Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  Looking downward Alice saw that the space that had been left by the electron which was excited from the lower level had been filled and that one of her companions in the valence level was missing. Before long the electron falling from above had dropped to the valence level and filled the vacant place. The atom was now back to its original state. Two electrons had exchanged levels, but as they were identical that was no difference at all.

  See end-of-chapter note 3

  "You will have noticed all the different colors of the photons which I emitted," said one of the electrons proudly. This remark tended to suggest that it was the electron which had fallen that had just spoken, but Alice was now too experienced with the effects of electron identity to fall into that trap. "That is the way that atoms emit light you know: when electrons change from one level to another. All the photons were of different energy, and hence different color, because the levels are all different distances apart. They are very closely spaced at the top of the well but are farther and farther apart as you get lower down. This level spacing is different in atoms of different types, so the set of photon energies is completely distinctive for each type of atom-as distinctive as a human fingerprint."

  Hardly had the eight electrons settled down, or got as settled as they could ever be while they were all in continuous frantic motion, when there was a tremor which seemed to run through the whole atom. "What was that!" cried Alice in some alarm.

  "It was an interaction of some sort. We have been separated from our Sodium partner and are drifting through the void as a free negative ion. But do not worry. I do not anticipate that we shall drift about aimlessly for very long. We shall very soon be back in business if the Exchange is agreeable."

  "What Exchange is that?" asked Alice. "Do you mean the Stock Exchange? I understand that controls business in my world."

  "In our case we mean the Electron Exchange. All of our activities are governed by electron interactions of some sort, so it is electron exchange which is significant. Perhaps you would like to visit the Exchange?"

  "Yes, I should think so," replied Alice. "How would I get there from here? Is it a long journey?"

  "Oh no. Not really. In fact it is not really a journey at all. As you are in an interacting atom, you are already there in a sense; you just need a different representation. It is all a question of how you look at things. Just follow me."

  As the electron had told her, they did not seem actually to go anywhere else, but still Alice found herself in company with an electron on the edge of a broad room. The floor was crowded with electrons which clustered around a large table in the middle of the room. It looked to Alice rather like one of the tables which she had seen in old war films, where commanders moved around various counters which represented aircraft, or ships, or armies. On this table also, she saw a great selection of counters which were being moved around into different groupings.

  She looked more closely at some of these counters and saw that they bore the same labels as the atom moorings on the Periodic Pier. In fact, as she looked really closely, she was no longer so sure that they were merely counters. They looked like reduced versions of the atoms which she had seen along the side of that jetty. "Perhaps they are the same," she thought. "Maybe those are the same atoms which I am seeing differently. I suppose that instead of the Periodic Pier, that would make this the Periodic Table."

  Around the side of the room, the walls carried rows of display screens on which she could see columns of numbers that changed as atoms were moved from group to group.

  "Are those the prices for the various atoms?" asked Alice.

  "Yes, after a fashion. Those numbers tell us the energies of the various electrons which are taking part in chemical combinations. They quote the binding energies of the electrons: the amount by which an electron's energy has been reduced below the value it would have if it were free. The larger is the value quoted, the lower is the potential energy that the electron has, and so the more stable and successful is the compound which it binds. The job of the Exchange is to make these binding energies as large as possible."

  "And is this all done by moving electrons from one atom to another?" queried Alice, who remembered the explanation she had been given of ionic bonding in Sodium Chloride.

  "Not always, no. Sometimes that is the most effective method and then the binding is done in that way. The Electron Exchange can get an advantage by moving electrons around because the electron states that are available within an atom are arranged in levels, or shells, with quite large gaps between. The binding energy for the last electron in a lower shell level is much greater than for the first electron that has to go into the next shell higher. This means that there is an easy method of improving the overall energy score for an atom which has only one electron in its highest shell. If this electron can move from its splendid but extravagant isolation into an almost full lower shell in some other atom, then there is almost certain to be an overall gain in binding energy.

  "It is equally true that, when an atom has but one space remaining in its highest occupied shell, this state will have an unusually low energy, and any electron which transfers into it will be likely to improve its energy balance sheet. It is generally true that atoms with just one electron too many or too few are the most active―the most likely to take part in transactions and to form compounds. Atoms with two electrons alone in a high state and those with only two spaces left in a lower one may engage in similar electron transfers, but the gain in binding energy for the second electron is usually a lot less than for the first one, and it is much less effective."

  "So what can an atom do if it has several electrons in its outer shell?" asked Alice, as this seemed to be expected of her.

