time. It can make static quantum images move, just like F=ma in Newtonian mechanics. Its importance is self-evident.
So, how does the Schrödinger equation describe the change of system state over time?
We know that the system state is described by the state vector (the first conclusion). We use the notation of Dirac and record the state vector as |Ψ. In this way, if you want to know how the system state changes over time, you want to know what values the state vector |Ψ will take at different times t. This is a function about time t, which we record as |Ψ(t) .
t takes different times, |Ψ(t) will have different values. Isn't this the law of how the state vector |Ψ changes with time? Therefore, the Schrödinger equation wants to describe the change of system state over time, which is to explain what kind of rules |Ψ(t) should follow. So, what kind of rules will it follow?
Since Schrödinger equation is a basic assumption of quantum mechanics and cannot be deduced from other conclusions, we can only rely on "guessing". Of course, this is not random guessing, but rather based on factual analysis, using rigorous logic and reasonable imagination to put forward some hypotheses, and then verify them with experiments.
Schrödinger mainly saw the "similarity between optics and mechanics ", and then extended some conclusions of optics to mechanics, and finally obtained the Schrödinger equation .
How did he do it?
First, Schrödinger noticed that geometric optics is the short wavelength limit of wave optics . This is easy to understand. When the wavelength of light becomes shorter and shorter, the light wave looks more and more like light, and wave optics naturally gradually approaches geometric optics.
Then, Schrödinger noticed that the e-function equation , which is the basic equation of geometric optics, is very similar to the Hamilton-Jacobian equation in analytical mechanics. So, Schrödinger thought: If geometric optics is the short-wavelength limit of wave optics, then will analytical mechanics similar to geometric optics also be the limit of some kind of wave mechanics ?
In other words, is it possible to say that our current mechanics is just "geometric mechanics", which is just the limit of some kind of wave mechanics (just like geometric optics is just the limit of wave optics)? Moreover, the short wavelength limit of a certain equation in wave mechanics happens to be the Hamilton-Jacobian equation in "geometric mechanics" ?
answer We all know that this kind of wave mechanics is quantum mechanics , and the short wavelength limit of the Schrödinger equation is the Hamilton-Jacobian equation.
Of course, this is not a coincidence. It does not mean that Schrödinger accidentally discovered an equation, and the limit of this equation happened to be the Hamilton-Jacobian equation. It's the other way around: Schrödinger was looking for something whose limit was the Hamilton-Jacobian equation, and then he found the Schrödinger equation, and the mechanics of this wave is quantum mechanics.
Logically speaking, this idea is very natural. As long as physicists notice the similarity between the e-function equation and the Hamilton-Jacobian equation and know the relationship between geometric optics and wave optics, it is natural to consider whether there is a kind of wave dynamics . So why didn’t it take until Schrödinger to seriously consider this?
In fact, Hamilton himself noticed the similarity between optics and mechanics, so some people say that Hamilton is only one step away from discovering the Schrödinger equation.
But, physics is not mathematics after all. It is responsible for reality. It does not mean that logically established things must exist in reality. At that time, there was widespread consensus on the wave nature of light, but who would think of mechanics and think that stones and apples also have wave properties? Moreover, classical mechanics also worked very well at that time, and people were full of confidence in it. Who would bother to fiddle with wave mechanics?
However, when it came to Schrödinger, the situation was completely different. Classical mechanics has been seriously challenged, and the quantum revolution is in full swing. De Broglie also proposed the revolutionary matter wave idea. At this time, considering the wave nature of general objects and whether there is a kind of wave mechanics, the existing mechanics is only the limit of wave mechanics and has a very realistic basis.
So, Schrödinger began to think, if the current mechanics is only the limit of some wave mechanics, then which wave equation will the current Hamilton-Jacobian equation be? Everyone knows the answer to
, it is the famous Schrödinger equation . In other words, if we let the Schrödinger equation take the short wavelength limit, that is, let the Planck constant h approach 0, it will return to the Hamilton-Jacobian equation in analytical mechanics.
So, if you want to understand the Schrödinger equation, it is best to first understand analytical mechanics .
6 Schrödinger equation
Of course, this article is about popular science quantum mechanics . Here I can only briefly talk about analytical mechanics to let everyone know why the Schrödinger equation is written like this. As for the specific content of analytical mechanics, we will talk about it later. If you are afraid of missing it, just keep an eye on my official account.
Simply put, analytical mechanics is a mechanical system that is completely equivalent to Newtonian mechanics. It has nothing new, but the description method is different from Newtonian mechanics.
The core of Newtonian mechanics is force . When we analyze the movement of an object, we must first analyze the force, and then use Newton's second law F=ma to calculate the movement of the object; the core of analytical mechanics is energy . We There is no need to conduct complex force analysis on the object. Just choose the appropriate generalized coordinates, find the Lagrangian L or Hamiltonian H of the system (knowing one of these two can find the other), and substitute it into The Granian equation or Hamiltonian equation can calculate the motion of the object.
Because the force is the vector , the size and the direction must be considered during analysis, while the energy is the scalar , so only the size should be considered. Therefore, when the environment is more complex and there are many constraints, analytical mechanics starting from energy is often much simpler.
Of course, if analytical mechanics and are just a more useful Newtonian mechanics, a Newtonian mechanics that handles complex problems more simply, we don't seem to have to spend a lot of energy studying it. The biggest advantage of analytical mechanics is that its method of dealing with problems can be easily extended beyond classical mechanics. Both electromagnetic fields and quantum mechanics can be handled in this way, but Newtonian mechanics cannot. This was something that the founders of analytical mechanics such as Lagrange and Hamilton never expected.
In other words, the method of dealing with problems in Newtonian mechanics and cannot be directly transferred to quantum mechanics. We will not analyze the force of objects in quantum mechanics, but use to analyze the mechanics of . set. In analytical mechanics, as long as you know the Hamiltonian H of the system, you can calculate the motion of the system by substituting it into the Hamiltonian equation. The same is true for quantum mechanics .
In other words, in quantum mechanics, if we know the Hamiltonian of the system and substitute it into an equation, we can know how the state of the system will change.
Under normal circumstances, the Hamiltonian H of the system is numerically equal to kinetic energy plus potential energy , which is the total energy of the system. Because energy is also a mechanical quantity, quantum mechanics uses operators to describe mechanical quantities. Therefore, after entering quantum mechanics, the Hamiltonian H has to follow the Romans and become the Hamiltonian operator H.
And we know that in quantum mechanics, it is Schrödinger equation that describes the change of system state |Ψ(t) with time. Therefore, if you can know how the system state changes with time by substituting the Hamiltonian operator H into an equation, then this equation is naturally the Schrödinger equation.
So, Schrödinger equation is such a thing: You give the Hamiltonian operator H(t) of the system, substitute it into the Schrödinger equation, and solve the equation to get the change of the system state over time|Ψ(t) . The specific form of
is as follows:
It can be seen that the main body of the Schrödinger equation is a relationship between the Hamiltonian operator H(t) and the system state changing over time |Ψ(t). i is an imaginary unit and ℏ is approximately Change Planck's constant (ℏ=h/2π), pronounced as h bar. This is a differential equation because it not only contains |Ψ(t), but also the derivation of |Ψ(t) with respect to time t (d/dt).
Knowing the Hamiltonian operator H(t) of the system, we can use to solve the Schrödinger equation to find |Ψ(t), which describes the change of the system state over time. Once you know the state of the system, you know the probability distribution, the average value of various mechanical quantities, and what will happen during measurement. Then you know everything. This is a general idea for analyzing many quantum mechanics problems.
So, we have The fourth extremely important conclusion: The change of system state over time |Ψ(t) obeys the Schrödinger equation. With it, static quantum images can become animated.
7 basic framework
So far, we have summarized four very important conclusions of and :
First, use state vectors to describe the system state;
Second, use operators to describe mechanical quantities, and different operators generally cannot Exchange order;
Third, measure a mechanical quantity, and the result is one of the eigenvalues of the operator of the mechanical quantity;
Fourth, the change of the system state over time obeys the Schrödinger equation .
With these conclusions, the general framework of quantum mechanics has been established.
We know how to describe the system state, and we also know how the system state changes over time, which is equivalent to knowing the state of the system at any time. Therefore, we can know the probability distribution, average mechanical quantity and measurement results of the system at any time, and we also know everything about the system.
Obviously, these four conclusions are not found randomly by me. They are the first four of the five basic assumptions of quantum mechanics. Their importance is self-evident. The last basic assumption is the so-called isomorphism principle . I won’t care about it here for now. We will talk about it later when it comes to multi-particles.
In this way, we start from Stern-Gerlach experiment and build up the basic framework of quantum mechanics step by step.
Seeing this, many people are probably wondering: Why does this seem to be different from the quantum mechanics I expected? In my impression, shouldn’t quantum mechanics be about discontinuity, uncertainty, blackbody radiation, the double-slit experiment, and Schrödinger’s cat? You have been talking about system states, state vectors and operators here. Is this still the quantum mechanics in my mind?
Of course it is!
Quantum mechanics is quantum mechanics. I can’t make up something else to lie to you. What we are doing now is to set up the basic framework of quantum mechanics. As for the things you are familiar with, you can deduce them from here. Learning quantum mechanics can’t just be about watching the excitement. We not only need to know what these phenomena are, but also how they come about.
Next, let’s take a look at how they emerge from the basic framework of quantum mechanics, .
8 An electron
Let’s look at the simplest example first: an electron.
In classical mechanics , an electron is like a small ball. You can tell where it is and what its speed is. It has a certain position and momentum at any time. When you press it, its motion state will change. How it changes, and its subsequent position and speed can all be calculated. If a bunch of electrons are allowed to pass through double slits, classical mechanics will think it is like a bunch of bullets shot through double slits, and there will definitely be no interference fringes.
When it comes to quantum mechanics , the situation is different. You can no longer say where the electron is, because when you say "where is the electron", it implies that the electron at this time has a definite position. After all, only when the location is certain can you tell where it is.
And we know that whether an electron has a definite position depends on its state: when it is in the position eigenstate, the position of the electron is determined by and . When measured, there is a definite value, and you can say where the electron is; in position superposition When in the state, the position of the electron is uncertain . There is a certain probability of being at the eigenvalue of each position during measurement. At this time, it is meaningless for you to say "where the electron is".
Therefore, we cannot take some concepts into quantum mechanics for granted. Some concepts are fine in classical mechanics , but they are not correct in quantum mechanics . We must slowly develop the habit of thinking about problems from a quantum framework, establish a systematic quantum concept, and gradually form a quantum mechanical way of thinking.
In the basic assumptions of quantum mechanics, we use state vectors to describe the system state and operators to describe mechanical quantities.Whether the position of an electron is determined depends on its state, so how to check its state?
In the Stern-Gerlach experiment , the spin of the silver atom can take two values. The corresponding states include the spin-up eigenstate, the spin-down eigenstate, and their superposition. state. The position of the electron can take multiple values of infinite , which corresponds to an infinite number of position eigenstates and their superposition states. We have to use the state vector in the infinite dimensional space to describe it. The
state vector is determined. It is determined whether the electron is in the position eigenstate or the position superposition state. It is also known whether there is a definite value when measuring the position. We can only talk about the position of electrons in this way, but we cannot directly say where the electrons are like classical mechanics. Now that we have talked about the position of
, what should you do if you still care about the momentum of and and want to know whether the momentum of electrons has a definite value? Similarly, if we want to know whether the momentum has a definite value, we look at whether the system is in the momentum eigenstate or the momentum superposition state , or we look at the state vector .
