2. The orthodox description of a measurement in quantum mechanics has the drawback that the process of making a measurement does not seem to be at all compatible with the rest of quantum theory. If the quantum theory is the true theory of atoms, as seems to be the case, and if the whole world is made of atoms, then presumably quantum theory should apply to the whole world and everything that is in it. That includes measuring instruments. Where a quantum system can give various values, its amplitude is a sum of states corresponding to each possible value. When the measuring device is itself a quantum system and there are various values which it could measure, it has no right to select just one of them. It ought to be in a state which is a sum of the amplitudes for all the possible results it might measure, and no unique observation could be made.
The conclusion you would draw from the above would seem to be either:
(a) We never actually observe anything
or
(b) Quantum theory is all nonsense.
Neither conclusion is really tenable (however tempting conclusion (b) might appear to be). We know perfectly well that we do observe things, but we cannot deny that quantum theory has an unbroken success rate at successfully describing all observations, while no alternative theory does as well. We cannot lightly abandon it.
lice walked with the Quantum Mechanic along the path away from the school. As they traveled the path grew wider and gradually changed to a well-surfaced road.
"I think the most curious thing you have shown me," remarked Alice, "was the way that you got those interference effects even when there was only one electron present. Is it true then that it makes no difference whether there are many electrons or only one?"
"It is certainly true that you may observe interference whether you have many electrons or only one at a time. However you cannot say that it makes no difference. There are some effects which you only see when you have many electrons. Take the Pauli Principle, for example...."
"Oh, I have heard of that," interrupted Alice. "I heard the electrons talk about it when I first came here. Would you tell me what it is, please?"
"It is a rule which applies when you have a lot of particles which are all the same-completely identical in every respect. If you would like to know more about it, it would be best if we were to call in here, since we happen to be passing, and they are very experienced in many-particle behavior."
Alice looked around at these words and found that, as they had been talking, they had come to a tall stone wall which ran along one side of the road. Immediately opposite them was a wide gateway. Impressive wrought-iron gates stood open between massive stone pillars with a coat-of-arms painted in the center of each. To the right of the gateway, visible above the wall, Alice saw a wooden board which carried the message:
In the center of the gateway stood an imposing figure, a large and exceedingly well-built man made even more massive in appearance by the flowing academic gown and the mortarboard which he wore. His round, florid face was copiously adorned with a bushy mustache and side whiskers. Firmly fastened in one screwed-up eye he wore a monocle on a wide black ribbon.
"That is the Principal," whispered the Mechanic into Alice's nearest ear.
"Do you mean the Pauli principle?" asked Alice rather wildly. She had been taken off-guard by his sudden appearance.
"No, no," hissed the Mechanic, "he is the Principal of the Academy. Though of course Pauli's principle is the principal principle of the Academy, he is its Principal." Alice wished that she had not asked.
They crossed the road and went up to this imposing personage. "Excuse me sir," began the Mechanic. "Would you be so kind as to tell my young friend here something about many-particle systems?"
"Of course, of course," boomed the Principal. "We have no shortage of particles here, dear me no. I shall be most happy to show you around."
He turned around with a billow of his flowing gown and led the way toward the Academy. As they walked up the drive Alice saw small figures dodging in and out among the shrubbery. At one point a figure popped above a bush and made a face at them. At least Alice thought it had. As usual it was very difficult to make out any detail. "Ignore him," growled the Principal. "That is only Electron Minor."
They arrived at the door of the Academy, which was housed in a dignified old house of vaguely Tudor appearance. Without pausing the Principal led them through the main door into a vaulted entrance hall and up a wide carved staircase. As they walked through the building, Alice could see small figures hiding behind the banister, dodging in and out of rooms, and running off down side corridors as they approached. "Ignore him," remarked the Principal again. "It is just Electron Minor. Particles will be particles!"
"But it cannot be Electron Minor if we saw him on the drive," protested Alice. "Surely it cannot be the one particle in both places. Are we talking about something like the case when an electron managed to go through both holes in your double-slit experiment?" she asked the Quantum Mechanic.
"No, it is not that; they do have many electrons here. But don't you see, the electrons are all exactly the same. They are completely identical to one another. There is no way to tell them apart, so naturally they are all Electron Minor."
"That is right," confirmed the Principal emphatically as he led them into his study, "and it is a problem, let me tell you. You may know how difficult it can be for teachers when they have two identical twins in their school and are unable to tell them apart. Well I have hundreds of completely identical particles. It makes checking the register a nightmare, I can tell you.
