Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  "How shall I get in then?" asked Alice. "I do not think I shall be able to get over the wall. I am sure it will be very effective at making me keep a polite distance," she argued hopefully. She was still not at all sure that she wanted to meet the Nuclear Family.

  "The coulomb barrier is acting to keep out only those same particles which have a positive electric charge. There are others that do not have any electric charge at all, and these particles can pass through easily. You are not carrying a charge at present, so you are liable to get in through the neutral particle entrance." He pointed toward a tall doorway in the bottom of the castle wall, which Alice had not noticed before. It was obligingly labeled: "Neutral Particles Only."

  Alice and her escort went over to the door and knocked loudly. "What are the nuclear particles like?" asked Alice cautiously. Are they much the same as the electrons I have already met?"

  "They are commonly considered by one and all to be bigger than any electrons and are known to be about two thousand times more massive." This answer did nothing at all to reduce Alice's feeling of nervousness as she heard slow, ponderous footsteps approaching the door from within. These grew louder until she fancied she could feel the ground tremble slightly with each footfall. Finally they stopped and the tall door began to swing slowly inward. Alice looked up nervously to catch her first sight of this monster which had summoned her. Finally the door was completely open and still she could see nothing. Were the nucleons invisible?

  "Here I am," snapped an irritated voice, from somewhere below the level of Alice's knees. Startled, she looked down and there, standing in front of her, was a small figure. It looked not unlike the electrons she had seen before, except that somehow there was an aura of power about it and, like her companion, it was wearing dark glasses. However, when Alice remembered how far she had shrunk on her way to Castle Rutherford, she realized that this figure must be far, far smaller than the electrons had appeared to her before.

  "I thought you told me that the nucleons were larger than the electrons!" she exclaimed, turning indignantly to the photon. She felt angry that she had been so deceived.

  "Why, most informed citizens agree that they are indeed larger and I am sure you would not wish to question my word over so small a matter. Of course the nucleons are much heavier than the electrons and so they are inclined to be that much more localized. As they are two thousand times heavier, they naturally have two thousand times more rest mass energy, and it is widely accepted that they are in the region of two thousand times more localized, even when they are having the same energy as an electron type guy. This means that they are apt to occupy less space and so they may seem to be smaller than the electrons, but informed opinion is that they are in actual fact larger.

  "Compared to the citizens of the Nucleus, the atomic electrons are such parties as have very little energy or momentum at all and are by no means well localized. They form considerable electron clouds which hang around in the vicinity of the nucleus and are very large indeed. They spread out over a volume which is hundreds of thousands of times farther across than the same nucleus." As Alice looked around she could see great gray clouds surrounding them, clouds which stretched away as far as the eye could see. It was strange to think that these were the electrons that she had seen so often before, but now seen from the viewpoint of a much more compact scale.

  The neutron which had greeted them (for such it was) was becoming increasingly impatient with this exchange. "Don't just stand there, whoever you may be," it snapped querulously. "Come closer so that I can identify you."

  "Why, he cannot see us," realized Alice. I do believe that he is blind!"

  "All neutrons are in such condition, as most people admit," replied her escort. "These parties are not such as have any interaction with photons, or hardly any, having no electric charge of their own. Neutrons are citizens who do not have much long-range interaction whatsoever, being only given to interactions of very short range indeed. Such a party is not much at recognizing others until they are close enough to touch."

  They moved up close to the neutron until he bumped into them. "Ah, there you are!" he exclaimed sharply. "Come in and let me shut the door. It is much cosier inside." He ignored the photon, of whom he was largely unaware. Alice was interested to note that the photon simply faded into the castle's fortifications, which were after all composed of the virtual photons emitted by the charge of the Nucleus.

  Alice followed the neutron into the Castle while he felt his way down a rough stone corridor. This passage was very narrow, but seemed obligingly to widen at their approach so that there was always just enough room to pass through. Alice found this behavior rather eerie, but she was never sufficiently sure that it was actually happening to make any comment. Now that she had met him, the nucleon whom she was following did not seem as threatening as she had feared. Impatient yes, but not in any way sinister. He reminded Alice of a distant uncle of hers.

  Together they entered a tall vaulted central chamber of bare stone. The walls rose sheer on every side and vanished into the shadows of the ceiling. Around the walls overhead were arched openings leading to various higher levels, vaguely reminiscent of the electron energy levels that Alice had seen in the atom outside. The floor area was of moderate size and was crowded with as many particles as it could contain, but as Alice and her companion entered she clearly observed that the massive stone walls drew back slightly to create just the right amount of extra space needed to accommodate the new occupants.

  Alice was quite sure of what she had seen this time and commented on the movement. "That is the effect of the self-consistent field within the castle," she was told.

  "Like electrons and all other particles, we nucleons have to occupy quantum states, and the available states here are controlled by the local potential well. In the case of the electrons in the atom, that potential well is provided by us. The electron states are fixed by the electrical potential and we control that potential. The atom is our territory and the potential energy of the electrons within it is controlled by their distance from the positive electric charge of the protons in the central Nucleus. By means of the electrical potential produced by this charge, we in the Nucleus control the electron states, and the electrons must fit into them as best they can. In our own case the situation is different. We ourselves provide the potential for our own nuclear states."

