The Fermi-Bose Academy
Alice 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 change."
"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 the 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."
"Right!" replied the Mechanic gaily. "Hadn't you noticed? We can talk quite confidently about observations, but what is really there to be observed, now that is quite another matter.
"Come along, though. It is time for the evening assembly of the Academy. You should find that quite interesting."
The Mechanic led her back into the main building and ushered her through the entrance hall into a huge room with a high vaulted roof. The great tiled floor was completely filled with a crowd of electrons, packed in as tightly as possible. Overhead, a wide ornate balcony ran around the edge of the vast hall, and on it Alice could see the vague distant figures of a few electrons hurrying to an exit. There was just one tiny space on the floor near the doorway by which they had entered, and an electron which had been following them darted in to it and immediately came to a halt, wedged in on every side by the dense crowd so that there was no room to move any farther.
See end-of-chapter note 3
"Why is it so crowded here?" cried Alice, overcome by the scale of the scene before her. "This is the valence level," answered one of the helpful electrons. "All the spaces on the valence level are full because the valence level is always full of electrons. None of us can move at all, as there are no states free to move into, you see."
"That is terrible," cried Alice. "How can any of you possibly move across the floor to get out if it is so crowded?" "We can't," said the electron with cheerful resignation. "But you can if you want to. You can go anywhere on the floor because there are no other Alices here, so there are plenty of Alice states free for you to move into. You will have no Pauli Exclusion problems at all." This still sounded very strange to Alice, but she tried to push her way into the tightly packed crowd and found, just as she had when she had tried to get into the full railway compartment earlier, that somehow she could move through without any trouble.
Alice made her way through the crowd of electrons toward a raised platform at the far end of the hall. On it stood the Principal, impressive as always in his gown and mortarboard. As she came closer Alice could hear his mellow voice booming out over the packed room.
"I know that you have all had a busy day today, but I trust that I do not need to remind you what an important role you must be prepared to play in the world. You electrons, each taking your place in your proper state, form the very fabric of everything we know. Some of you will be bound in atoms and will have to work away in your various levels, controlling all the details of chemical processes. Some of you may find your place within a crystalline solid. There you will be relatively free of attachment to any particular atom and may move around as far as the Pauli Principle and your fellow electrons allow. You may be in a conduction band, where you can move freely, and it will then be your task to rush around carrying your electric charges as part of an electric current. On the other hand, you may be in a valence band within a solid. Perhaps you will feel trapped there as there will be no states free for you to enter. Do not become discouraged. Not every electron may be in the highest energy states, and remember that the lowest levels must also be filled."
See end-of-chapter note 4
"As for you photons, you are the movers and shakers. Left to themselves the electrons would stay complacently in their proper states, and nothing would ever be done. It is your task to interact with the electrons at all times and to produce the transitions between states, the changes which make things happen."
At this point in the Principal's address, Alice became aware of the bright shapes of photons rushing though the crowd of electrons and of occasional flashes from different parts of the room. She turned around to see what was happening. It was difficult for her to see very far, because she was closely surrounded by so many electrons.
"This is really too bad!" Alice could not help exclaiming as she looked at all the captive figures, held fixed in position by the crush around them. "Is there no way in which anyone can move at all?"
"Only if we should get excited to the higher level," a voice answered. Alice could not see who had spoken. "But it doesn't really matter," she thought to herself. "Since they are all the same, then the same one as always must have spoken, I suppose." Just then there was a flash nearby and Alice saw that a photon had come rushing through the crowd and crashed into an electron. The electron soared upward and landed on the balcony, where he began running furiously toward the exit.
Alice was staring so hard at the retreating electron that she did not observe another photon rushing in her direction. There was a brilliant flash and she felt herself rising in the air. When she looked around she saw that she was now standing on the balcony also, looking down on the mass of electrons below. "This must be what the electron meant by being excited to the higher level. It doesn't seem all that exciting to me, but at least there is a lot more room here." She looked over the edge of the balcony at the floor beneath and could see occasional little flashes here and there, each one followed by an electron floating up from the floor and landing on the balcony, where he or she immediately began to run at high speed toward the exit. One of them landed on the balcony close to where Alice was standing.
