Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  "That carriage, as you can see, is not moving at the moment, so it has no kinetic energy, but it is high up so it has potential energy because of its position. Now as it starts to fall down into the dip it is losing height, so it loses some of that potential energy. This is converted to kinetic energy, so as it falls it goes faster and faster." Alice could vaguely hear the happy excited shrieks of the distant passengers in the carriage as it thundered down the track.

  "If the track were very smooth and the wheels ran without friction," the lecturer continued dispassionately, "then the carriage would come to rest again at exactly the same height." She leaned over and fiddled with something on her desk again. The distant figures on the roller coaster cried out in surprise as the next hump in the track suddenly surged up before them to a much greater height. Their carriage slowed and came to a complete stop before it had reached the top. "How did you do that?" exclaimed Alice in amazement. "Never underestimate the influence of a bank," muttered her companion. "Now see what happens."

  The carriage began to roll backward down the track, accompanied by more shrieks, still excited but not quite as happy as last time. It gathered speed until it shot through the lowest point and then climbed the opposite slope, slowing as it went. It came to rest just at the peak where Alice had first seen it and then began to slip back down once again.

  "This will go on indefinitely now, with the energy of the carriage changing from potential energy to kinetic energy and hack again, but you get the idea." The Manager pressed another button on her desk and the window closed on the scene.

  "That is the sort of obvious way in which you see energy in ClassicWorld. It will change from one form to another in a smooth continuous manner. You saw how the carriage got steadily faster as it rolled steadily down the slope, with no big jumps, and there are no obvious restrictions on the exact amount of energy which any object might have. Here in Quantumland it is often not like this. In many situations a particle is only allowed to have one of a restricted set of values and it can only accept or give up energy in large lumps, which we call quanta. In ClassicWorld all energy payments are made on the installment plan, with very frequent and very very tiny payments, but here they often have to be made as a lump sum.

  "As you saw, kinetic energy is a dramatic, showoff sort of energy―something which a body has just because it is moving. The more massive it is the more kinetic energy it has, and the faster it moves the more kinetic energy it has, but the amount does not depend at all on the direction in which it is moving, only on the speed. In this respect it is different from another important quantity which tells us how a particle moves. This is something we call momentum. Momentum is a sort of measure of a particle's obstinacy. Every particle is determined that it is going to keep on moving in exactly the same way as it was before, without any change at all. If something is moving fast it takes a lot of force to slow it down. It also takes a lot of force to make it move in a different direction, even if its speed does not change. Now a change in direction does not cause a particle to lose any of its precious kinetic energy, as this depends only on how fast it is traveling, but it still does not want to change because its momentum would have to be different. Particles are rather conservative that way.

  "It is all a question of what we call parameters," continued the Manager enthusiastically. "When you want to describe a particle, you have to use the right parameters. If you want to say where it is you must talk about its position and time, for example."

  "I would have thought that you would just need to say what its position was," objected Alice. "That will tell you where it is, surely?"

  "No, certainly not. You must give the time as well as the position. If you want to know where something is now, or where it will be tomorrow, it is no good my only telling you a position if that is where it was last week. You must know the position and the time, as things tend to move around you know. Just as if you want to know what a particle is doing you must describe that in terms of its momentum and energy, in general you need to give both position and time if you want to know where a particle is."

  "Here in Quantumland the parameters tend to be related. If you try to see where something is then that has an effect on its momentum, how fast it is moving. It is another form of the Heisenberg relation which I pointed out to you in the Bank."

  "Oh!" cried Alice, remembering a previous encounter. "Was that the reason that the electron I saw earlier could not stand still to let me see him without becoming all fuzzy?"

  "Yes, undoubtedly. The uncertainty relations affect all particles that way. They always seem a bit indefinite, and you can never pin them down too precisely.

  "I know what I shall do! I shall get the Uncertain Accountant to explain it to you," exclaimed the Manager. "His job is to try and balance the accounts, so he has to worry all the time about quantum fluctuations." She reached out an elegant finger and pressed yet another of the buttons with which her desk was so well supplied.

  There was a short pause, and then one of the doors which were spaced around the room opened and a figure entered. He looked rather like a picture of Ebenezer Scrooge from an illustrated copy of A Christmas Carol, except that he had a rather bemused expression on his face and an uncontrollable nervous twitch. He was carrying an enormous ledger whose covers bulged, not to say wriggled as if the contents were in continuous motion.

  "I believe I have done it," he cried triumphantly, twitching so violently that he almost dropped the book. "I have gotten the accounts to balance! Apart from the residual quantum fluctuations, of course," he added, less enthusiastically.