  "If you provide the potential in both cases, surely that makes the two cases the same," protested Alice.

  "No, it makes the two cases quite different. You see, in the atom the potential is provided mostly by the Nucleus so that the Nucleus controls the states although the nucleons do not themselves make use of them. The potential controls the states which give the probability distributions for the electrons, but the electrons which use them have little effect on the potential. The atomic potential is much the same wherever the electrons may happen to be."

  "For the Nucleus, on the other hand, the potential that we are now in is produced by the collective effort of all the nucleons within it. We have a very democratic system ourselves, though we rule the electrons autocratically. Our collective potential fixes the states which are available for us nucleons to occupy and so controls our probability distribution. This distribution subsequently controls the potential, as I said at the beginning. It is a vicious circle, as you might expect for the Nuclear Family, and you can see that the states we inhabit will naturally change as the distribution of nucleons changes."

  "Is the nuclear potential produced by the same electric charge as the potential which holds the electrons?" asked Alice, who thought that she should get this point clear in her mind.

  "Oh no, quite the reverse in fact. The electric charge in the nucleus is all carried by protons. You are bound to see some protons over there." He waved in the direction of the nearby particles. Alice glanced over and could see more neutrons, which looked just like her companion. Scattered among them were some other particles which looked distinctly more assertive. Wher
e the neutron had been slightly irritable, these appeared to be in a state of barely suppressed fury. "The protons all carry positive charges, and particles which have the same sort of charge repel one another, you know. Protons are forever flying into a temper with each other and threatening to rush off. It is very difficult to keep them together, I can tell you."

  "Don't the electrons have the same problem then? I should have thought that they would. If all electrons have a negative electric charge, then any two of them will have the same sort of charge and should repel one another."

  "That is quite true; they do repel one another. However, you must realize that the electrons are relatively spread out and diffuse, and their charges are widely separated, so the repulsion they produce is fairly weak. The attractive force from the concentrated positive charge in the Nucleus is able to keep them in order. The protons in the Nucleus are crowded close together, so their repulsive force is very strong. The electrical forces threaten to tear the Nucleus apart."

  See end-of-chapter note 1

  "In that case what does keep you all together?" asked Alice reasonably.

  "That is achieved by a completely different force, a strong force―in fact, the strong nuclear interaction is what it is called.

  "The strong nuclear interaction is very powerful. It is able to overcome the disruptive electrical repulsion within the nucleus, even though it has no obvious effects outside the nucleus. It is a short-range force you see. Inside the Nucleus the nuclear forces are dominant, but outside there is little sign of them, and all that anyone sees is the electrical field due to the positive charges carried by the protons. We nucleons hold firmly onto our immediate neighbors when they are within reach, but we are not really aware of those farther away in the crowd and have very little effect on them."

  Ever since she had entered the central hall of the castle, Alice had felt rather uncomfortable. Now she experienced a peculiarly eerie feeling and sensed that something was now in the chamber which had not been there just before. She looked around her and could see nothing. Then she looked upward toward the ceiling. She dimly perceived the great curved flank of some vast rounded shape which passed through the dim shadows of the soaring space above her head. It was obviously but a small part of some much larger object which looked vague and tenuous, like a ghost, and which was drifting through the surrounding walls as if they did not exist.

  Alice exclaimed aloud, and then she had to describe what she was seeing to the neutron, who had not been able to see it of course. "Ah, that will be an electron," he said. "They fill the entire volume of the atom you know, which means that they pass through the Nucleus as well as elsewhere. Electrons are completely unaffected by the strong interaction, so they are not aware of us when they do pass through. The nucleus is a tiny part of the volume occupied by electrons, so we do not see very much of them here. Well, actually I do not see them at all, but you know what I mean."

  "Is this strong interaction not caused by photons then?" Alice inquired. She had been told that photon exchange held atoms together, but she had understood that that was due to the interaction between electric charges, and she gathered that this was something quite different.

  "You are right, it is nothing to do with photons. It is caused by particle exchange―all interactions are―but it involves a different sort of particle. The strong interaction is in fact caused by the exchange of many different particles, the most evident of these being called pions. These are of necessity bosons, as they are created and destroyed during the exchange process. Pions have much greater mass than photons. Indeed photons do not have any mass at all, which makes them quite inexpensive to create, in energy terms. Pions have a mass about three hundred times that of an electron. They may still be created using an energy fluctuation, as allowed by the Heisenberg relation, but the fluctuation must be very large to provide the rest-mass energy of the pion, so it cannot last for long. In the available time the pions cannot get far from their source, so they can only be exchanged with particles which are close at hand, almost touching in fact. The strong interaction is consequently of very short range."