She looked down and could see a little electron-shaped hole where that electron had been a moment before. It was clearly visible, as the contrasting color of the tiled floor stood out sharply against the uniform background of closely packed electrons which covered the surface everywhere else. As she watched this space another electron nearby stepped smartly onto the gap which had just been created, although it could then move no further. Where this electron had been standing, however, there was now a hole and a more recently arrived electron stepped into that. "What a curious thing!" Alice said to herself. "I have become used to seeing electrons, but I did not expect to be able to see the presence of no electron quite so clearly!" She watched with interest as the movement along the balcony of the electron which had risen up to make the original hole was balanced by the movement of the electron-shaped hole as it progressed steadily across the floor in the other direction, toward the wide door by which she had originally come in.
See end-of-chapter note 5
When both electron and hole were out of sight, Alice herself walked along the balcony to the exit. She felt she had heard quite enough of the Principal's talk. She passed through the small door and found herself in a long corridor. Waiting for her by the door was the Quantum Mechanic. "How did you enjoy that?" he asked.
"Very well, thank you," replied Alice politely. She felt that it was expected of her. "It was most interesting to hear the Principal conducting the assembly."
"You say that," began the Mechanic, "but of course it was really the electrons which were doing the conducting, once they had been excited up to the conduction level. All electrons have an electric charge you know, so when they move around they cause an electric current to flow. The charge they carry happens to be negative, so the current flows in the opposite direction to the movement of the electrons, but that is a minor point. If all the states which any electron might reach are already full of electrons, as in the valence level, then there can be no movement and you have an electrical insulator. All the electrons and their charges are fixed in position in that case so there can be no electric current. In the present case you can get a current only when electrons have been carried up to the empty conduction level where they have plenty of room and can move easily. In that case you can get a current produced both by the electrons and by the holes they leave behind."
"But how can a hole give a current?" protested Alice. "A hole is something which isn't even there."
"First, you will agree that when the electrons are all present in the lower valence level, they cannot move and there is no current?" asked the Mechanic. The current is just the same as if there were no negatively charged electrons present."
"Well, yes," answered Alice. That sounded fair enough.
"Then you must admit that when there is one electron less, the current will look like that due to one less than no electrons. The hole in the valence level behaves as if it were a positive charge. You saw how the movement of the hole toward the door was actually due to a lot of electrons taking one step in the opposite direction. So the electric current produced by negatively charged electrons moving away from the door is the same as a positive charge moving toward the door would give. As I said, the photons produce a current both from the electrons they put into the conduction band and from the holes they leave behind."
"The photons seem to be rather a bother to the electrons," remarked Alice, deciding to change the subject.
"Well, they are certainly rather hyperactive, but then photons are naturally very bright. As the Principal says, particles will be particles. I expect that at the moment some of them are lasing electrons in the dorm."
"I am sorry," queried Alice, "but don't you mean hazing? I am sure that is the word that I have heard used to describe student pranks."
"No, it is definitely lasing. Come and see."
They walked on down the corridor to a door at the end. The Mechanic opened this door and they entered, closing the door behind them. They were now in a long room which was lined along both sides with bunk beds. Alice could see that many of the top bunks were occupied by electrons, but the lower bunks were for the most part empty. "You sometimes find them in the top bunks rather than the lower ones," remarked the Mechanic. "It is called population inversion. It is only when they are like that that lasing becomes practical."
It was not very long before a lone photon came running into the room. He rushed to one of the bunks and careened into the electron which occupied that elevated position. With a thump the electron plummeted down to the lower bunk, and Alice was startled to see that there were now two photons rushing together around the room. They moved in perfect unison so that they almost seemed as one. "That is an example of stimulated emission," the Mechanic murmured in Alice's ear. "The photon has caused the electron to make a transition to a lower level, and the energy released has created another photon. Now just watch and see how the lasing develops."
The two photons rushed up and down the long room. One collided with an electron, and then there were three photons and another electron in a lower level. As Alice watched, the photons interacted with more electrons, producing more photons. Occasionally she noticed a photon collide with an electron which had fallen to a lower bunk. When this happened the electron shot up to the higher bunk and the photon vanished, but as there were initially very few electrons in the lower bunks this did not happen often to begin with.