  "Very good," answered the Manager absently. "Now I should like you to take this little girl, Alice, here and explain to her about quantum uncertainty and fluctuations in the energy of a system and all that sort of thing." With a wave of farewell to Alice, the Manager turned back to her desk and began doing something particularly complicated with all the buttons on it. The Accountant led Alice out quickly before anything further could happen.

  They came to a much smaller, more cluttered office which contained a tall, old-fashioned desk covered in ledgers and with scraps of paper piled all over the floor. Alice looked at one of the open ledgers. The page was covered with columns of figures, much like other accounts ledgers she had seen, except that in this one the figures were continually changing slightly as she looked at them.

  "Right!" said the rather Victorian figure in front of Alice. "You want to know about Uncertainty do you, young lady?"

  "Yes please, if it is not too much trouble," replied Alice politely.

  "Well now," he began, seating himself at his desk. He steepled his fingers together in the traditional magisterial manner to increase the dignity of his appearance, but this was not a good idea as just then he gave such a particularly violent jerk that he got his fingers all tangled up, and he had to stop to unravel them.

  "Well now," he repeated, thrusting his hands deep into his pockets for safety. "The thing you must remember about energy is that it is conserved, which is to say that there is always the same amount of it. It may convert from one form to another but the total amount is always the same. At least it is if you take the long view," he added wistfully and sighed, staring mournfully into the distance.

  "Isn't it true in the short term then?" asked Alice, who felt that she should say something to keep the conversation going.

  "Well no, not entirely. In fact, not at all, if the term is short enough. You saw the Heisenberg relation on the notice outside in the Bank didn't you?"

  "Oh yes. I was told it gave the terms for the energy loans."

  "Well, so it does, in a way, but where do you think the energy for the loans comes from?"

  "Why, from the Bank of course."

  "Dear me, no!" said the Accountant, looking slightly horrified. "Most certainly not! It would be a fine thing if the Bank started lending out energy from its own stock!

  "No," he went on conspiratorially, looking around him carefully, "It is not widely known,
but the energy does not come from the Bank. In fact it does not really come from anywhere. It is a quantum fluctuation. The amount of energy that any given system has is not absolutely definite, but will vary up and down, and the shorter the time over which you measure it the more it is likely to vary.

  "In this respect energy is not really at all like money. Money is well conserved in the short term. If you want to have money for some purpose, you have to get it from somewhere, don't you? You may take it out of a bank account, or borrow it from someone, or you might even steal it!"

  "I wouldn't do that!" cried Alice indignantly, but the Accountant continued, ignoring her.

  "No matter where you get it, it has to come from somewhere. If you get more, then someone else has less. That is what happens in the immediate short term at any rate.

  "In the long term it is different; you may get inflation and find there is more and more money about. Everyone has more, but it does not seem to buy as much as it did. Energy is quite the reverse in a way. In the long term it is conserved, the total amount stays the same, and you get nothing like economic inflation. Every year you will need the same amount of energy on average to transfer from one state in an atom to another. In the short term, though, energy is not well conserved. A particle can pick up the energy it needs for some purpose without it having to come from anywhere else; it just appears as a quantum fluctuation. These fluctuations are a consequence of the uncertainly relation: The amount of energy you have is uncertain, and the shorter the time you have it the more uncertain the amount you have."

  "That sounds terribly confusing," said Alice.

  "You do not have to tell me!" answered her companion emphatically. "It is! How would you like to be an accountant when the figures you are trying to balance are fluctuating all the time?"

  "That sounds terrible," cried Alice sympathetically. "How do you manage?"

  "Well, I usually try to take as long as I possibly can to do the accounts. That helps a bit. The longer the period of time that I spend the smaller the residual fluctuations, you see. Unfortunately people will get impatient and come to me asking if I am planning to take forever to balance the accounts. That would be the only way to do it, you see," he went on earnestly. "The longer I take, the smaller the energy fluctuations, so if I did take forever, why then there would be no fluctuations and my accounts would balance perfectly," he cried triumphantly. "Unfortunately they just won't let me alone. Everyone is much too impatient and anxious to be off making transitions from one state to another all the time."

  "That is another thing that I wanted to ask about," remembered Alice. "What are all these states that I keep hearing of? Would you explain them to me please?"

  "I am not really the best person to do that. It is all part of Quantum Mechanics, so you really ought to go to the Mechanics Institute and ask them there."

  "That is what I was told before," said Alice. "If that is the best place to ask, would you please tell me how I might get there?"

  "I am afraid that I cannot actually tell you how to get there. That is not the way we do things here. But I can arrange that it is very probable that you will get there.

  He turned to the far wall of his office, which was covered with a dusty curtain. When he drew this aside with a sudden jerk, Alice could see a row of doors spaced along the wall. "Where does each of those lead?" she asked. "Does one of them lead to this Institute you were talking about?"