"The electrons are not so bad," he went on. "We just count them and see whether we have the correct total. At least the number of electrons is conserved, so we know how many we ought to have, but for the photons even that does not work. The photons are bosons, so they are not conserved you see. We may begin a class with thirty, say, and have fifty or more at the end of it. Or the number may drop to less than twenty-it is hard to predict. This all makes it very difficult for the staff."
Alice had spotted a new word in that remark. "Do you think you might explain that?" she asked hopefully. "Would you please tell me what a'boson' is?"
The Principal turned an even deeper red than he was before and spoke to the Mechanic. "I think it would be best if you took her to the beginners' Facts of Symmetry lesson, don't you? That should explain all about the Bosons and the Fermions."
"Right you are," replied the Mechanic. "Come along, Alice, I believe I can remember the way."
They walked down a corridor to a classroom and went in just as a lesson was beginning.
"Attention please," said the teacher. "Now as you well know, all you electrons are identical to one another and so are all you photons. This means that no one can tell when any two of you have changed places. As far as any observer could tell you might have changed places and so of course you will have to some degree. You all know that you have an associated wave function, or amplitude, and that this amplitude will be a superposition of all the things which you might be doing. Where there is no way of telling which things you are doing, then, as you know, you are doing all of them, or at any rate have an amplitude for every one of them. So you see, for any group of you it is impossible to tell when any two have changed places and this means that your overall wave function will be a superposition of all the amplitudes for which a different pair has swapped over. I hope that you have all made a note of that."
See end-of-chapter note 1
"Now the probability of making any observation is given by the square of your wave function, that is, the wave function multiplied by itself. As you are completely identical, it is obvious that when any two of you change places it can make no observable difference, so the square of your wave function cannot change. It might look as if there can be no change at all. Can anyone tell me what might change?"
One of the electrons put his hand up, or at least Alice assumed that was what had happened. She was not able to see at all clearly. "Please sir, the sign might c
hange."
"Very good, that is an excellent answer. I would make a note in your record that you had answered so well, except that unfortunately I cannot tell you apart from the others. Yes, as you know your amplitudes do not have to be positive. They may be either positive or negative, so that two amplitudes may cancel one another out when you have interference. This means that there are two cases in which the square of your amplitude would not be changed. It may be that the amplitude does not alter at all when two of you change places. In such a case the particles are bosons, like you photons. However, there is another possibility. When two of you exchange places, the amplitude may reverse. It changes between positive and negative. In this case the square is still positive and the probability distribution is unchanged, because multiplying the amplitude by itself will give two reversals, resulting in no change at all. This is what happens with fermions, such as you electrons. All particles fall into one or other of these two classes: They are either fermions or bosons.
"Now you may think that it does not matter much whether your amplitude reverses or not, especially as the probability distribution remains unchanged, but in fact it is very important indeed, particularly for fermions. The point is that if any two of you are in exactly the same state-that is, in the same place and doing the same thing-then if you exchange places, it is not only an unobservable change; it really is no change at all. In this case neither the probability distribution nor the amplitude can change. This is no problem for bosons, but for fermions, which always have to reverse their amplitude, such a situation is not allowed. For such particles you get the Pauli exclusion principle, which says that no two identical fermions may be doing exactly the same thing. They all have to be in different states."
See end-of-chapter note 2
"For bosons," as I said, "it is not a problem. Their amplitudes do not have to change at all when two of them change places, so they may be in the same state. In fact I can go further; not only may they be in the same state, but they positively like to be in the same state. Normally when you have a superposition of different states and square the amplitude to give the probability of observation, the individual states in the mixture are squared separately and contribute much the same to the overall probability. If you have two bosons in the same state, then when you square the two you get four. The two have contributed, not twice as much as one, but four times as much. If you had three particles in the same state they would contribute even more. The probability is much higher when there is a large number of bosons in one state, so they tend to get into the same state if at all possible. This is known as Bose condensation.
"So, there you have the difference between fermions and bosons. Fermions are individualistic, no two will ever do exactly the same thing, while bosons are very gregarious. They love to go around in gangs where each one behaves in exactly the same way as the others. As you will see later, it is this behavior and the interaction between you two types of particles which are responsible for the nature of the world. In many ways you are the rulers of the world."
At this point the Quantum Mechanic led Alice out of the classroom. "There you are then," he said. "That is the Pauli principle. It rules that no two fermions of the same type can ever be doing the same thing, so you can have one and one only in each state. The principle applies to all fermions of whatever type, but not to bosons. This means, among other things, that the number of fermions must be conserved. Fermions cannot just appear and disappear in a casual fashion."
"I should think not!" Alice said. "That would be ridiculous."
