The Heisenberg Bank
when Alice stepped through the doorway she found herself in a large pillared hall with marble walls. It looked much like other banks that she had seen only more intensely so, as it were. There was a line of cashiers' windows along the far wall, and the vast floor area was divided up by portable tape barriers so that the customers would be guided into regular lines as they waited to be served. At the moment however the place appeared to be quite empty of any customers at all. Apart from the cashiers behind their counters and a bank guard standing by the door, Alice did not see anyone.
As she had been advised to ask for information at the Bank, she began to walk purposefully toward the distant line of windows. "Now just you wait a minute!" called the guard by the door. "Where do you think you are going, young miss? Can't you see that there is a line?"
"I am sorry," replied Alice, "but actually I can't see a line. There are no people here."
"There certainly are, and a lot of them!" answered the guard emphatically. "We seem to have quite a rush of 'no people' today. Usually though we refer to them as virtual. I have seldom seen quite so many virtual particles waiting to get their energy loans."
Alice had a by-now-familiar feeling that things were not going to become all that much clearer very quickly. She looked over at the windows and saw that, although the room still appeared to be quite empty, the cashiers were all very busy. As she watched, she saw bright figures appear, one at a time, in front of one till or another and then rush quickly from the Bank. At one till she saw a pair of figures materialize together in front of a grill. One she recognized as an electron; the other was very similar, but was a sort of photographic negative of the first, opposite in every way to the electrons she had seen previously.
"That is a positron, an antielectron," murmured a voice in her ear. Alice looked around and saw a severe-looking, smartly dressed young woman.
"Who are you?" she asked.
"I am the Bank Manager," replied her companion. "I am in charge of the distribution of energy loans to all the virtual particles here. Most of them are photons, as you can see, but sometimes we get pairs of particles and antiparticles who come along together to ask for a loan, like the electron and positron pair that you were looking at just now."
"Why do they need an energy loan?" asked Alice. "And why can't I see them before they get it?"
"Well you see," replied the Manager, "in order for a particle to exist properly, so that it can be a free particle and able to move around and be observed normally and so on, it has to have, at the very least, a certain minimum energy which we call its rest mass energy. These poor virtual particles do not have even that energy. Most of them have no energy at all, so they do not really exist. Fortunately for them, they can get a loan of energy here at the Bank and this allows them to exist for a little while." She pointed to a notice on the wall, which read:
"That is called the Heisenberg relation. It governs all our transactions. The value ? is called Planck's constant, the correctly reduced value, of course. The relation gives the rate of exchange for our energy loans. The quantity ΔE is the amount of energy which is borrowed, and Δt is the period for which the loan is made you see."
"You mean," said Alice, trying to follow what the Manager was saying, "that it is like an exchange rate between different types of money, so that the more time there is, the more energy they can have?"
"Oh no! Quite the reverse! It is the energy and time multiplied together which are constant, so the greater the amount of energy, the shorter the amount of time they are allowed to keep it. If you want to see what I mean just look at that exotic particle and antiparticle which have just taken out a loan at window #7."
Alice looked where she was directed and observed a striking sight. In front of the window was a pair of figures; one was the opposite of the other, in much the same way as for the electron and positron that she had seen earlier. This pair, however, were bright, flamboyant figures, taking up so much space with their presence that they quite obscured the cashier behind them. Alice could not but be impressed by the extravagance of the two, but as she opened her mouth to make a comment they grew hazy and then vanished completely.
"That is an illustration of what I was saying," continued the Manager calmly. "That pair took out an enormous energy loan to support the huge rest mass that they needed for their lifestyle. Because the loan was so large, the repayment time was very short indeed, so short that they did not even manage to leave the counter before it had to be repaid. Because such heavy particles cannot get very far before they have to repay their energy loan, they are known in the trade as short-range particles," she added.
"Is the relation between time and energy the same for everyone then?" asked Alice, who felt that she might have discovered something definite at last.
"Yes indeed! The Planck constant is always the same whenever and wherever it applies. It is what is called a universal constant, which simply means it is always the same everywhere.
"We deal with energy at the Bank here," continued the Manager, "because energy acts as the currency here in Quantumland. As you would express your currency in pounds or dollars, the unit of energy that we use most of the time is called the eV. How much energy a particle has determines what it is able to do; how fast it can go, what state it can get into, how much it will be able to affect other systems, these all depend on the energy it has.