  Each of them could lead you almost anywhere, including, of course, to the Institute. But the point is that all of them will be very likely to lead you to the door of the Institute."

  "I do not understand," complained Alice, with an all-too-familiar feeling of increasing confusion. "What is the difference? If each of them can lead almost anywhere, it is the same as saying that they all could lead almost anywhere."

  "Not at all! It is a different thing entirely. If you were to go through any one door, why then you might end up almost anywhere, but if you go through them all at once then you will most probably end up where you want to be, at the peak of the interference pattern."

  "What nonsense!" cried Alice. "I cannot possibly go through all the doors at once. You can only go through one door at a time you see."

  "Ah, that is different! Of course, if I see you going through a door, then you will go through that door and no other, but if I do not see you, then it is quite possible for you to have gone through any door. In that case the general rule will apply."

  With a wave of his hand he indicated a large, striking notice which was fixed on the wall in front of his desk, where he could not avoid seeing it. It read:

  "That is one of the most basic rules that we have here. If it is possible to do several things, you do not just do one of them, you have to do them all. That way it saves having to make your mind up very often. So off you go, just go out through all the doors and when you have, then set off in every direction at once. You will find it is quite easy and very soon you will have got to the right place."

  "This is ridiculous!" protested Alice. " There is no way that I can go through several doors at once!"

  "How can you say that until you have tried? Have you never done two things at the same time?"

  "Well, of course I have" answered Alice. "I have watched television while I was doing my homework, but that is not the same thing at all. I have never gone in two directions at the same time."

  "I suggest then that you try it," replied the Accountant, rather huffily. "You never know whether you can do something until you try. That is the sort of negative thinking which is always holding back progress. If you want to get anywhere here you have to do everything that you possibly can and do it all at the same time. You do not have to worry about where it will take you, the interference will take care of that!"

  "How do you mean? What is interference?" cried Alice.

  "No time to explain. The Mechanics will tell you all about that. Now off you go and they will explain when you arrive."

  "This is really too bad!" thought Alice to herself. "Everyone I speak to rushes me on somewhere else and promises me that I will get an explanation as soon as I get there. I wish that someone would just explain things properly, once and for all! I am sure that I do not know how I can possible go several ways at the same time. It seems to me to be quite impossible, but he is so certain that I shall be able to manage it here that I had better try, I suppose."

  Alice opened a door and stepped through.

  Alice's Many Paths

  Alice stepped through the left-hand door and found herself in a small cobbled square with three narrow alleys leading out of it. She walked down the left-hand alley. Before she had gone very far, she found herself on the edge of a broad paved area. In the center rose a tall dark building with no windows on the lower levels. It looked very forbidding.

  Alice stepped through the left-hand door and found herself in a small cobbled square with three narrow alleys leading out of it. She walked down the right-hand alley. Before she had gone very far she came to a park, with weed-choked gravel paths winding between dismal drooping trees. Tall iron railings surrounded the park and a dank mist obscured the scenery within.

  Alice stepped through the left-hand door and found herself in a small cobbled square with three narrow alleys leading out of it. She walked down the middle alley. Before she had gone very far she came to another small square, in front of a rather shabby-looking building.

  Alice stepped through the right-hand door and found herself in a narrow alleyway with two others branching off it. She walked down the left-hand alley. Before she had gone very far she found herself on the edge of a broad paved area. In the center rose a tall, dark building with no windows on the lower levels. It looked very forbidding, and she had a distinct feeling that she ought not to be there.

  Alice stepped through the right-hand door and found herself in a narrow alleyway with two others branching off it. She walked down the right-hand alley. Before she had gone very far she came to a park, with weed-choke
d gravel paths winding between dismal drooping trees. Tall iron railings surrounded the park and a dank mist obscured the scenery within. She had a very strong feeling that she ought not to be there.

  Alice stepped through the right-hand door and found herself in a narrow alleyway with two others branching off it. She walked on down the central alley. Before she had gone very far she came to another small square, in front of a rather shabby-looking building. Somehow it seemed to her that this was the right place to be.

  Alice stepped through the center door and found herself facing a wall with three arched gateways which led to alleys beyond. She walked down the left-hand alley. Before she had gone very far she found herself on the edge of a broad paved area. In the center rose a tall, dark building with no windows on the lower levels. It looked very forbidding. She now felt very strongly that she ought not to be there.

  Alice stepped through the center door and found herself facing a wall with three arched gateways which led to alleys beyond. She did not walk down the right-hand alley at all, as that route somehow seemed to be completely wrong.

  Alice stepped through the center door and found herself facing a wall with three arched gateways which led to alleys beyond. She walked through the gateway to the central alley. Before she had gone very far she came to another small square, in front of a rather shabby-looking building. She now felt quite sure that this was the place where she ought to be.