Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  Although the helmet had gone, Alice's head was still full of the remarkable sights she had seen and she turned them over in her mind as she walked in silence beside the State Agent, along the shore of the infinite Void.

  Notes

  1. Within atoms, the allowed states for electrons have widely spaced energy levels and electrons may occupy only these levels. An electron can only transfer from one of these states if it goes to another (empty) one and in so doing its energy changes by a definite amount, the difference in the energies of the two states. An atom in its normal, or ground, state has its lowest energy levels uniformly filled with electrons, but there are levels of higher energy which are normally empty. When an electron is excited from its initial position it will end up in one of these empty higher levels or leave the atom completely.

  An electron which has been excited to a higher level can decay back to a level of lower energy if there is an empty state available. As the electron transfers to a level of lower energy, it must rid itself of the surplus energy, which it does by emitting a photon. This is how atoms come to give off light. Since the electrons all occupy definite states within the atom, any photon that is emitted can only have an energy equal to the difference of that possessed by the initial and final states of the electron. This gives a large number of possibilities, but nevertheless imposes a restriction on the energy which a photon may have. The photon energy is proportional to the frequency of the light and thus to its color, so the spectrum of the light produced by an atom consists of a set of colored "lines" of specific frequencies. The spectrum for a given type of atom is completely distinctive.

  Classical physics can give no explanation for these spectra.

  2. Virtual particles have a distinct fuzziness, both in time and in energy. This fuzziness shows itself as energy fluctuations, in which the particles behave as if they had more (or less) energy than they should. It can equally appear as an uncertainty in time. In a quantum system particles seem to be able to be in two places at the same time (or at least they have amplitudes which are).

  The particles can even turn time around. The physicist Richard Feynman explains antiparticles as being "particles traveling backward in time."* This explains the way in which the properties of antiparticles are opposite to those of the particles: A negative electric charge carried backward in time is equivalent to a positive charge moving toward the future. In both cases the positive charge in the future is being increased, and a negatively charged electron traveling toward the past is seen as a positively charged positron, which is its antiparticle.

  All particles have their antiparticles, as is to be expected if they are in effect the same particle behaving in a different way.

  lice walked with the State Agent along the edge of the Void, looking out over the shimmering tenuous surface which was continuously aboil with the activity of the virtual particles as they were born and died unnoticed.

  A little way out from the shore Alice saw a disturbance in the surface, some sort of circular depression in the general uniform level. Further out she could see other pits, and many of them were clumped together into groups. Some of the groups were very small and contained just a couple of the circular objects. Other collections were more extensive. She could see one group which contained a ring made up of six of the objects arranged in a circle, while others were attached around the outside. In the distance she could see some enormous assemblies spread across the surface. The largest contained many hundreds of the circular things, whatever they might be.

  As Alice watched, she saw photons soar intermittently from one or another of the shapes which were spread out before her. The brilliantly colored photons looked rather like flares fired from ships at sea.

  The Agent followed the direction of her gaze. "I see that you are watching the atoms as they swim in the Vacuum. Atoms provide us with much of our work in the electron state business, one way or another. You can see from here the various molecular partnerships that they have set up between them. These range from small two-atom businesses to huge organic conglomerates. Each different type of atom has its own distinctive spectrum of colors for the photons which it emits, so the photons act as signals which help you to identify the different types of atoms."

  See end-of-chapter note 1

  "I was wondering about all those things way out there," admitted Alice candidly. "I cannot see them very clearly from here. Is it possible to get any closer?"

  "If you want to look closely at atoms we ought to go along to Mendeleev Moorings. There you will see every type of atom on display, with all the different elements laid out in a regular order."

  The Agent led Alice along the shore until they came in sight of an extremely long, narrow jetty, which stretched far out over the Void. At the shore end there was an arched gate on top of which was a sign which read:

  "There you are," announced the Agent. "That is where the atoms lie at dock before they set out to form their different chemical compounds. We usually call it the 'Mendeleev Marina' or the 'Atomic Pier,' though sometimes people talk of the 'Quay of the Universe.' You will find each different kind of atom represented here."

  Together they walked beneath the sign and stepped onto the boards of the jetty. They strolled slowly out along the anchorage, while Alice looked at the long line of atoms moored in sequence to one side. Each of them appeared to her as a trumpet-shaped pit in the flat surface of the surrounding Void. The shape reminded her of the little whirlpool which she often saw forming over the drain whenever she emptied a bath, though these seemed to be quite still with no visible rotation. The surrounding surface of slick nothingness sloped down into each pit from the still flat level which stretched all around it. It sloped with almost imperceptible gradient at first, but ever more steeply as it funneled down toward the center. There were signs of activity taking place somewhere in the depths of the pit.

  "Why is there such a deep hole?" Alice asked curiously. "As we are looking at Nothing, I would expect it to be all flat and featureless."

  "That is a potential well," was the answer.