But there is a problem: if we want to see whether the position of the electron is determined, we need to see whether the state vector is the position eigenstate; if we want to see whether the momentum of the electron is determined, we need to see whether the state vector is It is not an eigenstate of momentum. There are two and state vectors here. What is their relationship? Is it the same state vector, or two different state vectors?
After a little thought, you will know: , they must be the same ! The
state vector describes the system state. If the system is already in a certain state, the state vector should be determined. At this time, it is your freedom to analyze position or momentum, and it does not affect the system, so the state vector describing the state of the system will naturally not change.
And you see, in Schrödinger equation uses |Ψ(t) to describe the system state. When time t is determined, |Ψ(t) is determined. In other words, the state vector is only related to time t, and has nothing to do with your analysis of position or momentum.
Besides, the mechanical quantities of electrons are not limited to momentum and position. Does it mean that one more mechanical quantity requires one more state vector? That doesn't make sense.
Therefore, they must be the same state vector in ! In other words, if you want to see whether the position of the electron is determined, you need to see whether the state vector is in the position eigenstate; if you want to see whether the momentum of the electron is determined, you need to see whether the same and state vector is in the momentum eigenstate. .
Then the question arises: If they are the same state vector, then how does this difference come about when analyzing position and momentum ?
9 Appearance
If the electron is in a certain state, the position says that the state vector is in the eigenstate, and there is a definite value when measuring the position; the momentum is wrong, the state vector is obviously in a superposition state, and there is no definite value when measuring the momentum. The position says that the state vector is in an eigenstate, and the momentum says that the state vector is in a superposition state. Neither of them accepts the other, and they all think that they are right and the other is wrong.
This reminds me of the story of blind men touching elephants : A group of blind men were touching an elephant. Someone touched the elephant's body and said the elephant was like a wall; someone touched the elephant's trunk and said the elephant was like a python. ;Someone touched the elephant's tail and said that the elephant was like a rope. The blind people quarreled, and no one was convinced. They all felt that they were right and the others were wrong.
Similarly, there is only and state vector. From the position perspective, the state vector is in the position eigenstate; from the momentum perspective, the state vector is in the momentum superposition state. They both use and , but they just look at the state vector from a different angle. What does
mean?
When it comes to vector , many people's first reaction is an arrow, which is a very abstract image.
If you want to concretize this abstract vector and describe it with a set of specific numbers, you have to do one thing first: establishes a coordinate system .After the
coordinate system is built, for example, if we build an Cartesian coordinate system , we can project the abstract vector into the coordinate system, and the coefficients projected onto each coordinate axis are the corresponding coordinates. Then, we can use specific numbers such as (1,2) to represent the original vector, and the abstract vector is concretized.
Of course, you can establish a Cartesian coordinate system, and naturally you can also establish an spherical coordinate system, or other coordinate systems. If the coordinate system is different, the projection of the same vector on the coordinate axis will be different, and the corresponding coordinates will also be different. The
state vector is also a vector, and of course it can also be decomposed into different coordinate systems.
In the Stern-Gerlach experiment , we use S0 to represent the spin-up eigenstate, S1 to represent the spin-down eigenstate, and then use S=S0+S1 to represent their superposition state, adjust The coefficients of S0 and S1 represent superposition states of different weights. Then, we found that if S0 is used as the abscissa and S1 is used as the ordinate, the state of the silver atom can be represented by an state vector in a two-dimensional space.
In the same way, if we do not consider the spin, but consider the position of the particle in space and time, we can also use a state vector to describe its state.
is different from spin. The position of a particle can generally take infinitely many values of , so that it has infinitely many position eigenstates. We have to use infinitely many eigenvectors |a1, |a2,... , |an,... to describe (eigenstate is also a state, and naturally it is also described by vector).
In the spin , we construct a two-dimensional coordinate system using S0 and S1 representing the spin eigenstates as the coordinate axes; when it comes to the position , we have to use infinite numbers representing the position eigenstates. The eigenvectors |a1, |a2,…,|an,… construct an infinite-dimensional coordinate system, and the state of the particle is described by the state vector in this infinite-dimensional space .
In other words, although the particles in only move in three-dimensional space, the state vector describing the state of the particles is not in the three-dimensional space, but in the infinite-dimensional space , which is easily confused by many beginners.
So, how can we get the eigenvector of the position?
As mentioned before, in quantum mechanics, we use operators to describe mechanical quantities (assumption 2), so we use the position operator to describe the position. Knowing the position operator A, solving its eigenequation A|Ψ=a|Ψ can obtain the eigenvector |Ψ describing the position eigenstate. We then use these eigenvectors as basis vectors to construct a position-related coordinate system.
decomposes the state vector into this coordinate system. If the state vector coincides with the coordinate axis, that is, it coincides with an eigenvector of position , it represents the position eigenstate; if the state vector does not coincide with the coordinate If the axes coincide, it represents a position superposition state. I believe this is not difficult to understand.
In the same way, we can also construct a coordinate system using the eigenvector of the momentum operator as the basis vector, and then decompose the state vector into this momentum-related coordinate system. If the state vector coincides with the coordinate axis, that is, it coincides with the eigenvector of a certain momentum , it represents the momentum eigenstate; if the state vector does not coincide with the coordinate axis, it represents the momentum superposition state.
Obviously, we use the position operator and the momentum operator to construct two coordinate systems of and different . When the state vector coincides with a certain coordinate axis in one coordinate system, it does not have to coincide with the coordinate axis in another coordinate system. In this way, a state vector can be an eigenstate at the position and a superposition state at the momentum. There is no contradiction.
Of course, there is a small question here: in N-dimensional space, can the eigenvector of a mechanical quantity operator form a basis vector to construct a coordinate system?
Whether a set of vectors can form a basis vector in N-dimensional space depends on whether they have N independent vectors. For example, in three-dimensional space, we need to see whether there are three independent vectors. Intuitively, it means whether these three vectors are coplanar.If they are coplanar, then vectors that are not on this plane cannot be expressed by them, and they cannot be called basis vectors.
Regarding this problem, although the mathematics is a bit troublesome, the result is very simple: those eigenvectors with different eigenvalues are orthogonal to each other, even if there are multiple eigenvectors corresponding to the same eigenvalue (simplified) and), we can always find a set of basis vectors. In a word: the eigenvectors corresponding to the mechanical quantity operators can always constitute a set of basis vectors in space, and you can safely use them to construct the coordinate system .
In quantum mechanics, selecting such a set of basis vectors is called selecting a representation . Because the basis vector we selected is the eigenvector of the positional operator, the established representation is called the positional representation , or the coordinate representation . If the selected basis vector is the eigenvector of the momentum operator, then the momentum representation is established.
In this case, the previous question becomes: facing the same state vector in , we can decompose it in the position representation . From the position perspective, the system is in the position eigenstate; we can also decompose it in the momentum representation From the perspective of momentum, the system is in a momentum superposition state, and there is no contradiction between the two.
0 Born's rule
After the representation is selected, we can project the state vector of abstract into the specific coordinate system, and then use the specific coordinates to represent the state vector. And we know that the state vector describes the state of the system (assumption 1), so after entering the specific representation, what is the physical meaning of each coordinate of the state vector?
In the Stern-Gerlach experiment , in order to describe the superposition state of silver atoms, we use S0 to represent the spin-up eigenstate, S1 to represent the spin-down eigenstate, and then use S=S0+ S1 represents the superposition state. If S0 is regarded as the horizontal axis and S1 is regarded as the vertical axis, then the coordinates of the vector S are (1,1). At this time, if we measure the spin of the silver atom, there will be a 50% probability of spin up and a 50% probability of spin down, with the same probability.
If we modify the coefficient and write the superposition state as S=0.6S0+0.8S1, the corresponding coordinates become (0.6, 0.8). At this time, the measured probability of spin-up is (0.6)²=0.36, and the probability of spin-down is (0.8)²=0.64. The two probabilities are different.
In other words, when we construct a coordinate system based on the eigenvector of a mechanical quantity operator, each coordinate axis corresponds to an eigenstate, and the coefficient of the state vector projected onto each coordinate axis ( The square of the coordinate) represents the probability that the measurement result is the eigenvalue corresponding to this eigenstate .
sounds a bit convoluted, but actually it’s very simple if you think about it. Our coordinate system is constructed based on the eigenvectors of mechanical quantities. The longer the projection of the state vector on a certain coordinate axis (the larger the coordinate), the greater the proportion of the eigenstate it "contains". The higher the probability that the measurement result is the corresponding eigenvalue of this eigenstate, the greater the probability. If the state vectors are all projected on a certain coordinate axis, and the projection on other coordinate axes is 0, then the probability that the measurement result is that this eigenstate corresponds to the eigenvalue is naturally 100%. This probabilistic interpretation of the
state vector was first proposed by Born , so it is also called Born's rule . Born also won the Nobel Prize in 1954.
Through Born's rule , we connect the coordinates of the state vector with the probability of the corresponding eigenvalue obtained during measurement.
1 wave function
With these understandings, we can discuss the problem under the specific representation of and .
is still the electron. When we consider the problem under the position representation , we actually construct a coordinate system based on the eigenvector of the electron's position operator, and then project the state vector that describes the electron state. into this coordinate system.
Now only consider the one-dimensional case of , that is, assuming that electrons only move in the x direction.If the electron is in the position eigenstate of x=1, it will be found at the position of x=1 during measurement. Because this is an eigenstate, we need to use an eigenvector to describe it, and the eigenvector is the basis vector of the coordinate system and will correspond to a coordinate axis. Therefore, the position eigenstate of x=1 will correspond to a coordinate axis in the coordinate system.
Of course, in addition to x=1, the position of the electron can also be in infinite places such as x=2, x=2.5, etc. Similarly, each position eigenstate will correspond to a coordinate axis in the coordinate system. In this way, there will be multiple infinite coordinate axes in this coordinate system.
Now, we project the state vector into this coordinate system with infinite coordinate axes. It will have an projection coefficient on each coordinate axis, which is the state vector on this coordinate axis. coordinates .
For example, x=1 is a coordinate axis, which represents the position eigenstate of x=1. The state vector has a projection coefficient on this coordinate axis, which is its coordinate on this axis, which we record as Ψ(1). In the same way, the state vector will also have a projection coefficient (coordinate) at x=2, x=2.5, which we record as Ψ(2), Ψ(2.5) respectively, and so on.
and Born's rule tells us: the square of the modulus of the projection coefficient of the state vector on the x=1 coordinate axis |Ψ(1)|², represents the electron found at x=1 during measurement. Prob . In the same way, |Ψ(2)|² represents the probability of finding an electron at x=2 during measurement. The electron's position The probability of finding an electron here.
In other words, Given a position x that an electron can take, we can all find a corresponding projection coefficient Ψ(x), so that |Ψ(x)|² represents the probability of finding an electron at x .
Given a position x, there is a number Ψ(x) corresponding to it. What is this mapping from number to number?
is the function ! It is a function we learned in junior high school.
Therefore, after entering the positional representation , the projection coefficient (coordinate) of the state vector on each coordinate axis is a function about the position x. We record it as Ψ(x). The name of this function is the famous wave function .
