Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  "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 t
he 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.