Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  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.)

  he Quantum Mechanic led Alice down the road and through a wrought iron gate into an attractive park. Beautiful flower beds, full with an abundance of assorted blooms, lined both sides of the path, giving a most pleasing effect as they strolled along in the warm summer weather. In the sky the sun shone brilliantly, pouring out its light upon the idyllic scene. Beside the path colorful butterflies flitted from bright blossom to bright blossom and a small stream burbled downhill over a bed of rounded pebbles, while here and there along its path the water poured over a miniature waterfall. Alice thought it all very pretty and was looking around her in delight when she saw another figure approaching on a converging path.

  The newcomer was plainly another little girl, but there was something very peculiar about her. She looked somewhat like Alice herself, but she resembled rather more the figure that Alice had occasionally seen in the negatives of her snapshots. Alice was reminded of the antielectrons which she had watched in the Bank. She noted to her surprise that, although the girl was coming toward her, she was facing in the opposite direction and walking along backward.

  Alice was so absorbed in the remarkable appearance of this strange girl that she did not consider how quickly they were approaching one another. Before she had fully realized what was happening they had collided. There was a blinding flash which quite took her senses away. When she recovered from this she found herself walking alone down the path along which the other girl had come. Looking back she could see that the reverse-girl was walking away, still backward, along Alice's original path. Now, however, she was accompanied by another negative figure which was companionably walking along backward beside her. This second figure resembled her previous companion, the Quantum Mechanic.

  When she looked around her, Alice was startled to discover that her surroundings had altered every bit as dramatically. Everything seemed to be reversed. In the sky the sun glowered darkly, draining the light from the scene below. Beside the path, dull butterflies flitted backward from dark blossom to dark blossom and a small stream burbled uphill over a bed of round pebbles, while here and there the water soared to the top of a miniature waterfall. Alice had never experienced anything like this before.

  She was so fascinated by this remarkable scene that she did not observe that once again a small girl was bearing rapidly down upon her in reverse. Alice looked around just as they collided, with yet another blinding flash. When she recovered from her shock, she saw the girl was backing away along the path by which Alice had just come. She noticed furthermore that the scenery had now returned to normal. "Curiouser and curiouser," said Alice to herself. "The first collision somehow managed to make the whole countryside reverse itself, while the second one has put it back to normal. I'm sure I do not see how that could possibly happen. How can my colliding with that girl, however violently, affect the stream and the sun? It does not make any sense at all." Alice continued to debate with herself about the meaning of her recent experience. It had been so very remarkable that she paid hardly any attention when she heard a sharp detonation to one side of her, while shortly after an extremely energetic photon rushed across the path.

  Alice had not reached any satisfactory explanation of her recent experience when the path led her out of the park and onto a wide, level plain. This seemed devoid of any feature, apart from a large, utilitarian building which stood facing her a little way ahead.

  When she got close she could see that the building had a name board mounted centrally on the front, a little above the level of Alice's head. At one end this sign bore the words "State Agent." at the other "Virtual Realtor." In the center of the vast blank frontage were a door and a small window, which was full of notices.

  As Alice could see no one outside, she opened the door and went in. Immediately inside the door was a short counter and behind this a huge room, almost empty apart from what seemed to be tiers of shelves rising up into the shadows in the distance. In the center of the room, a single figure was visible sitting at a desk and talking into a telephone. When he saw Alice he rose and hurried over.

  He rested his hands on the counter and smiled widely in a toothy and rather insincere manner. "Come in, come in," he said, ignoring the fact that Alice had already come in. "What may I have the pleasure of showing you? Perhaps you are planning to move into your very first state? I am confident that we will be able to give you every satisfaction."

  To tell you the truth," began Alice, not that she had been at all tempted to lie about it, "I am not actually looking for anything. I was told that you would be able to tell me something about how electrons and other particles move between states."

  "Well, you have certainly come to the right place. We have long been established in the particle transit
ion business. If you would care to come with me to one of our locations I will endeavor to clarify the situation to your complete satisfaction."