Many friends are very confused about state vector and wave function , because some places say "use state vectors to describe the system state", and some places say "use wave functions to describe the system state", which makes them dizzy. . Obviously one is a vector and the other is a function, but it seems impossible to hit them. Why does the system state seem to be described by both the state vector and the wave function?
The reason is here, because the wave function is bound to the specific representation. Only when we select a specific representation, establish a specific coordinate system, and project the state vector to the coefficient of the specific coordinate system is the wave function .
Therefore, we are right to say "use state vectors to describe the system state", and it is also right to say "use wave functions to describe the system state". Just like we can either say vector a, or we can decompose it into a coordinate system and say this is vector (1,2).
established the position representation . The projection coefficient of the state vector in this specific coordinate system is the wave function Ψ(x). The square of the wave function's module |Ψ(x)|² represents the discovery of this at position x. Probability of electrons. For example, Ψ(1)=0.1 means that the probability of finding an electron at x=1 is 0.1²=0.01, Ψ(2)=0.2 means that the probability of finding an electron at x=2 is 0.2²=0.04, etc. Wait, so the problem becomes concrete.
Of course, you can create a position representation, and naturally you can also create an momentum representation . We can also construct a coordinate system based on the eigenvector of the momentum operator, and then decompose the state vector into this coordinate system.In this way, the projection coefficient of the state vector is the wave function under the momentum image, and the square of its module represents the probability that the electron has this momentum when measured.
Obviously, between different representations, is equivalent to . You can discuss problems in terms of position representation or momentum representation, just like you can choose either Cartesian coordinate system or spherical coordinate system. The same state vector can correspond to the wave function under the position representation or the wave function under the momentum representation. The only difference between them is Fourier transform .
Because everyone usually has more contact with positional representation, some people mistakenly think that quantum mechanics is quantum mechanics under positional representation. He is not very clear about the relationship between position representation and momentum representation , nor is he very clear about the difference between wave function and state vector , so he is always confused.
OK, now we enter the position to represent .
2 Position representation
After entering the position representation, we can use the wave function to replace the original state vector . And we also know that the change of the system state over time follows the Schrödinger equation (assumption four), and the original Schrödinger equation uses the state vector |Ψ(t) to describe the system state:
So, now we The wave function can be used to replace the state vector in the original equation.
Because the Schrödinger equation describes the change of the system state over time, we use the wave function Ψ(x) to describe the system state. The change of the wave function with time t is naturally Ψ(x,t). Therefore, under the position representation, we can use the wave function Ψ(x,t) to replace the original state vector |Ψ(t).
But this is not enough. In order to make the Schrödinger equation more specific, we also expand the Hamiltonian operator H(t).
Regarding the Hamiltonian operator, we talked a little bit before. Here, everyone only needs to know: Generally speaking, if we know the Hamiltonian operator of the system, we will know the situation of the system itself (such as the number, quality and interaction between particles) and the location of the system. external conditions (such as the external electromagnetic field where the particles are located). Basically, knowing the Hamiltonian operator of the system, we know everything about the system.
In classical mechanics , if there is no energy exchange between the system and the outside world, the Hamiltonian H of the system can generally be written as kinetic energy (P²/2m) plus potential energy V, which is numerically equal to the total of the system. Energy :
When it comes to quantum mechanics , mechanical quantities must be described by operators. Then, the Hamiltonian closely connected with the energy naturally needs to be operatorized, and the result of operatorization is the Hamiltonian operator H in the Schrödinger equation.
Obviously, if the Hamiltonian H of the system can be written as kinetic energy (P²/2m) plus potential energy V, if we want to operatorize it, we must operatorize the mechanical quantity inside, that is, the momentum P. Under the positional representation , the result of the momentum P operator is -iℏ∂/∂x. We don't care why it looks like this for now, but everyone should remember that this is only the form of the momentum operator in the position representation, and it does not look like this in other representations.
So, we have gathered all the conditions to write the Schrödinger equation under the positional representation : use the wave function Ψ(x,t) to replace the state vector |Ψ(t), and expand the Hamiltonian operator H into the most common form (P²/2m+V), and found the momentum operator under the position representation (-iℏ∂/∂x).
Then, we can rewrite the Schrödinger equation under the positional representation (only considering the one-dimensional case):
This equation is longer than the original one and looks a bit more complicated. However, it only replaces |Ψ(t) with Ψ(x,t) and expands the Hamiltonian operator H(t).Their core difference is: the original equation of is the general Schrödinger equation without a specified representation. Now this is the Schrödinger equation under the position representation .
Let’s take a look at this equation. i and ℏ are constants, and m is mass. If the potential energy function (generally referred to as potential function ) V(x,t) is determined, then the only unknown quantity is the wave function. Is Ψ(x,t)? One equation has one unknown quantity. By solving the equation, we can get the wave function Ψ(x,t).
In other words, for the Schrödinger equation under the positional representation , as long as the potential function V(x,t) is given, we can solve a corresponding wave function Ψ(x,t) ) (Whether an exact solution can be found is another matter).
Knowing the wave function Ψ(x,t) of the particle, we can know the probability of finding the particle |Ψ(x,t)|² (Born's rule) at any time t and any position x. Once the probability distribution is determined, the average value of the mechanical quantity is also determined. It is in this sense that we say that the wave function completely describes the system state.
In Newtonian mechanics , if an external force is given to an object, the object will have an acceleration and its state will change accordingly. In quantum mechanics , we no longer use "force" to describe external influences, but use potential (energy) functions. For example, if Newtonian mechanics talks about gravity, we will talk about gravitational potential energy here; if Newtonian mechanics talks about elasticity, we will talk about elastic potential energy.
analytical mechanics is a system with energy as the core. It is different from Newtonian mechanics with force as the core. Quantum mechanics follows the logic of analytical mechanics, so what appears in the Schrödinger equation is the potential (energy) function instead of force.
Therefore, as long as we determine the potential function , we can obtain the wave function describing the state of the particle by solving the Schrödinger equation, and then know the various conditions of the particle. In fact, when everyone first starts learning quantum mechanics, a large part of their work is to solve the Schrödinger equation under various potential functions.
For example, for a free-falling particle, its potential energy is gravitational potential energy -mgx, so the potential function V(x,t) is -mgx (excluding time t). We substitute -mgx into the Schrödinger equation and solve the equation to obtain the wave function Ψ(x,t) that describes the particle state. Then, we can know the probability of finding this particle somewhere in 1 second, 2 seconds, n seconds and the average value of various mechanical quantities.
Similarly, for a simple harmonic oscillator, its potential function is V(x)=mω²x²/2 (also not including time t). We substitute it into the Schrödinger equation and after solving the wave function Ψ(x,t), we can also obtain various information about it.
In other words, if we want to understand a quantum system, we usually have to do two things first: First, find the potential function V(x,t) of the system; second, substitute the potential function into the Schrödinger equation and solve the equation Find the wave function Ψ(x,t) that describes the state of the system.
Generally speaking, it is relatively easy to find the potential function. However, the Schrödinger equation is a partial differential equation, so it is not that easy to solve. In fact, we can only solve the Schrödinger equation accurately in very few cases, and more often, we can only use some approximate methods.
In this way, I believe everyone has a general understanding of the basic framework of quantum mechanics, and the general method of quantum mechanics dealing with problems. Then, we can analyze specific problems in this way, and get whatever conclusions we get. This is how the counterintuitive and incredible quantum mechanical properties that everyone is familiar with come from. If you don’t believe it, let’s take a look.
3 discontinuity problem
First, let’s look at a topic that everyone likes to hear: discontinuity .
Many quantum mechanics science popularizations start from blackbody radiation , and tell you that it was Planck that creatively viewed the spread of energy as parts rather than continuous, which solved the problem of blackbody radiation. Thus created quantum mechanics.
Of course, Planck only regarded this as a mathematical technique at the time, and did not really think that the propagation of energy was discontinuous. Only later did Einstein regard this as a physical reality. Later, Bohr initially solved the problem of hydrogen atoms by assuming that the orbits of electrons are discrete and cannot continuously absorb and release energy.
In short, if you look at the early development history of quantum mechanics alone, many people will mistakenly think that quantum mechanics makes everything discrete and discontinuous. It seems that as long as we discretize some things, problems that cannot be explained by classical mechanics will be solved. It seems that discontinuity is the core of quantum mechanics.
Some students also think that if you want to establish quantum mechanics, do you just need to discretize everything in classical mechanics and make all classical mechanics discontinuous?
However, if you look at the quantum mechanics we are talking about here, the whole article talks about using vectors to describe the state of the system, using operators to describe mechanical quantities, using the Schrödinger equation to describe the change of state vectors over time, etc., there is no such thing. What to mention about continuity and discontinuity.
Some students go further. They feel that there are discontinuities everywhere in quantum mechanics. Therefore, time and space in quantum mechanics must also be discontinuous. It happened that he also knew the concepts of Planck time and Planck length , so he cut time and space into pieces in his mind, thinking that this was quantum mechanics, and then said that he could easily Solved Zeno's Paradox .
I have to say that if you just read some popular science books on quantum mechanics, and then make some extensions based on them that you think are reasonable, and add some imagination, it is very normal to come to such a conclusion. However, if you learn a little bit about quantum mechanics systematically, you will know that this inference is very wrong.
The simplest evidence is if you look at the Schrödinger equation, what appears in it is the partial derivatives ∂/∂t and ∂/∂x with respect to time t and space x. What does it mean to derive? Derivative means that it must be continuous. I believe everyone still remembers that " must be continuous if it is differentiable, but continuous may not necessarily lead to ".
The Schrödinger equation contains partial derivative operations for time and space . This is clearly telling us: In quantum mechanics, we assume that time and space are continuous. Otherwise, the Schrödinger equation will be meaningless .
Indeed, in some quantum gravity theories, such as loop quantum gravity , time and space are considered to be discontinuous, but this is not what we often call quantum mechanics. It belongs to the frontier exploration field of quantum gravity. The theory itself still has many problems and has not yet reached a consensus.
Quantum mechanics, which everyone often talks about, is very mature in theory and has been tested by countless experiments. It assumes that time and space are continuous .
In other words, although in quantum mechanics can have discontinuous things (such as energy) in , the background stage of time and space is still continuous. Moreover, we say that energy can be discontinuous in , not that must be discontinuous in . It can still be continuous in some cases. Therefore, simple and crude ideas like "everything in quantum mechanics is discontinuous " should be given up as soon as possible ~
Then, since time and space in quantum mechanics are both continuous, and Energy can be discontinuous, so how does this discontinuity arise?
4 Intuition and Counterintuition
At this point, I want to emphasize a very important thing to everyone: When learning quantum mechanics, we must look at the world from a quantum perspective, not a classical perspective. We should not always think that the quantum world is strange, so we have to use classical images that we are more familiar with to make analogies. Quantum mechanics is something more fundamental. What needs to be explained is not quantum mechanics, but classical mechanics.
What we should really ask is not why quantum mechanics is strange, but how do the various phenomena of classical mechanics emerge from quantum mechanics? What we should really wonder is not why the quantum world is like this, but why the classical world can be like this?
Quantum mechanics has been around for a hundred years. Faced with this extremely successful theory that has profoundly changed our thinking and life, it stands to reason that we should feel that it is already natural. But the fact is completely opposite: when many people mention quantum mechanics, their first reaction is still counterintuitive and counterintuitive. They think this theory is weird, difficult to understand, and unreasonable!