"I do not think you can say that, you know, because bosons do appear and disappear. Their number is not conserved at all. You can argue that the number of fermions must be definite if there is one and only one in each state, since a particular number of occupied states implies that there is that particular number of fermions to occupy them. The argument does not hold for bosons, since you can have as many as you like in any state. In practice the number of bosons is not at all constant.
"If you just look out this window here," he said suddenly as they were passing, "you can see the difference between fermions and bosons quite well."
Alice gazed through the window and saw that a group of electrons and photons were being drilled on the Academy field. The photons were doing very well, wheeling and reversing in perfect synchronism with no differences between any of them. The group of electrons, however, were behaving in a manner which was obviously driving the drill sergeant to despair. Some were marching forward, but at different speeds. Some were marching to the right and to the left, or even backward. A few were jumping up and down or doing headstands and one was lying flat on his back, staring at the sky.
"He is in the ground state," said the Mechanic, looking over Alice's shoulder. "I expect the other electrons wish that they could join him there, but only one of them is allowed you see. Unless the other had an opposite direction of spin, of course-that would make a sufficient difference between them.
"You can clearly see the difference between the fermions and bosons here. The photons are bosons, so it is easy for them to do the same thing. Indeed, they positively like to be the same as one another, so they are very good at marching in step. The electrons, on the other hand, are fermions and so the Pauli exclusion principle stops any two of them from being in the same state. They have to behave differently from one another."
"You often talk about the electrons being in states," Alice remarked. "Would you please explain to me just what is a state?"
"Once again," responded the Mechanic, "the best way will be for you to sit in on one of the classes here. The Academy teaches world leaders, since it is the interaction of electrons and photons that rules the physical world, by and large. If they are to be world rulers, they have to go to Statecraft classes naturally. Come along and let us see one."
He led Alice down to a large low building at the back of the Academy. When they went inside Alice could see that it was some sort of workshop. A number of electrons were working away at different benches. Alice went over to watch one group, who were busily erecting a set of fences around the edge of the bench. Alice could see there were various structures on the bench, and as the students moved the fences around, these structures all changed.
"What are they doing?" Alice asked her companion.
"They are setting up the boundary conditions for the states. States are controlled largely by the constraints which hedge them in. In general, what you can do is governed by what you cannot do and the restrictions serve to define the possible states. It is very much like the notes you can get from an organ pipe. For a pipe of a given length you can produce only a limited number of notes. If you change the length of the organ pipe, then you will change the notes. Quantum states are given by the amplitude or wave function which the system can have, and this is much like the sound wave in an organ pipe.
"As you have already discovered, you usually cannot say what an electron is really doing, because if you observe it, to check you will select out one particular amplitude and reduce the amplitudes to that one alone. The only time when you can be really certain about your electron is when it has a single amplitude instead of a superposition and when your observation can give but one value. In that case the probability of your seeing that value from your measurement is 100 percent and for any other result the probability is zero-it won't happen. When you make the observation, then you will see the expected result. In such a case, the reduction of the amplitude to that for your observed result has made no difference at all, as you were already in such a state. The state is not changed by the observation, and it is called a stationary state. In this class electrons are setting up stationary states."
Alice walked around the table, looking at the states which the electrons were crafting. They looked to her like a series of boxes, eight in all. There was one very large one, one slightly smaller than the large one, and six tiny ones of much the same size. She turned a corner of the table and was surprised to see that th
e states had changed completely. Now they had the appearance of a number of stands, rather like cake stands, on tall pedestals. There were two which were much wider than the others; four of the same widths, but with successively taller pedestals; and two small ones. She walked quickly around another corner of the table. Now she saw that the center of the table was occupied with a large board to which were fastened a number of coat hooks. There were two rows of three and isolated single hooks top and bottom. "Goodness, whatever is happening?" she asked her companion. "I keep seeing the states quite differently when I look at them from different directions."
"Well, of course you do," replied the Quantum Mechanic. "You are seeing different representations of the states. The nature of a state depends on how you observe it. The very existence of a stationary state relies upon some observation for which it always produces a definite result, but a state cannot give definite results for all observations you can make. For example, the Heisenberg relations prevent you from seeing the position and the momentum of an electron at the same time, so a stationary state for one observation will not be a stationary state for the other. The observations which you use to describe the states are called its representation.
"The nature of a state may be very different, depending on how you observe it. Indeed the very identity of the different states can change. The states that you see in one representation may not be the same as the ones in another representation. As you may have noticed just now, the one thing which must remain constant is the number of the states. If you can put one of the electrons in each state then you must always have the same number of states to contain them all, even though the individual states may have changed."
"That seems very vague to me," complained Alice. "It sounds as if you cannot be at all sure what is really there."
Alice in Quantumland: An Allegory of Quantum Physics Page 8