  Alice understood this to mean that he would explain things, so she came around the counter and followed him to one of the sets of shelves, or whatever they were. Either they were a long way away and very large, or perhaps she and the State Agent shrank as they came closer, but however it happened Alice found as she drew near that they now looked much more like a tall block of apartment buildings. These bore a sign which read:

  They were very open at the front, and she could see electrons moving around on each level.

  "There you have a good example of quality states built on wellspaced energy levels. Each one is occupied by the permitted number of electrons, up to the highest occupied level. Above that there are many vacant states, but there is currently no room for any more electrons on the lower levels. When an electron is a sitting tenant in a state, there is of course no room for another electron. Usually if he is left to himself an electron has no inclination to move from a state once he has settled into it. However, if we wait for a little we may be lucky and see some forced movement."

  Alice stood and looked at the edifice, and after a short wait she saw a photon rush into the front. There was a commotion, and one of the electrons in the lowest level soared up and out of sight. Alice looked around to see where the photon had come from. Parked nearby there was a small truck with, painted on its side, the slogan:

  "I am in luck," cried the State Agent joyfully. "A photon has given its energy to an electron in the lowest level and excited it right up to one of the vacant levels at the top. It is not so often that we get a removal from the ground state. That leaves a very attractive vacancy. I must see to it at once."

  He rushed off and soon came back carrying a notice board on a post, which he planted in the ground. The notice read:

  Hardly had he got the board in position than one of the electrons in the second level gave a short cry and toppled down to the empty state. Once there he settled in and carried on as if nothing untoward had happened. As he fell Alice saw a photon rush out. Since the electron had not fallen very far, the energy carried by this photon was much less than the energy carried by the photon which had released the original electron.

  The State Agent sighed, picked up a paintbrush from a pot he had brought out when he fetched the sign, and proceeded to cross out the word "Ground" and write "Second" in its place. The paint was hardly dry when Alice heard another sharp cry. An electron in the third level had fallen into the empty place in the second. The State Agent cursed and altered his board again to read "Third." He slammed the brush back into the paint pot and glared at the building.

  There was another sharp cry. An electron from still higher up had fallen into the third level. The State Agent tore his notice from its post, flung it on the ground and stamped on it.

  See end-of-chapter note 1

  "Excuse me," said Alice, rather hesitant at interrupting this display of passion. "I had thought you said that the electrons would stay in their states indefinitely if they were left alone, but those ones appear to have fallen down quite spontaneously."

  "So it may seem," replied the Agent, quite glad to be distracted from his momentary fit of temper. "Actually all of those electron transitions have in fact been stimulated by photons, but you did not notice them because they were virtual photons. Virtual photons play a very important part in all electron interactions. They not only give these apparently spontaneous transitions between states, but they also help to create the states themselves in the first instance. So you see, the very particles which keep an electron in its steady state are also the ones which force the electron to leave it.

  "Before I tell you about virtual particles we ought to look at normal particles, the ones which are not virtual. They are commonly known as real particles. The distinctive thing about them is that there is a very strict relation between their particular masses and the energy and momentum that they can have. That is what the notice there is about."

  The agent pointed to a small sticker, printed on fluorescent green paper, which was attached to the front of the building. It said:

  "They are certainly very fond of notices here," Alice thought to herself. "That one does sound rather suggestive, though I must admit that I do not know what it means."

  "The mass shell," continued the agent, as if answering her thoughts, "is the region where energy and momentum are related in the strict way required for real particles. It is the straight and narrow path followed by conventional, hidebound particles.

  "If you want to be a force in the community and to push things around, then you have to be able to transfer momentum. If you want something to move from where it is, or if you want to keep something from moving away, then you must transfer momentum. In each case you are concerned with moving, and moving means momentum. Whether you want to start a movement or to stop one makes little difference. It is changes in momentum which deflect objects from their paths and make things change, and it is control of momentum which makes particles take a particular path, for that matter.