However, have you ever thought about what exactly are you talking about when you say that quantum mechanics is counterintuitive to ? You can be counter-intuitive, which means you already had an intuition before. After you have a set of intuitions about the world, you discover some phenomena that are inconsistent with these intuitions, and then you become counterintuitive.
For most people, this intuition is the intuition formed by studying Newtonian mechanics in middle school.
Therefore, when they tried to incorporate various phenomena in the quantum world into the original layout and tried to understand quantum phenomena using the thinking and habits of Newtonian mechanics, they found that they could not understand it, so they felt counter-intuitive.
This kind of thing is normal. If a person has accumulated a lot of experience, after encountering new things, he will naturally hope that the original experience can still be used. Therefore, in the early days of quantum mechanics, those physics masters also hoped to solve problems within the classical framework. They intentionally or unintentionally retained many ideas and concepts of classical physics. After about a quarter of a century of arduous exploration, they formed Comparative systematic quantum mechanics.
Probably the history of the first 25 years of quantum mechanics is too exciting. Various characters have appeared in turn, and various ideas have launched round after round of impacts on classical physics. There are two forces within quantum mechanics, matrix mechanics and wave mechanics, and there is also the debate between Bohr and Einstein . It is perfect to use it as a book.
This has triggered a more serious problem: Most of the popular science books on quantum mechanics on the market today are about the history of quantum mechanics in the past 25 years..
They started with Planck and blackbody radiation, talked about Einstein and the photoelectric effect, talked about Bohr and hydrogen atoms, talked about Heisenberg and the mysterious matrix, talked about de Broglie and matter waves, talked about Schrödinger's Mysterious Girl and Schrödinger's equation, coupled with the small debates between matrix mechanics and wave mechanics, and the big debate between Bohr and Einstein, a wonderful popular science book on quantum mechanics is completed.
A book written like this is good as History of Quantum Mechanics . However, if you regard it as a popular science book on quantum mechanics and hope to learn the thinking of quantum mechanics from here, and understand the basic framework of quantum mechanics and the general methods of dealing with problems, it will be very easy to run into problems.
The reason has also been mentioned. The history of quantum mechanics in the first 25 years itself is full of all kinds of confusion, and the masters also mixed all kinds of classical things when thinking about problems. Looking at quantum mechanics from a classical perspective is naturally counterintuitive, strange, and even weird. If you want to learn quantum mechanics, but instead of learning how to see the world from a quantum perspective, you learn a bunch of counter-intuitive and weird , this is not a good thing.
For example, the discontinuity here. After reading the history of the first 25 years of quantum mechanics, many people are extremely impressed by this discontinuity. Therefore, it is easy for him to think that quantum mechanics means that everything is discontinuous, time is discontinuous, and space is discontinuous. He thinks that quantum mechanics can be obtained by discretizing all classical mechanics, and then starts all kinds of random thoughts.
5 wave-particle duality
Similar to wave-particle duality , this is also a very typical attempt to use classical thinking to explain quantum phenomena.When we talk about waves in classical mechanics, we think of things like water waves; when we talk about particles, we think of things like peas.
However, in quantum mechanics , if you also talk about particle nature , it only means that it has certain properties such as mass and charge. An electron does not behave like a pea at all, it is not determined at all. Orbital; when you talk about volatility in quantum mechanics, that only means that it has coherent superposition. It does not mean that there is really something like a water wave in space.
The result of this is that, you see, we first try our best to let readers accept that any particle has wave-particle duality : an electron is both a wave and a particle. It sometimes looks like a wave and sometimes looks like a particle. When we measure it in the form of waves, it behaves like waves ; when we measure it in the form of particles, it behaves like particles .
After everyone was confused by this, but only remembered that "electrons are both waves and particles", you came to tell readers: I'm sorry, this wave we talk about in quantum mechanics is not classical. Wave; particle in quantum mechanics, it is not a classical particle .
Reader: ...
You can imagine that after such a round of science popularization, readers will not be confused? Could he not find quantum mechanics mysterious and mysterious, counter-intuitive and weird? If your mind is a little bigger, you can continue to use the wave-particle duality: electrons are both waves and particles, both yin and yang, and the five elements of yin and yang complement each other... This can easily lead to punching Schrödinger and stepping on Heisenberg. A left hook knocked down Bohr and Einstein .
In the final analysis, wave-particle duality is the product of the early development of quantum mechanics. In that chaotic stage, people tried to use as many classical concepts as possible to describe quantum mechanics. Before quantum mechanics was established, people did need such a crutch. However, more than a hundred years after quantum mechanics was established, do we still need to use the crutch of a hundred years ago step by step?
In the article, we talk about using state vectors to describe the state of the system, using operators to describe mechanical quantities, and using the Schrödinger equation to describe changes in the system state over time. There is no mention of wave-particle duality throughout the article, and it is unnecessary.
In classical mechanics , waves and particles are two entities that cannot coexist, and it is natural to distinguish them. But when it comes to quantum mechanics , as long as we start from the basic framework of quantum mechanics, we will find that particles have definite mass and charge, and the wave function describing the state of the particle has coherent superposition. It is very natural. There is no need to deliberately mention it. The wave-particle duality that people easily confuse. After learning quantum field theory , everyone will find it more natural.
Of course, if you insist on using wave-particle duality , it is not impossible. However, you must know what you are talking about when you talk about wave-particle duality. You must know the difference between the wave nature and particle nature of quantum mechanics and those of classical mechanics.
We all know that quantum mechanics is a more profound theory than classical mechanics. Quantum mechanics can describe things that classical mechanics can describe, and quantum mechanics can also describe things that classical mechanics cannot.In this case, why should we care about classical mechanics when we study quantum mechanics? Why do we still do things like " to understand quantum mechanics from the perspective of classical mechanics", which is absurd, useless and easy to create all kinds of confusion?
Can’t we study quantum mechanics openly and think about quantum problems in a quantum way? The we want to do is not "How to understand quantum mechanics from a classical perspective", but the other way around: If our lower-level world is quantum, how do various phenomena in the classical world emerge ?
If there is no discontinuity in the basic assumptions of quantum mechanics, then how does the energy discontinuity we often talk about emerge? How can we explain the single-electron double-slit interference experiment without using semi-classical and semi-quantum things like wave-particle duality ? The quantum world is full of various probabilities and uncertainties, why does the macro world seem not to have one? How to start from quantum mechanics and give a complete and self-consistent description of the physical world? etc.
This is a series of very grand topics, which we will discuss later. In this article, we will first set up the basic framework of quantum mechanics and learn the general methods of dealing with problems in quantum mechanics. Once we have clarified these, our minds will have completed a format from classical to quantum. Then, you will feel that quantum mechanics is natural and no longer counter-intuitive, because current quantum mechanics is your intuition.
Therefore, we must gradually try to think about quantum mechanical problems in a quantum way. Haven't we already found the basic postulates of quantum mechanics? Just start from here.
So, let’s start from here and see why can be discontinuous? Remember again, it says "can", not "must". Is the energy of
6 continuous?
Suppose there is a particle here, and we want to see whether its energy is continuous. First of all, we need to realize: when we say this sentence, what are we talking about?
In classical mechanics , the kinetic energy of a particle is related to its speed, and the speed of a particle can take continuous values. It can be 1, 1.6, or any other real number. Therefore, the particle's Kinetic energy can also take on continuous values. Similarly, the potential energy of the particle can also take continuous values, because the potential energy depends on the position, and the position can take continuous values.
Therefore, in classical mechanics, the kinetic energy and potential energy of particles can take continuous values, and of course the total energy of the particles can take continuous values. There is nothing to say about this.
After reaching quantum mechanics , if you still want to find kinetic energy through the speed of particles, you will find that this path is blocked. The reason is also very simple. The speed of classical mechanics refers to the displacement change per unit time. The particle is at point A at the moment and reaches point B one second later. We divide the distance between the two points AB by the time to get the speed and then the kinetic energy.
However, in quantum mechanics, can we still say that the particle is at point A at this moment?
can’t! Only when the particle is in the eigenstate of position A, can we say that the particle must be at point A. If the particle is in the position superposition state , then there is a certain probability that it is at point A, and there is a certain probability that it is at point B, point C, and so on. Therefore, the particle does not have a definite position under normal circumstances, so you cannot say that it is at point A at this moment. In the same way, you have no reason to say that it will be at point B the next second. The position of
is uncertain, so how to determine the speed of the particles?
Therefore, we cannot talk about the kinetic energy of particles like classical mechanics, nor can we talk about the continuity of energy like classical mechanics. We must discard the experience of classical mechanics and consider issues directly from the framework of quantum mechanics.
We know that in quantum mechanics, operators are used to describe mechanical quantities (assumption 2). energy and are also mechanical quantities, so naturally they have to be described by operators. What operators should be used? As mentioned earlier, use Hamiltonian operator . In classical mechanics, the energy of a particle is generally equal to the Hamiltonian. After we operatorize it, we get the Hamiltonian operator H in the Schrödinger equation. And we also know that the result of measuring a mechanical quantity is one of the eigenvalues of the corresponding operator (Assumption 3).
Therefore, if we want to judge whether the energy of a particle is continuous, it is not based on whether its speed is continuous like classical mechanics, but whether the eigenvalue of the Hamiltonian operator is continuous..
As mentioned before, the Hamiltonian H in classical mechanics is generally written as kinetic energy (P²/2m) plus potential energy V:
Under the position representation, the operator corresponding to the momentum P looks like this -iℏ∂/ ∂x (I don’t care why it looks like this), substitute it in, and you will get the Hamiltonian operator H under the position representation:
In other words, if we want to see whether the energy is continuous, we have to look at this Hamiltonian operator Whether the eigenvalue of symbol H is continuous.
wants to see whether the eigenvalue of an operator is continuous. As mentioned before, solve the eigenequation of this operator (A|Ψ=a|Ψ, where a is the eigenvalue of operator A, |Ψ is the corresponding eigenstate).
So, our question now becomes: Where can I find the intrinsic equation of the Hamiltonian operator H on ?
7 Stationary Schrödinger equation
If you want to find the eigenequation of the Hamiltonian operator, you have to first find an equation containing the Hamiltonian operator , right? Let's take a look at the Schrödinger equation under the positional representation :
Does the Hamiltonian operator H look a bit like the right side of the Schrödinger equation (nonsense, the right side of the Schrödinger equation that has not entered the representation is the Hamiltonian operator, isn't it different~) ?
If we can put Ψ out like algebraic multiplication, then is the only Hamiltonian operator H left on the right side of this equation? In other words, if Ψ can be put forward, the right side of the Schrödinger equation of position representation can be written as HΨ, and we can see the Hamiltonian operator H.
But unfortunately, the right side of this equation is not algebraic multiplication. The wave function Ψ(x,t) and the potential function V(x,t) under the position representation are also related to time t, and The multivariate functions related to space x cannot be mentioned casually.
Therefore, if you want to propose Ψ, you must first find a way to separate the time and space parts of the wave function Ψ(x,t) and the potential function V(x,t). How to do this?
Let’s first look at the potential function . The current potential function V(x,t) is related to both time t and space x. So how can we separate them? Simple, we just assume that the potential function does not depend on time t. In other words, we only consider the potential function V(x) that does not depend on time t and is related to space x.