See end-of-chapter note 6
Soon the room was crowded with a horde of identical photons, all rushing to-and-fro in perfect synchronism. There were now almost as many electrons in the lower bunks as in the upper ones, so that collisions were as likely to excite an electron to a higher position, with the loss of one of the photons, as to create a new one. The stream of photons poured out through the door at the end of the dormitory and down the corridor as a tight coherent beam of light. Before they had gone halfway down the corridor they collided with the massive form of the Principal who was walking toward them.
He immediately stopped, drew himself up to his full height, and spread his thick black gown to either side, so that he presented a dense black body, effectively blocking the corridor. The photons struck the inky black material and vanished completely. The Principal stood there for a moment, looking both hot and bothered and mopping the perspiration from his ruddy face with a handkerchief.
"I will not tolerate this sort of behavior," he puffed. "I have warned them before that any photons who carry on in this way will be instantly absorbed. It is hot work, though, since the energy released has to go somewhere, and it usually ends up as heat."
"Excuse me," said Alice. "Could you tell me where all those photons have gone?"
"Why, they have not gone anywhere, my dear. They have been absorbed. They are no more."
"Oh dear, how tragic," cried Alice, who felt sorry for the poor little photons who had been so abruptly snuffed out.
"Not at all, not at all. It is all part of being a nonconserved particle. Photons are like that. Easy come, easy go. They are always being created and destroyed. It is nothing very serious."
"I am sure that it must be for the photon," cried Alice.
"Well, I am not even so sure about that. I do not think it makes much difference to a photon how long it seems to us that it exists. They travel at the speed of light, you see, as after all they are light. For anything traveling at that speed, time will actually stand still. So, however long they seem to us to survive, for them no time at all will pass. The entire history of the universe would pass in a flash for a photon. I suppose that is why they never seem to get bored.
"As I said in the assembly, photons have many important parts to play in exciting electrons from one state to another and indeed in creating the interactions which make the states in the first instance. In order to do this, it is necessary that they be created and destroyed very frequently; it is part of the job, you might say. Creating interactions is more the task of virtual photons, though. We do not deal much with them here. If you are interested in states and how one goes about moving from one to another, then you should visit the State Agent. Your friend there will show you the way."
The Principal escorted them out of the Academy and back down the drive to the gate. As they walked on down the road, Alice turned back once to wave to the Principal, who was standing solidly in the center of the gateway where she had first seen him.
Notes
1. If you have many particles you will have some sort of amplitude for each of them and an overall amplitude which will describe the whole system of particles. If the particles are all different from one another then you know (or can know) the state each is in. The overall amplitude is just the product of the amplitudes for each particle separately.
If the particles are identical to one another, then things get more complicated. Electrons (or photons) are completely identical. There is no way to distinguish one from another. When you have seen one, you have seen them all. If two electrons were interchanged between the states they occupied, there is no way that you would ever be able to tell. The total amplitude is, as usual, a mixture of all the indistinguishable amplitudes, which now includes all those permutations in which particles have been interchanged between two states.
Interchanging two identical particles makes no difference to what you can observe, which means it makes no difference to the probability distribution that you get when you multiply the amplitude by itself. This could mean that the amplitude itself does not change either, or it could mean that the amplitude changes sign, for example, going from positive to negative. This is equivalent to multiplying the amplitude by -1. When you multiply the amplitude by itself to get the probability amplitude, then this factor -1 is also multiplied by itself to give a factor of +1, which produces no change in the probability. The change in sign sounds like a trivial academic point, but it has amazing consequences.
2. There is no obvious reason why an amplitude should change sign just because it cannot be shown that it may not, but Nature seems to follow the rule that anything not forbidden is compulsory and to take up all her options. There are particles for which the amplitude does change sign when two of them are interchanged. They are called fermions, and electrons provide an example. There are also particles for which the amplitude does not change in any way when two are interchanged. These are called bosons, and photons are of this type.
Does it really matter whether the sign of the amplitude for a system of particles does or does not change sign when two of them are interchanged between states? Surprisingly, it does. It matters a great deal.
You cannot have two fermions in the same state. If two bosons were in the same state and you happened to interchange them, then it really would make no difference at all-it could not give even a change of sign. Such amplitudes are not allowed for fermions. This is an example of the Pauli principle, which says that no two fermions may ever be in the same state. Fermions are the ultimate individualists; no two may conform completely.