  "What sort of well is that?" Alice continued curiously. "I know about garden wells which supply water and about oil wells, and I vaguely remember seeing something about a treacle well in a book I was reading recently, but what do you get in a potential well?"

  "Why, the source of the potential, of course. You have to have a source to provide the water in a garden well. Here there is an electric charge as the source of electrical potential in the potential well. You should know by now what is in the well. It contains virtual photons. They provide the electrical attraction which would make the potential energy of a negative charge drop farther and farther below the surrounding vacuum level as it moves toward the potential source at the center of the atom. The potential source actually creates the well, you see."

  The first pit was fairly shallow, but Alice could see that the others became successively deeper the farther they were positioned along the pier. The jetty stretched away ahead of her into the distance, with atom after atom moored along the side. Beside each one was a small notice to mark the mooring. The first of these read: 1H; the second, 2He; the third, 3Li. Each position had a different label. "Will these atoms all set out from here eventually to combine into groups like the ones already out on the surface of the Void?" asked Alice.

  "Most of them will certainly, but there are a few which will not, like the one just here for example."

  They paused beside an atom which carried the notice: 10Ne. "That is an atom of a Noble Gas element. They are an aristocratic lot and that means that they refuse to engage in commerce of any sort. They keep to themselves. They are perfectly satisfied with the way they are and will not mix with anyone else. They always travel around in splendid isolation. You never see them take part in any sort of compound."

  They walked a little farther and the Agent explained that, even apart from the aloof Noble atoms, there was considerable variation in the enthusiasm with which different elements jo
ined into compounds. "For example, this is a particularly active concern," he remarked, as they came to a notice which read: 17Cl.

  Alice decided that it was time to examine one of these atoms more closely, so she tentatively extended one foot off the edge of the jetty. To her delight she did not sink. Her foot stood on a tiny dimple in the surface, rather like some pond-skating insects she had once watched. When she tried to walk toward the atom, however, she discovered that there was no friction in the void. The surface was extremely slippery, and she was quite unable to keep her footing. With a small cry she skidded down the increasingly steep slope and toppled into the deep pit.

  As she fell, Alice found that she had plenty of time to look around her. The sides of the well became ever steeper as they closed in upon her, and she soon noticed that she was falling through the ghostly outline of a series of rooms which had low, closely spaced ceilings. The first few rooms were very low indeed, scarcely tall enough for a doll's house, but as she fell the rooms became steadily taller. Initially they were all completely empty and deserted, but then she came to a room which contained a large, round table surrounded by chairs. On the floor below this she could see desks and filing cabinets, as if she was passing through some sort of office.

  As the time passed she became increasingly amazed to find that she was still falling, without any sign of reaching the bottom. Down, down, down; would the fall never come to an end?

  Alice gradually began to realize that her fall was not going to come to an end. She had not reached the bottom of the hole, but she was not getting any lower. She was floating quite unsupported in the center of the funnel, on a level with one of the shadowy rooms. She looked around her and noted that she was not alone. Near her were two electrons who were involved in a hectic flurry of activity. Around them she could discern the faint outline of an extremely tiny and cramped office. "Excuse me," she called. "Do you think that you could stop for a moment and tell ine where I am?"

  "No room, no room," they called.

  "I beg your pardon, what do you mean?" cried Alice, to whom this reply did not seem particularly relevant.

  "There is not enough room here for us to slow down at all, let alone stop," they answered her. "As you know, when the position of a particle is restricted, the Heisenberg relation forces its momentum to be large, and it is so cramped here that we have no choice but to keep moving. If we had as much room as they have on some of the higher levels we could afford to move in a more leisurely way, but not here. This is the lowest level, you see, so we must expect to be kept busy."

  "Really?" inquired Alice. "What is it that you do that is so important?"

  "We do not do anything in particular. No one is particularly interested in what the electrons in the ground state are doing, just as long as we keep moving."

  "In that case, do you think you could tell me where I am, without stopping?" Alice asked. "For I do not know where I have come to. What is preventing any of us from falling farther down into the well."

  "You are in the lowest level of a chlorine atom, as we have already told you. Here, we are so close to the potential source that there is very little room, so we have to move very quickly as our momentum is forced to be high. This means that our kinetic energy is also high. None of us is in a particularly virtual state, you see. Electrons have secure positions in atoms, with very good tenure. Most atoms have been around for a long time and the quantum energy fluctuations are small, so for us electrons the energy and momentum are properly related.

  "You probably know that when an electron, or anything else, falls farther into a potential, it loses potential energy, and this will be converted to kinetic energy," they went on.

  "Yes, that was explained to me when I visited the Heisenberg Bank," agreed Alice.

  "Here in this potential well, though, when we get closer to the center, there is less and less room, so we need to have more kinetic energy. If we were to fall even closer we would need to have more kinetic energy than we can possibly get from converting potential energy, so we are unable to fall any further. In fact, paradoxically, we just do not have enough energy to be able to fall any lower and we cannot borrow the energy as a quantum fluctuation because we would need it for a long time.