Let’s think about the situations we usually encounter: the gravitational potential energy of an object is only related to the height (it has nothing to do with time), and the elastic potential energy of a spring is only related to its position (it has nothing to do with time). When we do electromagnetics questions, we usually do it first Given an electromagnetic field (which does not change with time). It can be seen that the potential function V(x) of that does not depend on time t is very common. We will consider this simple case first, and it will not be too late to consider more complex ones later.
potential function has been solved, but what about wave function ?
In order to also separate the time and space parts of the wave function, we write the wave function Ψ(x,t) as the product of ψ(x) which only contains position and φ(t) which only contains time:
Of course , you may ask why the wave function is written in this form? Indeed, the wave function that can be written in this form is only part of and very little of . But as you will see later, more and general solutions can be constructed through these few solutions.Therefore, it is very meaningful for us to look for this small set of solutions first.
Therefore, we assume that the potential function V does not depend on time , and write the wave function Ψ(x,t) in the form ψ(x)φ(t), and combine the time and space of the Schrödinger equation Partially separated.
Then, we will substitute the new form of the wave function ψ(x)φ(t) into the Schrödinger equation under the position representation . Through a simple derivation and substitution of , anyone who understands it will understand it. It doesn’t matter if you don’t understand . Working, the original Schrödinger equation becomes like this:
In order to facilitate the distinction, we use the uppercase Ψ(x,t) to represent the wave function of that contains both time and space, and only contains the space of . The part is represented by lowercase ψ(x), and the part of that only contains time is represented by φ(t).
You can see that since Ψ(x,t) is split into the form of multiplying ψ(x) and φ(t), the partial derivatives ∂/∂x and ∂/∂t in the original equation become Ordinary derivation d/dx, d/dt, this form is simple. In this way, the left side of the equation is really only related to time t, and the right side of the equation is only related to space x (because the potential function V on the right does not depend on time, and ψ(x) does not include time).
Something related to time (the left side of the equation) is equal to something related to space (the right side of the equation). It seems impossible. How can two unrelated functions be equal?
However, they still have the same possibility, that is: , they are all equal to a constant !
If you think about it, the things on the left change with time, maybe a value at 8 o'clock, and a value at 9 o'clock; the things on the right change with location, maybe a value in Beijing, and a value in Wuhan. There is no relationship between the left and right sides. If you force them to be equal now, then they can only be equal to an constant . Let's record this constant as E.
Therefore, the above equation can be split into two:
The first equation is very simple and easy to solve. Regardless of it here, let’s focus on the second equation. If you multiply both the left and right sides of equation 2 by ψ, it can be written like this:
This equation has a very famous name, called stationary Schrödinger equation . Why is
called fixed state ? On the surface, "sedation" should mean not moving and not changing with time. However, here we only assume that the potential function V does not depend on time. Although the wave function Ψ(x,t) is written in the form of Ψ(x)φ(t), it is still related to time φ(t). It seems impossible to talk about it. Go to "Ding".
However, let’s think about Born’s rule: |Ψ(x,t)|² represents the probability of finding a particle at position x at time t. In other words, although the wave function Ψ(x,t) is related to time t, the wave function itself does not correspond to any physical reality. What is truly physically meaningful is the square of the module of the wave function |Ψ(x,t)| ², which represents the probability that we find particles at a certain time and place.
However, when we calculated |Ψ(x,t)|², we found that the time factor actually canceled each other out during the calculation process, and the final result had nothing to do with time. More specifically, |Ψ(x,t)|² is equal to |ψ(x)|², which is only related to the space part.
Therefore, when the potential function V does not depend on time, although the wave function Ψ(x,t) itself is related to time, the probability distribution |Ψ(x,t)|²=|ψ(x)|² is related to time Nothing to do with . In this way, the average value of any mechanical quantity has nothing to do with time, so we say that this is " steady state ", which is a state in which the probability distribution and the average value of the mechanical quantity do not change with time.
8 energy eigenstate
Now that we understand the meaning of steady state , let’s ask about the meaning of the constant E. What is the constant E that makes time and space equal?
Everyone knows that in physics, we generally use E to represent energy. So does this constant E have anything to do with energy?
has something to do with it! This E is exactly the system's energy .
Why? Let’s take a look at the stationary Schrödinger equation :
. The ψ here is only related to the space x and is a one-variable function ψ(x). In this case, we can put out ψ on the left side of the equation, and the remaining part is the Hamiltonian operator H.
Therefore, we can write the stationary Schrödinger equation into a very simplified form of Hψ=Eψ. Warm reminder, H here is the Hamiltonian operator, which is an operator, and E is a number. Please don’t write off ψ with a stroke of your pen and make a joke about H=E~
Many people should still remember that we talked about operators when we talked about “Using Operators to Describe Mechanical Quantities (Assumption 2)” The eigenequation : If the mechanical quantity is described by operator A, then when the system is in the eigenstate ψ of the mechanical quantity, the value of the mechanical quantity is determined by . No matter how many times you measure, the measurement result will be eigenvalue a, and the corresponding eigenequation is Aψ=aψ.
Let’s take a look at the stationary Schrödinger equation Hψ=Eψ. Is it very similar to the operator’s eigenequation (Aψ=aψ)? Generally speaking, the operator corresponding to energy is the Hamiltonian operator H. If ψ is an energy eigenstate, then isn't Hψ=Eψ the eigenequation of energy ?
But the question is: Is this ψ of an eigenstate of energy? ?
If ψ is not an energy eigenstate, then the stationary Schrödinger equation Hψ=Eψ cannot be regarded as an energy eigenstate. Therefore, how to judge whether this ψ is an energy eigenstate?
First, let’s recall how this ψ comes from: We assume that the potential function V does not depend on time , and then split the wave function Ψ(x,t) into the product of the time and space parts ψ(x)φ( t), and this ψ is the space part.
At first glance, this ψ seems to have nothing to do with the energy eigenstate, but just looking at it is not enough, we still have to calculate it.
If ψ is really an energy eigenstate, then E is the corresponding energy eigenvalue . At this time, if you measure the energy of the system, the measurement result must be the eigenvalue E, and the average value must also be E.
Therefore, if you want to prove that ψ is an energy eigenstate, you must first prove that the average value of the Hamiltonian operator H in the state ψ is equal to E. If the average value is not equal to E, then this is definitely not an eigenstate. Through calculation, we found that the average value of Hamiltonian operator H in state ψ is indeed equal to E.
Of course, it is not enough that the light average is equal to E, because the energy eigenstate means: no matter how many times you measure, the result is E. Now you only say that the average value of Hamiltonian operator H in state ψ is E. What if this E is the average of 0.5E and 1.5E? In other words, if we measure the energy of a particle, there is a 50% probability that it is 0.5E, and a 50% probability that it is 1.5E, so that the average value is still E. But obviously, this is not an eigenstate of energy.
Therefore, in addition to the mean being equal to E, we also need to ensure that it has no dispersion and no fluctuations. In statistical language, both the variance and standard deviation must be 0. Through calculation, the standard deviation of the Hamiltonian operator H in the state ψ is indeed 0 (the calculation process is omitted, I only talk about the ideas, it is best for everyone to do the calculations by yourself). The average value of
is equal to E, and the standard deviation is 0. In this way, we can ensure that the result of each measurement is E, and we can determine that ψ is an eigenstate. Therefore, we can say openly: When the potential function V does not depend on time, the state described by the stationary Schrödinger equation Hψ=Eψ is exactly the eigenstate of energy, and the stationary Schrödinger equation is the eigenequation of energy.And this constant E is nothing else. It is the energy of the system under the eigenstate ψ. You’re done!
In other words, if the potential function V does not depend on time, the system is in the steady state , which is the energy eigenstate . In this state, measuring the total energy of the system will always result in a definite value E.
Why does the potential function not depend on time , but the total energy is determined? Let me give you a simple example and everyone will understand.
An apple falls. When the apple falls, the gravitational potential energy is converted into kinetic energy. But everyone knows that Apple's total energy (kinetic energy + gravitational potential energy) has not changed during this process. It is conserved and has a certain value E. Why is energy conserved when an apple falls? Because the apple's gravitational potential energy mgh does not depend on time, it is only related to the height h of the apple. In other words, making the apple's potential energy function mgh independent of time results in energy conservation, causing the apple's total energy to always be a constant value E.
If the apple's potential function V depends on time , then the sum of its kinetic energy and potential energy is no longer a fixed value (the simplest, when the apple is stationary, the kinetic energy remains unchanged, but the potential energy changes with time, so the total energy It must also change with time and is no longer conserved), and the total energy is no longer a fixed value E.
The implication here is: Apple system also has energy exchange with external systems. For example, if we pull an apple up and down with a rope, the sum of the apple's kinetic energy and gravitational potential energy is definitely not a fixed value. Because our hands do work on the apple, there is an energy exchange between the apple and us.
In this way, do you understand the meaning of the stationary Schrödinger equation Hψ=Eψ?
9 Potential function
Didn’t we talk about the continuity of energy earlier? Why do we spend so much time talking about the stationary Schrödinger equation ?
Because energy is also a mechanical quantity, and mechanical quantities are described by operators, the value of the mechanical quantity is one of the eigenvalues corresponding to the operator. Therefore, if you want to know what values energy can take, you have to know what eigenvalues the corresponding Hamiltonian operator has; if you want to know what eigenvalues the Hamiltonian operator has, you have to know its eigenvalues. What is equation .
Now, we have found the eigenequation of the Hamiltonian operator H, and found that it is actually the stationary Schrödinger equation Hψ=Eψ. Therefore, we can continue to discuss the issue of continuity of energy.
Let’s take a look at the stationary Schrödinger equation , which is the energy eigenequation :
From the equation, a state ψ (energy eigenstate) of the system corresponds to an energy E (energy eigenvalue) . If you want to know the energy E, you must first know the system state ψ.
So, how do we know the wave function ψ that describes the state of the system?
I have talked about this before: can solve the Schrödinger equation just ! By the way, although the wave function mentioned at the beginning refers to Ψ(x,t) related to time t, but customarily, we also regard the ψ(x) in the stationary Schrödinger equation as only related to space x. It's called the wave function, as long as everyone knows it.
In other words, if we want to know how the energy of a particle takes a value, whether it is continuous or discrete, we have to know how the wave function ψ that describes the particle state can take a value. If you want to know how to take the value of the wave function ψ, you have to solve the stationary Schrödinger equation .
In the stationary Schrödinger equation, in addition to the energy E and the wave function ψ, there is also an undetermined potential function V. In other words, different potential functions of (such as different electromagnetic fields) will have different solutions, and thus different wave functions ψ and different energy values will be obtained.
Therefore, we cannot generally say whether energy in quantum mechanics is continuous or discrete, but we must treat it differently according to different potential functions .
0 free particles
As always, we start from the easy to the difficult, starting with the simplest. So what kind of potential function is the simplest? Of course, it is the potential function V=0, that is, when there are no external constraints.
In Newtonian mechanics, if the total external force is 0, the particle will do the simplest rest or uniform linear motion. When it comes to quantum mechanics, if the potential function is 0, how will the particles move?
Obviously, when the potential function V is constant equal to 0, it is still independent of time. Then, we can continue to use stationary Schrödinger equation to deal with the problem.