  At this point a disturbance broke out. Two of the protons had had a sudden and violent argument and were threatening to storm off in opposite directions. Neutrons rushed in to separate the contestants and keep them well apart, so diluting the strength of their mutual repulsion. While the neutrons crowded between the protons to increase their separation, they also grasped them firmly to hold them within the Nucleus.

  "You see how we neutrons are necessary to hold the Nucleus together, particularly in the larger nuclei," remarked a neutron. "In a Nucleus every proton repels every other proton, not just those immediately next to them, as is the case for the strong interaction. The repulsion rises rapidly with the number of protons in the Nucleus, and this means that heavy nuclei, which have a large number of protons, need proportionately more neutrons to keep them well away from one another so that their repulsion does not overwhelm the attractive force exerted by their immediate neighbors.

  "The Family of nucleons comes from two distinct clans, the protons and the neutrons. The lineage displayed on the wall over there shows how they combine." He indicated a large diagram hanging on the wall, among various other symbols and heraldic decorations. This showed a large and fanciful drawing of a proton and a neutron at the top two corners of the chart. Down the center were listed all the different nuclei in which the Family were involved. Alice saw that they were identified by the same labels that she had seen marking the different atoms at the Mendeleev Marina. On close examination she noted that the labels were slightly different: There was another number given for each one. Now the nuclei were given as, 1H1, 2He4, 3Li7 and so on.

  From the original proton and neutron at the top of the picture, lines were drawn to the various nuclei listed. There was one line from the proton to the 1H1 nucleus and no line at all from the neutron. To the 2He4 nucleus there were two lines from the proton and two from the neutron. Thereafter many nuclei had approximately equal numbers of lines from the proton and from the neutron. As Alice looked toward the bottom of the chart she saw that each nucleus depicted there had many more neutron lines than proton ones.

  "That chart shows how the different nuclei are populated from the two distinct clans of nucleons. The first number tells you the number of protons involved. This is the same as the number of electrons which can be controlled and hence decides the chemical behavior of the atom. The second number gives the total number of nucleons which populate a given Nucleus.

  "Lighter nuclei have the same numbers of protons as of neutrons. A carbon nucleus, for instance, contains six protons and six neutrons. The repulsion given by six protons, each repelled by every single one of the five other protons, is still not enough to overcome the attraction caused by the strong interaction. Here in our uranium Nucleus, on the other hand, we have 92 protons. The repulsive force between all the different pairs of protons is now very large, so a relatively large number of neutrons is needed to keep the protons apart and dilute their electrical repulsion. In our Nucleus we have all of 143 neutrons. The number of neutrons need not be quite the same in every uranium nucleus. For a given element the number of protons is always the same, since this fixes the number of electrons and hence the chemical behavior, but the number of neutrons does not have much effect on the chemistry of the atom and can vary slightly from one Nucleus to another. Nuclei of an element which have different numbers of neutrons are known as isotopes. We have 143 neutrons in this Nucleus, as I said, but many uranium nuclei have 146, which makes them a little more stable."

  "I have heard of stability before," said Alice. "I thought that atoms were completely unvarying, and, although they might take part in different compounds, the atoms themselves last forever."

  "Not entirely. The walls of the nuclear potential barrier serve to keep us inside, just as the coulomb barrier keeps other protons out. Occasionally, however, there is penetration, and the Nucleus is changed in some way. I
t works both ways; particles outside the Nucleus might break in, or some from among our complement may try to escape.

  "The reason that protons and neutrons stay in the Nucleus is the same as the reason that electrons stay in the atom: because they require less energy where they are than they would if they were outside. The decrease in energy from the value they would have outside the Nucleus is called the nuclear binding energy, or BE. There are energy levels for nucleons within the Nucleus in much the same way as for electrons in the atom, and, as neutrons are not identical to protons, these levels may be filled with neutrons and with protons independently. Because the levelfilling process is the same for neutrons and protons, stable nuclei tend to have equal numbers of the two types. For the heavier nuclei, which have larger numbers of protons, the proportion of neutrons is greater, as I have described already. For all nuclei there is a ratio of protons to neutrons which gives the most stable atom. An excess of either type will give a tendency to instability and some form of decay. I am forced to admit that, in uranium, the repulsion between the protons is so great that the Nucleus is barely stable at the best of times. Any disruption of the balance between protons and neutrons could well be disastrous."

  Suddenly an alarm trumpet sounded and a strident voice echoed through the vaulted chamber. "Alert! Alert! Condition Alpha. We have an escape attempt in process."

  Alice looked around to see if she could see any cause for this alarm. Everything looked much as before. There was considerable movement among the assembled nucleons, but then they, like other particles which she had encountered, were always in continual agitation, so that was nothing new. As she watched carefully she noticed that a small group of particles, two protons and two neutrons, were moving together through the crowd, holding tightly to one another. They would rush up to the wall, collide with it and bounce back, and rush across the chamber to collide with the opposite wall. Alice was strongly reminded of the person she had seen trying to penetrate his locked door when she first arrived in Quantumland.