  "There are only two states on this level so there is only room for two electrons, one being in a spin-up and one in a spin-down state. There are more states available as you move up to higher energy levels, so you will find more electrons on the levels above. The next two levels can hold up to eight electrons on each level. In any atom, the lowest levels, the ones with the lowest potential energy, are the first to be filled. The Pauli principle allows only one electron in each state, so when all the states on one level already have an electron, any extra electrons have no choice but to move into levels higher up. The levels are filled from the bottom until all the electrons are accommodated. The highest level which contains any electrons is called the valence level. That is where the valence electrons live, though there are plenty of unoccupied states higher up in the attic. The valence electrons make all the decisions and control the compounds our atom can join. If you want to discover how an atom operates, your best plan would be to go up and talk to them."

  See end-of-chapter note 2

  "How shall I get up to that level from here?" asked Alice.

  "Well, if you were an electron you would have to wait here until you were excited to the higher level by a photon which could give you the extra energy you would need. In your case, however, I expect that you can be carried up by the Ladder Operator."

  "Don't you mean the elevator operator?" queried Alice. "I have been in an elevator in a large department store and it had an operator who took people from floor to floor, but I have never heard of a ladder that needed one."

  When she looked around, however, she could see a sort of ladder with very widely separated rungs. Beside it stood a rather indistinct figure. "May I ask who you are?" said Alice curiously.

  "I am the Ladder Operator. I am not a physical creature, but merely a mathematical construct. It is my job to transform a system from one state to a higher or lower one." He performed some complicated operation which Alice completely failed to understand but resulted in her being carried rung by rung up to the higher level.

  In due course Alice arrived at the level on which she had seen the large, round table. This level contained more electrons than the first. She managed to count eight in all, though with some difficulty. As with all the electrons she had seen so far, they were moving energetically around. Several of them were circling around the table, some in one direction and some in the other. The others were not obviously rotating but were nonetheless in motion. None of them was sitting quietly on any of the chairs around the table, but they were leaping up and down, and some were stepping on and off the table. The electrons were never still, though on this level they were not moving quite so frantically as they had been on the lowest one.

  "Hello Alice," they cried as she appeared. "Come, and let us show you how a reliable, medium-sized atom operates. The way in which Chlorine Corporation conducts its business is decided by us seven electrons in the valence level."

  "But there are eight of you!" protested Alice.

  "That is because we have entered into a partnership with another atom, Sodium Syndicate, to form a sodium chloride molecule. Working together in this way we like to think that we are the Salt of the Earth. An atom runs much more harmoniously when all of its levels that hold any electrons are filled completely. On our own we have only seven electrons in the valence level and Sodium has but the one, although there is room for eight. It helps both of us if the Sodium valence electron comes over here to sit on our valence level and give us a full board. This means of course that we now have an extra electron, and so we have a negative charge. The Sodium atom has an electron less than normal, which gives them a positive charge, and the electrical force between these opposite charges holds the two atoms together. That is known as ionic bonding between the atoms an
d is one of the common forms of corporate structure."

  "That sounds very cooperative on both sides," agreed Alice tactfully. "Which of you is the electron that has come over from the Sodium atom then?" she asked.

  "I am," they all cried, talking together. They paused for a moment and looked at one another. "No, he is the one," they now said, still speaking in perfect unison. Alice realized that there was absolutely no point in asking any question which tried to distinguish between the identical electrons.

  "Could you explain to me, please, why you say that the sodium atom has a positive electric charge when it has lost one of its electrons," she asked instead. "Surely it still has quite a few electrons left, and they will presumably have negative charges also."

  "That is quite true, all we electrons do have the same amount of negative charge, as we are all identical. Normally in an atom this charge is balanced and neutralized by an equal amount of positive charge carried by the Nucleus. Atoms are usually neutral, with no net electrical charge either way. So you see, when an atom has one electron more than usual, it will be negatively charged. It is known as a negative ion. If it has one electron fewer than normal, the positive charge on the Nucleus will dominate, and the atom becomes a positive ion."

  "I see," said Alice thoughtfully, "but what is this Nucleus that you are talking about?"

  "Every atom has one," was the evasive answer, "but you do not want to know too much about it. You really don't!" he added nervously.

  At this point the conversation was interrupted by a faint cry which started somewhere below them, passed through the valence level close at hand, and finally stopped somewhere above. Alice looked up and saw that it was due to an electron which had apparently been excited by a photon from its position in a lower level and was now looking uncomfortably remote in one of the empty higher levels. The electron wandered rather slowly around the high wide level until eventually it gave a brief cry and toppled to the level below. As it did a photon rushed out of the atom, carrying away the energy released by the fall. Alice watched with interest as the electron fell in succession from one level to the next, in each case emitting a photon. As the lower energy levels were more widely separated than those above, each fall was farther than the one before, so the photons created had higher energy arising from each successive fall. As their energy increased, the color of the light moved farther toward the blue end of the spectrum.