In the steady state Schrödinger equation, if V=0, the equation becomes like this:
This is a very simple differential equation. We can easily write its general solution . At this time, the wave function ψ is long Like this (if you don’t know how to solve it, read the book yourself, I won’t teach you how to solve differential equations here~):
What does this solution mean? Everyone has learned trigonometric functions in middle school. One like Asinkx is a sine wave. The larger A is, the higher the sine wave oscillates, and the greater the distance between the wave crest and the trough; the larger K is, the denser the sine wave is, and the smaller the distance between the two wave crests is.
Obviously, if A and k are not subject to any restrictions and can take any value, then the image of this sine wave can also change at will. It can be arbitrarily high or arbitrarily dense, similar to cosine wave Bcoskx.
Therefore, we solve the stationary Schrödinger equation with potential function V=0, and the obtained wave function ψ(x) is a superposition of the sine wave Asinkx and the cosine wave Bcoskx, that is, ψ(x)=Asinkx+Bcoskx. Since the potential function V is 0 everywhere, there are no other constraints on the particles, so we have no other conditions to constrain the values of A, B, and k. In other words, A, B, and k can take any value of . We can ignore
A and B, but this k is closely connected with the energy E: the larger the
k, the denser the wave, the greater the corresponding energy E.
Now, we say that this k can take any value, then naturally this E can also take any value. That is to say, when the potential function V=0, the energy E of this free particle can take any positive real number, which is obviously the continuous of .
So, we got the first conclusion: The energy value of the free particle (potential function V=0) is continuous, and it can take any positive energy value .
Are you a little surprised? Maybe in your impression, energy in quantum mechanics must be discontinuous. But we didn't expect that our first conclusion was that the energy of the simplest free particle was actually continuous .
Everyone should remember that "whether energy is continuous " is not a basic assumption of quantum mechanics. The basic assumptions are the state vectors, operators, measurements, and Schrödinger equation mentioned earlier. We start from these assumptions. If the energy is calculated as continuous , it is continuous. If the energy is calculated as discrete , it is discrete. That's it.
So here comes the question, where does the discontinuous energy that everyone is familiar with, the energy in bits and pieces come from?
1 One-dimensional infinite deep square potential well
If you think about it, the reason why the energy E of the free particle is continuous is because it has no constraints on the wave function ψ (x) = Asinkx + Bcoskx, so A, B, and k can take any value. . What if we add some restrictions? If I don't let k take any value, does the corresponding energy E also not be able to take any value? Would it therefore become discontinuous?
Utopias are useless, we have to use calculations to speak. We add a very simple restriction to the free particles: lock the particles in a "dungeon" and prevent them from getting out. What does
mean? Isn’t the potential function of free particles everywhere equal to 0, and no one cares about it everywhere? Now I add two copper and iron walls on the left and right sides to close it up.
As shown above, in the range from 0 to a, the potential function V is still equal to 0, and the particles are still free within this range. However, outside this range, that is, where it is less than 0 and greater than a, the potential function V is infinite , and the particles should not even think about it.
is like a trap, because it is one-dimensional and square, and the potential function outside the trap is infinite, so it is called one-dimensional infinite deep square potential well .
So, what kind of restrictions will such a potential well place on the wave function? Within the potential well, that is, within the range from 0 to a, the potential function is still 0, which is no different from the situation of free particles. However, outside the potential well, the potential function is infinite and the particles cannot "get out". This is different.
In classical mechanics , when we say that a particle cannot go out, it means that its position coordinates cannot leave that range. But when it comes to quantum mechanics , under normal circumstances, particles have no definite position at all, only the probability of finding a particle at a certain position |ψ(x)|². Now the potential function outside the potential well is infinite, and we say that particles cannot get out, which means that the probability of finding particles outside the potential well is 0, that is, |ψ(x)|²=0, that is, ψ(x)=0.
Since x=0 and x=a are the left and right boundaries of the potential well, the wave functions in these two places must also be 0: ψ(0)=0, ψ(a)=0. So, we have two constraints.
So, what changes will these two constraints bring to the system? What changes will it make to the energy E of the particles? Let’s look at them one by one.
first look at the first ψ(0)=0, because ψ(x)=Asinkx+Bcoskx, so ψ(0)=Asin0+Bcos0=B (because sin0=0, cos0=1). If ψ(0)=0, then we get B=0. In this way, the wave function ψ(x) only has the first term ψ(x) = Asinkx.
If the wave function ψ(x)=Asinkx, and the second condition tells us ψ(a)=0, then we get Asinka=0. What does this mean?
As mentioned before, the image of the sine wave sinx is like this:
Therefore, there are two possibilities for Asinka=0: A=0 or sinka=0.
A=0 is a very boring situation, because B is already equal to 0. If you add A=0, then the entire wave function ψ(x)=0. Translated: the probability of finding particles anywhere is 0, which means there are no particles. Therefore, this is a mediocre solution and does not fit the current situation. What is really interesting about
is the latter solution, which is the case of sinka=0. Let's take a look at the image of the sine function sinx. Its value can be 0. Do you see that it has many intersections with the x-axis? These intersection points are where sinka equals 0.
In other words, if we want sinka=0, we only need to let ka take the places where the sine function intersects the x-axis. Friends who have studied trigonometric functions in middle school all know that where the sine function intersects the x-axis, only the positive semi-axis is considered, which happens to be π, 2π, 3π,...
In this way, ka cannot take any value at will, but only It can take π, 2π, 3π, etc., and it can be written in a more compact form:
. And we know that this k is directly related to the energy E of the particle. When solving the stationary Schrödinger equation with potential function V=0, in order to make the form simpler, we made a simple replacement for the energy E:
Now that the value of k is known, the value of energy E can be simply replaced. ：
Therefore, this energy E is really discrete , because n here can only take natural numbers such as 1, 2, and 3.Now, do you understand where this discrete energy comes from?
2 Discontinuity
When free particles , the potential function V is 0 everywhere. It does not have any restrictions on the wave function ψ (x), so k can take any value, and the corresponding energy E can also take continuous values. However, when the particle is no longer free but is constrained in a finite-width potential well, it cannot run around and k cannot take any value. Therefore, the corresponding energy E cannot take any value, that is, it is discontinuous.
In the one-dimensional infinite deep square potential well , we require that the wave function ψ takes the value 0 on both sides of the potential well, that is, ψ(0)=ψ(a)=0, which is equivalent to fixing a Both ends of the rope. Therefore, between 0 and a, this rope can be bent into a waveform, or it can be bent into two waveforms or three waveforms, like the following picture:
Because ψ(x) represents the system state (energy intrinsic state), therefore, each possible waveform of represents a possible state of the system, corresponding to a certain energy E.
In classical mechanics , we use the position and momentum of a particle to describe its state. Even if we lock the particle in a cell and limit its range of activities, its position and momentum in the cell can still change continuously, and its energy can also change continuously. It can still move around continuously in the cell without anyone caring about it.
But when it comes to quantum mechanics , this cell not only limits its range of activities, but also limits its state and energy, so that it can no longer take values at will.
In a one-dimensional infinitely deep square potential well, the wave function obtained by solving the stationary Schrödinger equation is a sine wave. As a wave, it has its own arrogance and pride. Even if it is behind bars and its range of activities is restricted, it still wants to maintain the appearance of a wave. Therefore, the state and energy of particles appear in discretization .
In this way, do you have a deeper understanding of discontinuity in quantum mechanics?
3 Hydrogen atoms
In the basic assumptions of quantum mechanics, we do not make any assumptions about whether energy is continuous. We only use state vectors to describe the state of the system, and use the Schrödinger equation to describe the change of the system state with time.
When the potential function V does not depend on time , the system is in the steady state (energy eigenstate). At this time, the measured energy has a certain value. Only when energy has a definite value can we talk about whether the value of energy is continuous or discrete. If the system is in the energy superposition state , and there is no definite energy value, then this question is meaningless.
After the potential function is determined, we can solve the stationary Schrödinger equation to get the wave function describing the system state, and then get the energy situation, and then we know whether the value of the energy is continuous or discrete.
When the potential function V=0, the particle is completely free, and its energy is continuous ; when the potential function is not 0, but a one-dimensional infinitely deep square potential well, the particle's energy becomes discrete . If we change the environment and change the potential function, the operation process is still the same. We substitute the corresponding potential function into the Schrödinger equation to solve, and then analyze the value of the energy based on the wave function.
For example, we know that hydrogen atom is composed of a proton and an extranuclear electron. So, what values can the energy of this electron take? Is it continuous or discrete?
Similarly, to analyze the behavior of electrons, we need to know its potential function.And we know very well that electrons and protons will attract each other. According to Coulomb's law, this potential function V can be written as:
Then, we substitute this potential function into the stationary Schrödinger equation. After a series of equations that we think are very complicated, but in quantum mechanics With a relatively simple calculation, you can get the energy that the electron in the hydrogen atom can take:
This is the famous Bohr formula . Bohr obtained this formula from his model and became famous all over the world. Now, we can derive it very naturally from the Schrödinger equation .
I won’t go into this solution process, it will be written in any quantum mechanics textbook. But the result is obvious. Like the one-dimensional infinitely deep square potential well, the energy value that the electron under the Coulomb potential can take is the same as discrete . It can only take some specific values. n=1 is the lowest state of of energy, also called ground state , and other situations are called excited state .
4 Atomic Model
In the history of quantum mechanics, the problem of hydrogen atoms has always been important. Now that we know how hydrogen atoms are treated in quantum mechanics, we might as well go back and see how classical mechanics handles hydrogen atoms and see what difficulties it encounters. This is also very helpful for our in-depth understanding of quantum mechanics. benefit.
On the eve of the quantum revolution, there are four major problems that plague classical mechanics: including the familiar blackbody radiation and photoelectric effect , as well as the less familiar atomic spectrum and atomic stability issues. The latter two questions are related to the atomic model , and the hydrogen atom is the simplest atom, so it is very important.
Speaking of atomic models, the first ones to appear are Thomson . He believed that an atom is a sphere, with positively charged substances evenly distributed in the sphere, and negatively charged electrons embedded in the sphere one by one. This model is called the " jujube cake model ".
But soon, Thomson's model was slapped in the face by his student Rutherford . When Rutherford bombarded gold foil with alpha particles, he found that most of the alpha particles passed through the gold foil, but a very small number of alpha particles actually bounced back.
What does this mean? If the positively charged matter in the atom is uniformly distributed, then bombarding the atom with alpha particles will be like bombarding a cake with a bullet, and it will never bounce back. Now that a very small amount of alpha particles have been bounced back, it means that there is a very small amount of very hard stuff inside the atom.
After repeated experiments and thinking, Rutherford believed that positively charged matter can only be concentrated in a very small range, and the mass of atoms is mainly concentrated here. This is what we call the nucleus. In this way, the positively charged nucleus is like the sun, and the negatively charged electrons are like planets orbiting the sun. Rutherford's atomic model is called " planetary model ".
Although the planetary model is in good agreement with experiments , there is a huge theory problem: if the electrons are really rotating around the nucleus, then according to the classical electromagnetic theory, the electrons will continuously release energy as they rotate. In this case, when the energy of the electron is exhausted, it should fall into the nucleus, and the atom will be destroyed.
But we all know that the world is stable, atoms are not destroyed, and electrons do not fall into the nucleus. Then the question arises: Why can atoms remain stable? Why don't electrons fall into the nucleus due to the continuous release of energy?
This is the stability problem of atoms, which cannot be answered by classical physics.