  "Such an atom has to change to a different kind of binding, one which is known as a covalent bond. An atom such as carbon, for instance, has four electrons in its outer shell. This means it has four electrons too many to be an empty shell and four electrons too few to be a full one. It is too nicely balanced to gain anything by actually transferring electrons to or from another atom, so instead it shares them. It turns out that if the electrons from two atoms are in a superposition of states such that each may be in either atom, then the energy of the two atoms may be lowered and this serves to bind them together.

  "The ionic bond, in which an electron is completely moved from one atom to another, can only work between very different atoms, one of which has an electron too many, the other an electron too few. The covalent bond, on the other hand, can work when both atoms are of the same type. The most remarkable example is given by the covalent bonding of carbon atoms, the basis of the huge Organic Conglomerates." Alice could sense an atmosphere of awed respect emanating from the electron manipulators around the table as the Organics were mentioned.

  "A carbon atom has four electrons in its outer, or valence, level. If each of these electrons is combined with electrons from other atoms, then all of the eight electron states contribute to the superposition and the shell is effectively filled. In this way a carbon atom can attach itself to as many as four other atoms, which may also be carbon. The carbon atom may also exchange two of its electrons with another carbon atom to give a double bond, in which case it will not be connected to so many other atoms, though the connection will be stronger.

  "The ionic bond at its strongest connects but one atom to one other, so it does not produce large molecules. Where there are two electrons to transfer, things can get more complex. Even then the situation does not compare with that of carbon, where one atom may connect to four others and each of those may be connected to several others. Carbon-based compounds can form into enormous organic molecules of great complexity, which may contain hundreds of atoms in all."

  "Do all of the different atom types that I can see there form compounds in the ways you describe?" Alice asked.

  "Yes, apart from the noble gases. With the noble gases, the atoms have filled valence shells already and so do not stand to gain by any electron transfer
s. All of the others do form compounds to a degree, though some are more active than others and some are encountered much more often. The chlorine atom which you visited, for example, is very active. It will form compounds with the simplest atom, hydrogen, which employs only one electron in total, and also with the largest natural element, uranium. That is a very large establishment indeed. It employs almost a hun dred electrons, but only the ones in the outer valence level really affect its chemical behavior. It is so large that there have been rumors that its Nucleus is unstable," he added confidentially.

  "I wanted to ask about that," said Alice firmly. "You have mentioned the nucleus again. Please, would you tell me: What is the nucleus?"

  All of the electrons looked somehow uncomfortable, but reluctantly answered. "The Nucleus is the hidden master of the atom. We electrons conduct all the business of forming chemical compounds and emitting light from the atom and so on, but it is the Nucleus which really controls the sort of atom we are. It makes the final policy decisions, and fixes the number of electrons that we can have and the levels that are available to put them in. The Nucleus contains the nuclear family, the hidden underground of Organized Charge."

  Alarmed by this outburst of candor, the electrons around the room all tried to shrink unobtrusively into one corner, or at least as far as they were able without becoming too localized. Too late, the harm was done! Alice became aware of a new menacing presence nearby.

  Amongst the scurrying electrons there was now a hulking shape, looming over Alice and her companions. She realized that it was a photon, but distinctly more massive than any she had seen before. Like all the photons she had seen it was glowing, but in a peculiarly dim and furtive way. Alice also noticed that, surprisingly for something which was itself the essence of light, this photon was wearing a pair of very dark glasses.

  "It is a heavy virtual photon," gasped the electrons, "Very heavy, a long way off its mass shell. It is one of the enforcers for the Nucleus. Photons such as him transmit the Nucleus's electrical control to its client electrons."

  "I hear that someone is asking questions," said the photon, in a menacing tone. "The nucleons are the sort of particles that do not like to hear that questions are being asked by any other person whatever. I am taking that same person for a little ride to meet certain parties, or rather certain particles. They want to meet her very badly indeed."

  This did not sound like a very promising start for a new encounter, and Alice was considering whether she might safely refuse. She could never quite make out, in thinking it over afterward, how it was that they began: All she could remember was that they were running side by side and the photon was continuously crying "faster," and Alice felt that she could not go faster, though she had no breath left to say so. They rushed across the surface of the table and dived into one of the atoms represented on its surface. It was one of the uranium atoms and it grew enormously as it rushed up to meet them.

  The most curious part of the experience once they were inside the atom was that the things around them never changed in position: However fast they went they never seemed to pass anything. What Alice did note was that her surroundings, the busy electrons and the outlines of the levels which contained them seemed to be getting steadily larger as she ran.

  "Is everything really growing, or am I shrinking?" thought poor puzzled Alice.

  "Faster!" cried the photon. "Faster! Do not try to talk."

  Alice felt that she would never be able to talk again; she was getting so out of breath: and still the photon cried "Faster! Faster!" and dragged her along.