The Pauli principle is extremely important and is vital for the existence of atoms and of matter as we know it. Bosons are not governed by the Pauli principle-quite the reverse, in fact.
If each particle is in a different state and you square the overall amplitude to calculate the probability distribution for the particles, then each particle separately contributes much the same amount to the total probability. If you have two particles in the same state and square that, you get four times the contribution from only two particles. Each has contributed proportionately more, so that having two particles in the same state is more probable than having each in a different state. Having three or four particles in the same state is even more probable, and so on. This increased probability for having many bosons in the same state gives the phenomenon of boson condensation: Bosons like to get together in the same state. Bosons are easily led; they are inherently gregarious.
Boson condensation is seen, for example, in the operation of a laser.
3. Electrical forces involving electrons can operate to hold atoms together, as discussed in Chapter 7, but they do not give rise to any repulsion which would push the atoms apart; so why do atoms keep a fairly uniform distance from one another? Why are solids incompressible? Why are the atoms not pulled into one another, so that a block of lead would end up as one very heavy object of atomic size? Once again it is a consequence of the Pauli principle, which says that two electrons cannot be in the same state.
Since the atoms of a given type are all the same, each has the same set of states. Does this not put the equivalent electrons in each atom into the same state, which is not allowed? Actually, as the atoms are in different positions, the states are slightly different. If you were to superimpose the atoms, then the states would be the same, and the Pauli principle forbids this. The atoms are kept apart by what is known as Fermi pressure, but which is really the intense refusal of the electrons in one atom to be the same as their neighbor. Matter is incompressible because of the extreme individualism of electrons.
4. In a solid the electron states from the individual atoms have combined together to form a large number of electron states which belong to the solid as a whole. These states are grouped into energy bands, within which the energy levels of the states are so close together as to be almost continuous. Corresponding to the larger energy-level separations in the individual atoms, there are gaps in the energy bands of the solid. The lower bands are full of electrons which have come from the lower levels in the atoms. The highest of these full bands is called the valence band, and above it, separated by a band gap which contains no states, is another band: the conduction band. This band is either completely empty or, at most, only partly full.
In the valence band the electrons cannot move. Any electron movement requires that electrons should change from one state to another, and there are no empty states for the electrons to go into. If an electric potential was put across a material, it would apply a force to the electrons in the valence band, but they could not move. If there were no electrons in the conduction band, the material would act as an electrical insulator.
5. If an electron in the full valence band is given enough energy, either by collision with a photon or even by a chance concentration of thermal energy, then the electron may rise across the band gap into the higher conduction band. As there are plenty of empty states in this band the electron can now move around, and an electric potential will produce conduction. Further, there is now a free space in the valence level, where the electron used to be. Another electron may move into this hole, and so on. There will be a hole in the otherwise full valence band, and it will be moving in the opposite direction to the electron movement. This hole behaves very much like a particle with positive charge.
The above describes the behavior of semiconductor materials: materials such as silicon, which is widely used in electronics. Electric current is carried both by electrons in the conduction level and by holes in the valence level.
6. When a photon which has the correct energy interacts with an electron in an atom, it may produce a transition from one energy level to another, as described in Chapter 6. In most cases the transition will be from a lower to a higher energy level, since usually the lower levels will all be full. The photon is equally capable of producing a transition from a higher to a lower level, if the lower level is empty.
Should a substance happen to have a lot of electrons in a higher level, and a lower level is mostly empty (a condition known as population inversion), then a photon can cause an electron to transfer from a higher state to a lower one. This change releases energy and creates a new photon, in addition to the one which caused the transfer. This photon can in turn induce more electrons to fall to a lower state.
In a laser the light produced is reflected back-and-forth from mirrors at either end of the cavity, causing more photon emissions each time it passes repeatedly through the material. A little of this light escapes through the mirrors, which are not perfect reflectors, and gives an intense narrow beam: laser light. As the photons were emitted in direct response to the photons already present, the light is all "in step," or in phase, and has unique properties for producing interference effects on a large scale, as may be seen in holograms. (Not all holograms need laser light, but it helps.)