Rutherford could not solve the problem, so he threw it to his student Bohr . Bohr tinkered for a while, and after fully absorbing the quantization ideas of Planck and Einstein, he proposed a brand new atomic model.
Bohr believed that the orbit of an electron cannot be chosen arbitrarily, it can only be in some specific orbits.When the electron is in these specific orbits, the electron does not emit or absorb energy (so it does not crash). It only emits and absorbs energy when it jumps from one orbit to another.
Bohr's model is a mixture of classical and quantum . It includes the concept of quantized orbits and the classical model of electrons rotating around the nucleus. Theoretically, such a "stitch monster" must be full of flaws (no one believed it at the time), and this model cannot explain more complex atoms.
However, physicists are more interested in whether your model can explain experimental phenomena than theory. When more and more experiments came on Bohr's side, everyone gradually accepted the main idea of of Bohr's model and admitted that there was indeed some correct in it. At the same time, everyone is also looking forward to a more perfect theory, hoping that the Bohr model can be derived from there and explain things that the Bohr model cannot explain.
About ten years later, with the full establishment of quantum mechanics, everything became clear. So, how does current quantum mechanics view Bohr model ?
First of all, we must make it clear: in quantum mechanics, electrons are without the concept of orbits. What is a track? The electron is here this second and there the next second. Its position at each moment can be accurately calculated. This is the orbit. However, in quantum mechanics, there is no definite position of electrons in the general state. We can only calculate the probability of finding electrons in various places, so we can't talk about orbits at all.
But we also know that Bohr's model is consistent with experiments, and it must contain some correct things. So, if there is no definite orbit in quantum mechanics, what is the orbit Bohr said?
Previously, we have solved the Schrödinger equation under the Coulomb potential , and obtained the Bohr formula :
Every possible E here represents a possible state of the electron. Yes, this is actually the "orbit" Bohr mentioned.
Each "orbit" is actually a stationary state, an energy eigenstate. Because the states and energies that electrons can take under the Coulomb potential are all discrete , Bohr felt that electrons could only stay in some specific and discrete "orbitals".
Why don't electrons fall into the nucleus? Because among these allowed energies E, there is a minimum value, which is the ground state energy when n=1 (the energy here takes a negative value, and the negative sign means that the electron is bound by the nucleus, E1=-13.6eV, E2=- 3.4eV...), the energy of the electron cannot be smaller than it, so it cannot fall into the nucleus.
In this way, do you have a deeper understanding of atomic issues?
5 Double-slit experiment
I write this article mainly to help everyone set up the basic framework of quantum mechanics and let everyone know how to look at problems from the perspective of quantum mechanics.
Many people think quantum mechanics is strange, weird, and even scary. The fundamental reason is: They do not look at quantum issues from a quantum perspective. Intentionally or unintentionally, they retained many classical concepts and thinking, and looked at the quantum world from a semi-classical and semi-quantum perspective. It is only strange that does not feel strange.
In the early days of the quantum revolution, before the quantum building was built, it is understandable that those masters used more familiar classical thinking to think about problems. They encountered obstacles everywhere, and after various arduous explorations, they established a mature quantum mechanics framework. Today, more than a hundred years later, do we still look at problems from a semi-classical and semi-quantum perspective, and continue to crawl and wallow in the quagmire of the early quantum stage?
Many people think quantum mechanics is strange and think that no one can understand quantum mechanics. They are proud to say that many physics masters say the same thing.But please believe me, most people find quantum mechanics strange, simply because they lack the most basic understanding of the basic concepts and framework of quantum mechanics. They are stuck in a semi-classical and semi-quantum quagmire and cannot get out of it. The strange thing in the eyes of the physics master is not the same thing at all.
It’s just like mathematics. Some people say that solving quadratic equations is too difficult. Some people say that the Riemann Hypothesis is too difficult. They all say that mathematics is difficult, but can this be the same thing? If everyone sets up the framework of quantum mechanics and learns to look at problems from a quantum perspective, then many things that originally seemed very counterintuitive and incredible will become very natural.
For example, the single-electron double-slit interference experiment , which has been described by countless popular science articles as terrifying, terrifying to think about, and subversive of three views, is an ordinary experiment from the perspective of quantum mechanics.
Why do so many people think the double-slit experiment is scary? Because they looked at this experiment from the classic perspective of .
From the classic perspective , there are two "weird" aspects of the single-electron double-slit interference experiment: First, the familiar interference experiments involve a large number of particles, and the interference between different particles is easy to understand. However, now we only emit one electron at a time. Over time, interference patterns can still appear on the screen, which is difficult to understand.
only emits one electron at a time, who are you interfering with? How can there be an interference pattern if there is no interfering object? It's as if every electron has consciousness and knows where the electrons in front and behind it are going. This kind of atmosphere looks very weird when paired with some horror music.
What is even more "weird" is the second of : when we release electrons one by one, an interference pattern will slowly appear on the screen. However, once we add a detector after the gap to see which gap the electrons passed through, the interference fringes disappear.
From a classical perspective, there was originally an interference pattern here. When I "looked" at where the electrons were going to pass, the interference pattern disappeared. As if consciousness can affect the experiment, or electrons can read my mind, if I exaggerate the atmosphere here, it will not be weird, but terrifying.
I went to the Internet to search for "double-slit experiment". Take a look at these hot search terms:
What horrors, scams, and truths are there? Even more exaggerated, even "double-slit experiment saw ghosts" popped up. A scientific experiment found a bunch of these things, but no one could find them.
Of course, from the classic perspective of and , the double-slit experiment is indeed very weird and terrifying. However, from the quantum perspective , you will find that this is a very natural experiment, and what it embodies is nothing more than some of the most basic characteristics of quantum mechanics.
First of all, why does an interference pattern appear every time an electron is emitted?
In quantum mechanics, we use wave functions (state vectors) to describe the state of electrons, and this state is that can be superimposed on . In other words, if ψ1 is a possible state of the electron and ψ2 is also a possible state of the electron, then their linear superposition ψ=ψ1+ψ2 is also a possible state of the electron (ψ1 and ψ2 can have different coefficients in front of them) , this is called state superposition principle .
This should feel natural to everyone. In the Stern-Gerlach experiment, the silver atom can be in the spin-up eigenstate ψ1 or the spin-down eigenstate ψ2. Then, it can also be in the spin-up and spin-down eigenstates. The downward superposition state ψ=ψ1+ψ2 is normal.
Moreover, we also know that the probability of measuring a mechanical quantity is linked to the square of the module of the wave function |ψ|².
Then, we will find that: the probability corresponding to the superposition state |ψ|²=|ψ1+ψ2|² is not equal to the sum of the probabilities of the original states |ψ1|²+|ψ2|², there is still a difference between them With a cross-term , primary school mathematics teachers will often emphasize that "the square of the sum is not equal to the sum of the squares." And this cross-term is the reason for the interference.
In fact, the interference of waves in classical mechanics and is also due to cross terms. Because the intensity of the wave is also related to the square of and , the intensity of the superposition of two light waves is and is not equal to the sum of the intensity of each light wave of (the intensity is related to the square, there will be more cross terms), and the light and dark we see The degree is related to the intensity of light, so interference fringes appear.
In quantum mechanics, the probability of superposition of two wave functions is not equal to the sum of the probabilities of each wave function (|ψ1+ψ2|²≠|ψ1|²+|ψ2|²), so the probability distribution of the superposition state The image is not a simple superposition of the original two probability images, so there is an interference of probability on . As time goes by, more particles will accumulate in places with high probability, so the probability interference image becomes a real interference image.
In other words, there is no difference between single-electron double-slit interference in quantum mechanics and classical interference, both because of superposition . In classical mechanics, two waves can be superimposed, and in quantum mechanics, the two wave functions (state vectors) describing the state of the system can also be superimposed, and their observables (intensity and probability) are all square-related , so after superposition There will be one more cross-term , and then an interference pattern will appear.
As for "the interference pattern disappears after just one glance", that's even simpler. No matter what you use to look at, be it human eyes, instruments or a dog, as long as we know which gap the electrons pass through, we are essentially completing an measurement of through interaction with the system. Measurement in quantum mechanics will change the state of the system. It will change the system from its original state to an eigenstate of the measured mechanical quantity. This is something we are all too familiar with.
So, when you measure which gap an electron will pass through, this operation changes the state of the electron, changing the electron from its original state to a certain eigenstate. When the state changes, the probability distribution also changes, so the interference pattern naturally disappears. Some books say that the double-slit interference of a single electron is the interference of the electron itself with itself. In fact, it is said that this is the interference between the two states of the electron (the state of passing through gap 1 and the state of passing through gap 2). The measurement process changes the state of the electrons, thus destroying the interference pattern.
It can be seen that if we establish the quantum mechanical framework, from a quantum perspective, the double-slit experiment is very simple and natural. It is nothing more than saying " system status can be superimposed, and measurement will change the system status ". What is so strange about these basic conclusions? Where is the horror at all? If you insist on looking at the problem from a classic perspective, and then scare yourself, saying oh my god, it’s so scary, and your world view will be shattered, then what else can I say?
Of course, this is just a very brief introduction to the double-slit experiment (I will write a separate article to discuss it in detail later). The purpose is to let everyone know: if we learn to look at problems from a quantum perspective, many things you thought were strange, weird, Horrible questions will become very natural. , you think the double-slit experiment is scary, it is no different from the ancients who found lightning scary. , once you master the correct perspective of looking at these problems, they are all very natural phenomena.
6 Uncertainty Principle
In addition, many people think that Uncertainty Principle is also mysterious, but in fact it is also very natural. You will quickly understand when you look at the illustration in Griffith's "Introduction to Quantum Mechanics":
In the picture above, it is difficult for you to tell where the wave is, but it is easy to tell the distance between the two wave crests (that is, wavelength); in the picture below, you can easily tell where the wave is, but you can't tell what its wavelength is.
That is, if the wavelength is more precise (above), the position of the wave is less precise; if the position of the wave is more precise, the wavelength is less precise (below).
In quantum mechanics, we use wave functions to describe the state of the system, and there is a simple relationship between wavelength λ and momentum p: p=h/λ.Substituting momentum for the wavelength in the above figure, we have: the more precise the momentum, the less precise the position; the more precise the position, the less precise the momentum.
In addition, we can also see that the more certain the position of a wave, the more uncertain its wavelength. This is an intrinsic attribute of the system, and it has nothing to do with whether you measure or not. Heisenberg initially thought that the measurement interfered with other physical quantities, leading to inaccurate measurements, but later he learned that this was not the case.
Regarding the uncertainty principle, I will briefly talk about it here, because this article made me unexpectedly discover: It turns out that the article on the public account can only write 0,000 words at most, and no more can be posted! I am already frantically testing the edge of the limit. I will talk about it in more detail later ~
7 Interpretation of Quantum Mechanics
There are many exciting topics in the quantum world, such as Schrödinger’s cat, the debate between Bohr and Einstein, and Bell’s inequality , many world theory, Dirac equation, quantum field theory, quantum entanglement, quantum communication and quantum computing, etc., I won’t talk about it here. But everyone should be clear that the prerequisite for us to happily discuss these topics is that you have mastered the basic framework of quantum mechanics and know how to think about problems from a quantum perspective, otherwise you will just watch the fun.
For example, many people know the debate between Bohr and Einstein , but few people know what they are arguing about. Some people just label Einstein as "anti-quantum mechanics" and believe that Einstein first participated in the establishment of quantum mechanics, became conservative after becoming an authority, and then began to oppose quantum mechanics. That is too superficial.