  "Are we nearly there?" Alice managed to pant out at last.

  "Nearly there!" the photon repeated. "Why, we are there all the time and no other place, but we are not sufficiently localized, not hardly. Faster!" They ran on for a time in silence, going faster and faster while the surrounding scene ballooned in size around them, spreading upward and outward until everything she had seen before was too large to be readily appreciated.

  "Now! Now!" cried the photon. "Faster! Faster! Your momentum is now nearly so large as will localize you within the Nucleus." They went so fast that they seemed to skim through the air, until suddenly, just as Alice was getting quite exhausted, she found herself standing in front of a tall, dark tower which rose smoothly in front of her, curving up from the ground and narrowing steadily as it rose. It was dark and featureless at the lower levels, though somewhere at the top Alice could see that it finished in a confusion of turrets and battlements. The overall effect Alice found extremely forbidding.

  "There you see Castle Rutherford, the home of the Nuclear Family," said the heavy virtual photon.

  Notes

  1. Atoms had been found to contain light negative electrons and, later, were found to contain a positively charged nucleus. This suggested that they might be tiny versions of the solar system, with planetary electrons in orbit around a nuclear sun. The notion gave rise to fantasies in which the electrons were indeed miniature planets, with still more miniature people living on them, and so on ad infinitum. Unfortunately for such schemes, the "solar system" picture is clearly wrong.

  • The only reason planets do not fall directly into the sun is because they are orbiting around it. There is definite evidence that many electrons do not have any rotation the nucleus.

  • According to classical physics, the orbiting electrons within atoms should radiate energy and their motion should run down. Something as small as an atom would do this rather quickly, in less than a millionth of a second, and atoms do not collapse in this way. (The solar system is, in fact, running down, but rather slowly, on a time scale of millions of years.)

  2. Because of the Pauli principle you can have only one electron in each state. As electrons are available in spin-up and spin-down versions, this effectively doubles the number of states. Electrons will fall into the atomic states because they have lower energy there and it is a general rule that things tend to fall to lower energy levels (as you may discover by holding a cup over a tiled floor and releasing it). Any atom has a large number of levels which could hold electrons; in fact, the number of states is infinite, though the upper ones are very close together in energy. An atom will continue to attract electrons into its levels only until it contains the correct number to compensate the positive charge of its nucleus, after which the atom no longer has any surplus positive charge with which to attract further electrons. When an atom has reached its full complement of electrons, it will in almost all cases contain more than there are room for in the state of lowest energy. Some electrons must then be accommodated in states of higher energy.

  3. When people looked at the light given off by atoms of a single type, they found that the spectrum was not a uniform spread of colors like a rainbow, but a set of sharp lines, each of a distinct color. Every type of atom showed these line spectra, which were a complete mystery to classical physics.

  The set of energy levels for the electrons is unique to any given type of atom. When electrons transfer from one level to another they emit photons which have an energy that corresponds to the difference in the energy of the two levels. As the energy of photons is proportional to the frequency and color of the light, this gives an optical line spectrum for the atoms which is as distinctive as a fingerprint.

  The explanation of the existence of a line spectrum was the first major success of the developing quantum theory. The theory fitted the observed line frequencies and predicted other line spectra which had not been seen. These where all found in due course and showed that the quantum theory could not readily be dismissed.

  lice stood gazing up at the dark heights of Castle Rutherford as it loomed overhead. "Where did that come from?" she asked her companion. "How did we get here from the atom's potential well?"

  "I have to tell you that during no time are we going anywhere. We are remaining strictly in the vicinity of the atom, but are now somewhat localized in its center or indeed rather more than somewhat. What you see before you is the bot
tom of the same potential well. Do you not recognize that same item?"

  "No, I certainly do not!" replied Alice emphatically. "The potential well was a well; it was a hole which went downward. This is a tower which goes upward. Quite a different thing."

  "It is by no means so very different when you think about it," replied the photon. "The Nucleus is producing an electric field, and this same Nucleus gives a negative potential energy to any negative electrons which are in the locality. When you are mixing in such company, as with electrons and so on, you are naturally seeing the potential as a pit going downward. Nuclear particles like protons are such particles as carry a positive charge at all times, so if guys like these should come calling unexpectedly, they are liable to find their potential energy is rising more than somewhat as they approach the Nucleus. This will usually make characters like this keep a polite distance, and the field acts like a barrier. In fact, it is for this reason it is called the coulomb barrier. The nucleons are apt to hate having uninvited visitors. If you are mixing with characters of this sort, you are seeing what they are seeing, which is a high potential wall around the Nucleus."