"Not all particles are completely destitute like the ones that are lining up. Many of them do have sufficient energy of their own, and in that case they can keep it for as long as they like. Those are the ones which you may see moving around outside. Any particle which needs to have a mass has to have energy just to exist at all."
She pointed at another framed notice on the wall, which read:
"If a particle wants to have mass then it must find the energy to support it somehow. If it has any energy left over then it can use it to do other things. Not that all particles bother with mass," she added. "There are some free-and-easy, bohemian particles which do not have any rest mass at all. They are not tied down like most particles who have to provide for their mass, so they can make use of even small amounts of energy. Photons are a case in point. A photon has no rest mass, so a photon at rest would not weigh anything at all. Mind you, you do not normally find photons at rest; they are forever rushing about at the speed of light, as photons are what light is made of you see. Light is not a smooth continuous stream. It is made up of a lot of quanta, little packets of energy, so that the flow of light is lumpy. These quanta, or particles, of light are called photons. Practically everything comes in quanta of some size. This gives quantum physics its name, you know. Look at all those photons leaving the Bank now. Basically photons are all the same, exactly like one another in the way that electrons are all the same, but you may notice that many of these photons seem quite different. That is because they have different amounts of energy. Some of them have very little energy, like those radio frequency photons going out now."
Alice looked down at a crowd of photons which were rushing past her, flowing around her feet and on out through the door. As they went, she heard snatches of music, dramatic voices, and something about "doing lunch on Thursday." "I didn't know that radio waves were made up from photons," admitted Alice. "Oh yes indeed. They are very long wavelength photons of course, with low frequency and very little energy. They are very gregarious because if they are to have any noticeable effect you need a lot of them at once. Friendly little fellows aren't they?" smiled Alice's companion. "Visible photons now, the ones which make up the light that people use to see by, they have higher frequency and more energy. One of those can have quite a noticeable effect. The really affluent ones though, the big spenders, are the X-ray and gamma photons. Each one of those carries a lot of energy around with it and they can really make their presence felt on their surroundings if they choose to interact."
"That is certainly very interesting," said Alice, not entirely untruthfully, "but I still feel confused about the whole idea of energy. Can you tell me what energy actually is?"
"Well now," replied the Manager with satisfaction, "that is a very sensible question to ask. Unfortunately it is not a very easy one to answer. Come into my office and I will try to give you an explanation."
The Manager led Alice briskly across the tiled floor of the main hall and through an unobtrusive but rather forbidding door in one corner. Within was a large modern office. Motioning Alice to sit on a deep comfortable chair placed in front of the wide desk, the Manager went round and sat in the chair behind it.
"Well," she began, "energy is a little bit like money in your world and it is not too easy to say exactly what that is either."
"I should think that was quite easy," responded Alice. "Money is coins, like my pocket money, or it can be bank notes."
"That is cash, which is certainly one form of money, but money does not have to be in notes and coins. It can be in a savings account, for example, or in stocks and shares, or even invested in a building. In much the same way energy can take many forms, which seem quite different from one another.
"The most obvious form is kinetic energy," said the Manager, as she settled more comfortably into her chair and her voice took on the complacent tone of someone who is about to give a long lecture to a captive audience.
"A particle, or any other object for that matter, will have kinetic energy if it is moving about. Kinetic just means moving, you know. There are other forms of energy as well. There is potential energy, such as the gravitational energy which a stone has if it is up a hill and so is in a position to roll down it. You can also have electrical energy, or chemical energy, which is just potential energy which the electrons have when they are inside atoms. Then, as I have already mentioned, there is the rest mass energy which many particles must have just to exist, so that they can have some mass. One form of energy can convert into another, just as you can pay cash into your deposit account. I can illustrate that for you if you will just look through the round window." She leaned over and pressed a button on her desk, and a round window on the wall in front of Alice opened up. Through it Alice could see a fairground roller coaster. As she watched, a carriage climbed to the top of one hump and paused there momentarily before it rushed down the far side.
"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-choked 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.
Alice looked more closely at the building. On a faded board by the door she could make out the words "Mechanics Institute." This was indeed where she had intended to come!