In order to understand what Bohr and Einstein were arguing about, we must first understand one thing, a very important but easily overlooked thing: The formal theory of quantum mechanics (or the understanding of quantum mechanics The mathematical description of quantum mechanics (also called naked quantum mechanics) and the interpretation of quantum mechanics are different. We must distinguish between the two . What does
mean? We observe various phenomena in nature, discover physical laws, and then describe them using mathematical language. At the beginning, as long as the theory can give correct predictions and the calculation results can be consistent with the experiments, does not ask what kind of physical reality is represented behind these mathematical languages.
For example, after de Broglie proposed the matter wave hypothesis, Schrödinger found the corresponding wave equation, which is the famous Schrödinger equation . Through the Schrödinger equation, we can well describe various quantum phenomena. But what exactly is the solution to the Schrödinger equation, that is, the wave function? But no one agrees.
In other words, although we use wave functions to describe the state of the system, and this works very well. But what exactly is this wave function? Does it describe the real state of particles ( real ), or is it just a tool for us to understand particles, describing only our understanding of particles ( non-real )? This is actually a philosophical ontological issue, and I did not mention a word about such issues in the article.
The quantum mechanical framework I introduce here is actually just a set of mathematical descriptions of quantum mechanics. We can say that it is the formal theory of quantum mechanics or bare quantum mechanics . If we want to ask about the physical image behind this mathematical language, it involves the interpretation of quantum mechanics .
The so-called interpretation of is to interpret the physical images behind a set of mathematical languages. We use state vectors to describe the state of the system, operators to describe mechanical quantities, and Schrödinger's equation to describe the change of the system state over time. These are all mathematical descriptions of quantum phenomena and are the formal theory of quantum mechanics. This is recognized by everyone, whether Einstein or Bohr.
However, if we want to know what kind of physical world this mathematical language corresponds to, and what the wave function is, interpretation will appear.Faced with the same set of formal theories, interpretations can be diverse, so the differences between Bohr and Einstein emerged.
The Copenhagen interpretation headed by Bohr believes that the wave function does not describe the true state of particles. It is just a tool for us to understand the quantum world. The wave function only has epistemological significance. When we measure, the wave function collapses instantaneously. Moreover, although the evolution of the system state obeys the Schrödinger equation, the process of measurement leading to the collapse of the wave function does not obey the Schrödinger equation...
There are many other views on the Copenhagen interpretation, so I will not list them all here. Through such an interpretation, Bohr and others constructed a relatively complete quantum picture. In this way, everyone will have a specific picture in their mind when dealing with quantum mechanical problems.
Of course, although the Copenhagen quantum image is consistent with the experiment, it also has many theoretical problems: the wave function collapses instantly during the measurement process, and this process does not satisfy the Schrödinger equation. How does the collapse process occur? Measurement is so important here, so what kind of behavior can be considered measurement? Why are there two types of evolution processes, one that obeys the Schrödinger equation and the other that does not? The quantum world is so different from the classical world. If you draw a line between them, where is this line?
More importantly, Copenhagen interpretation says that the wave function does not describe the true state of the electron, it is just a cognitive tool. They believe that there is no real electron state at all. Only when we find electrons when measuring, can we say that electrons exist. Therefore, from the perspective of Copenhagen, is our measurement process that creates electronic . Electronics do not exist when you do not measure them.
This statement completely angered Einstein . He said: " Doesn't the moon cease to exist when we don't look at the moon? ". Everyone is more familiar with Einstein's other sentence "God does not throw dice", but in fact, Einstein was more concerned about the existence of the moon than whether he threw dice. We often see the debate between Bohr and Einstein in popular science books. What Einstein opposed was not quantum mechanics (no one opposed the formal theory of quantum mechanics). What he opposed was the Copenhagen interpretation of quantum mechanics .
Einstein hated the Copenhagen Interpretation (so did Schrödinger and de Broglie), so he found some loopholes in the Copenhagen Interpretation while looking for some new interpretations. However, although the Copenhagen Interpretation had many problems, it was consistent with the experiments, and its competitors were too weak at the time. Einstein's superb fault-finding skills continued to patch up the Copenhagen Interpretation. Coupled with the authority of Bohr, Heisenberg, and Born in the quantum field, Einstein will only be unhappy with it until his death, and there is no good way to deal with it.
Two years after Einstein's death, a man named Everett proposed a new interpretation of quantum mechanics: the many-worlds interpretation .
This is an extremely simple explanation in theory, but seems to be extremely "absurd" in reasoning. Many Worlds can even be said to be an interpretation of that does not need to be interpreted as , because it has two basic assumptions: first, the system state is described by the state vector; second, the evolution of the state vector over time obeys the Schrödinger equation (it can be seen that it is similar to ours) The formal theory discussed here is not the same, so many worlds is not just an interpretation, it is an independent theory).
It does not need any of the additional assumptions of the Copenhagen interpretation (collapse caused by measurement, boundary issues between quantum and classical, etc.), nor does Born's rule. These things are not assumptions in many worlds, but conclusions . It can also be consistent with all experiments, and there is no problem of "if you don't look at the moon, the moon does not exist".
In multi-world interpretation (theory) , the wave function describes the true state of particles (reality). Measurement is just the interaction between the instrument and the system. The measurement process also obeys the Schrödinger equation. There is no wave function. Collapse . It has many other viewpoints, which together constitute a complete picture of quantum mechanics, but it is obvious that this is a completely different picture from the Copenhagen interpretation of . The details of
will not be discussed here, but will be discussed later. However, from here we can at least see that: there is wave function collapse in the Copenhagen Interpretation, but there is no wave function collapse in the Many Worlds Interpretation; in the Copenhagen Interpretation, the wave function does not describe the true state of particles, but in the Many Worlds Interpretation, the wave function describes the true state of particles. ; There is a quantum-classical boundary problem in the Copenhagen interpretation, but not in the many-worlds interpretation...
There are many differences between these two interpretations, but they are both consistent with experiments. Who do you think I should listen to?
Copenhagen interpretation is sometimes called the orthodox interpretation, and many textbooks are also written in Copenhagen form. Today, multi-world interpretation of also has many supporters. However, whether it is Copenhagen, many worlds, or any other interpretation, the proportion of supporters is very low. The choice of more physicists is: don’t interpret! Don’t interpret! Don’t interpret!
They just use the formal theory of quantum mechanics to do calculations, as long as they can be calculated and useful! As for the physical image behind it, screw Bohr and Einstein, I don't believe any of them, they are the shut up and calculate faction. Of course, shutting up about calculations doesn't mean they don't care about interpretation. No physicist really doesn't care about the images behind quantum theory. However, the existing explanations were not very convincing, and no explanation was particularly convincing, so they simply ignored it.
Therefore, many quantum mechanics textbooks will also consciously avoid the problem of interpreting . They only introduce the formal theory of quantum mechanics , only introduce how we use mathematical language to describe quantum phenomena, and only introduce this set of concepts that everyone recognizes. s things.
formal theory does not talk about whether the wave function collapses at all. It only says that the measurement result is one of the eigenvalues of the corresponding operator. As for what happened during the measurement process, whether the wave function collapsed or the world split, it doesn't care.
Some friends may be confused: I have been studying physics for so long, why do I only seem to have interpretation problems in quantum mechanics, but not at all when I study other theories? For example, when we study Newtonian mechanics, there is no explanation.
Newtonian mechanics certainly has interpretations, but in Newtonian mechanics we use real numbers and functions in three-dimensional space to describe particles and fields. This description has very direct spatial meaning. Therefore, everyone can achieve broad consensus on what concepts in Newtonian mechanics represent what physical meaning. When a stone falls, the mathematical formula describing the process is like this, and the physical image in everyone's mind is also like this. No one has any objection.
However, in quantum mechanics , we use vectors and operators in Hilbert space to describe the system state and mechanical quantities. This is a very abstract mathematical structure. Hilbert space is not the three-dimensional space we come into contact with every day. As a result, it is more troublesome to correspond mathematical concepts with physical reality. Therefore, some people think that the wave function describes reality, some people think that it does not; some people think that the wave function collapses during measurement, some people think that it does not collapse, and so on. There is no consensus on
, which also shows that our understanding of the quantum world is not deep enough. With the advancement of theory and experiment, we may be able to distinguish different interpretations in the future, clarify many things that we still don’t understand, and form a quantum mechanical picture that everyone can agree on. By then, naturally no one will mention any quantum mechanical interpretation.
Quantum mechanics interpretation is a very grand and profound topic. It is not only related to physics , but also to philosophy . It can be said that Einstein has been thinking about it for the rest of his life.
In this article, we only need to know that there is such a thing as quantum mechanics interpretation , know the relationship between formal theory and interpretation, and know that what we introduce here is only formal theory of quantum mechanics. We will discuss more issues about the interpretation of quantum mechanics later.
In this way, the article is coming to an end.
8 Conclusion
In classical mechanics, system state , observable quantity and observation result are all the same, and we do not need to distinguish them deliberately. When it comes to quantum mechanics, in order to describe the Stern-Gerlach experiment and other quantum phenomena, we must distinguish between the three.
We use state vectors to describe the system state and operators to describe mechanical quantities. The measurement result is one of the eigenvalues of the corresponding operator. The change of the system state over time obeys the Schrödinger equation.
In order to concretize the abstract state vector, we need to establish a coordinate system. Then, we found that the coordinate system established with the eigenvectors of the mechanical quantity operators as basis vectors is excellent. Selecting such a set of basis vectors is called selecting an representation . The one established with the eigenvector of the position operator as the basis vector is called position representation , and the one established with the eigenvector of the momentum operator as the basis vector is called momentum representation . They can interact with each other through Fourier transform. Convert. After
selects the representation, we can project the state vector into a specific coordinate system. The projection coefficient (coordinate) is the wave function . Therefore, in addition to the state vector, the wave function can also be used to describe the state of the system.
Then, we also wrote the Schrödinger equation under the position representation. By solving the equation, we can get the wave function. To solve the Schrödinger equation, you must first determine the potential function .
If the potential function does not depend on time , the probability distribution does not change with time, and the average value of the mechanical quantity does not change with time. This state is called steady state . Because the energy in the steady state has a certain value, the steady state is also the energy eigenstate . Energy has a definite value. Solving the stationary Schrödinger equation can obtain the energy that the system can take. In this way, it can be known at a glance whether the energy is continuous or discrete.
So, we know the reason why energy is discontinuous in quantum mechanics, and we also know the general method of dealing with problems in quantum mechanics. After mastering the way of thinking in quantum mechanics, you will find that many familiar quantum mechanical properties (such as energy can be discontinuous) can be deduced, and many problems that everyone thinks are strange, weird, and even scary (such as the double-slit interference experiment) will become Be very natural.
has established the basic framework of quantum mechanics and the general method of dealing with quantum mechanical problems. The purpose of this article has been achieved. Due to space limitations, many topics that everyone is very interested in can only be briefly mentioned here. We will talk about it later. If you are afraid of missing it, just keep an eye on my official account "Long Tail Technology".
Finally, we also distinguished between the formal theory of quantum mechanics and the interpretation of . These things will lead to many super exciting topics later. However, the prerequisite for understanding them is that the formal theory of quantum mechanics has been clarified.
Article source: Long Tail Technology
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