  "On the mass shell, you cannot have momentum without supplying the appropriate kinetic energy that befits your mass. A really massive particle, one with a lot of energy already invested in its rest mass, does not need so much extra kinetic energy to provide it with a given amount of momentum as would a lighter particle. All real particles must have the appropriate amount of energy if they are to have momentum. This is true even for photons, which do not have any rest mass at all."

  The Agent reached into his pocket and took out a number of legallooking documents. "The conditions are quite precise. Provided that real particles obey them, they are free, free of any energy debt. They can move around as they wish, as far as they wish. They are quite free to come and go. You may have seen the rule: 'What is not forbidden is compulsory,"' he remarked.

  "Yes, I did," replied Alice, anxious to air her knowledge. "I saw that in the Heisenberg Bank, and the manager told me about momentum and...."

  "There is another rule," continued the agent enthusiastically, without actually stopping to listen to Alice's reply. "It says 'What is forbidden had better be done pretty quickly.' This is the rule followed by the virtual particles. These are not usually discussed much in polite, classical society, but they have a very important part to play in the world. Virtual particles behave in ways that classical laws say are simply not allowed."

  "How can that be?" asked Alice, a little naively. "Surely if something is not allowed then no particle will be able to do it."

  The agent heard her then and answered her question. It is the quantum fluctuations which permit it," he said. "If you have been to the Bank, you will remember that particles may have a loan of energy for a short time. The larger the amount of energy the shorter the time of course. You may have heard the expression The difficult we do immediately, the impossible takes a little longer.' Well, in quantum mechanics the impossible does not take a little longer, but it does last a little shorter. Virtual particles can enjoy all the benefits of energy which they do not possess, on a short-term free trial. This includes being able to transfer momentum."

  "It must be a rather short free trial," said Alice thoughtfully.

  "Oh, it is; it is. But it is something for nothing you see, so they all want it. You will have a better appreciation of virtual particles once you have seen them."

  "But I can't see them," complained Alice. "Surely that is the point."

  "You cannot see them at the moment," the Agent replied sternly, "but you will when you put on my virtual reality helmet." He walked quickly away in the direction from which they had come, and Alice hoped that she had not offended him. She was relieved when he returned soon after, carrying a large and highly technical looking helmet. This had a transparent visor which entirely covered the front, and there was a long cable attached to a socket at the back. The cable snaked away along the path by which he had come until
it was lost from sight in the distance. "Here it is," he said triumphantly, "a marvel of modern technology. Just put this on, and you will see the world of virtual particles."

  Alice felt a little nervous as she contemplated the helmet. It was large, and it looked very complicated and even, she felt, a little sinister. However, if this was going to reveal the virtual particles she had heard mentioned so often, she was prepared to try it. She put the helmet on her head. It was very heavy. The Agent reached across to the helmet and made some adjustment at the side of her head, where Alice was unable to see. The view through the visor clouded over with little sparkling dots and....

  When her view through the visor cleared, it had dramatically changed. Alice could still see the electrons in their various levels, but now instead of their appearing to be within a tall building she saw them as enmeshed in a network of vivid lines which joined one electron to another, so that they looked as much as anything like flies caught in some great spider's web of shining strands. As she looked more carefully at these strands she could see that they were actually composed of photons, but photons distinctly different from the ones she had seen before at the Academy.

  All the photons which she had met before had been moving very rapidly, but they had at least been moving in a normal fashion. They had started at one position and a little time later they were at a new position, even if their positions were never precisely defined, while in the intervening period they passed through all the points between the two positions. It had never occurred to Alice that it was possible to travel in any other way, but some of these virtual photons seemed to manage it. As she looked at them she found it very difficult to say in which direction they were moving, or indeed if they were moving at all in any normal fashion. A given strand in the net, which represented the behavior of one photon, would seem to appear at the same moment at the positions of both the electrons which it joined, without apparently moving in the normal way from one to another. This link would then fade while others appeared elsewhere in the great mesh of photons which coupled together the electric charges of all the electrons.