When it was quiet again Alice opened her eyes and lowered her hands. She found there was no sign of the crowd of electrons which had been clustering around her and that she was walking out of the station entrance all alone. Looking around she found herself in a street which at first sight seemed quite normal. She turned left and began to walk along the sidewalk.
Before she had gone very far she came across a figure standing dejectedly in front of a doorway and searching though his pockets. The figure was short and very pale. His face was difficult to make out distinctly, as was the case for everyone Alice had met recently, but he did look, Alice thought, rather like a rabbit. "Oh dear, oh dear, I am late and I cannot find my keys anywhere. I must get inside straightaway!" So saying he stepped back a few paces and then ran quickly toward the door.
He ran so very fast that Alice was not able to see him in any one position, but saw instead a string of afterimages which showed him at all the different positions he passed through along his path. These extended from his starting point to the door, but there, instead of stopping as Alice would have expected, they continued on into the door, getting smaller and smaller until they were too small to be seen. Alice had scarcely had time to register this strange series of images when he rebounded backward just as rapidly, once again leaving a series of images. This time they ended abruptly with the unfortunate person sprawled on his back in the gutter. Apparently in no way discouraged, he picked himself up and raced toward the door again. Again there was the series of afterimages, shrinking away into the door, and again he bounced off and ended up on his back.
As Alice hurried toward him he repeated this action several more times, throwing himself at the door and then falling back again. "Stop, stop," cried Alice. "You must not do that; you will surely hurt yourself."
The person stopped his running and looked at Alice. "Why, hello my dear. I must do this I'm afraid. I am locked out and I must get inside quickly, so I have no choice but to try and tunnel through the barrier."
Alice looked at the door, which was very large and solid. "I do not think you have much chance of getting through that by running at it," she said. "Are you trying to break it down?"
"Oh no, certainly not! I do not want to destroy my beautiful door. I just want to tunnel through it. I am afraid that what you say is true, though. The probability of my managing to get through is indeed not very high at all, but I have to try." As he said this he charged at the door again. Alice gave him up as a bad job and walked off, just as he came staggering back once more.
After she had walked a few paces, Alice could not resist looking back to see if by any chance he had abandoned his efforts, and she saw again the series of images rushing toward the door and shrinking down when they got to it. She waited for the rebound. Previously this had followed immediately after, but this time it did not happen. The door stood there looking solid and rather deserted, but there was no sign of her acquaintance. After a few seconds had passed with nothing happening, Alice heard a rattling of bolts and chains from behind the door and then it swung open. Her vanished companion looked out and waved to her. "I was really in luck!" he called. The probability of penetrating a barrier this thick is very small indeed, and I was amazingly fortunate to get through so quickly." He closed the door with a solid thump and that seemed to end the encounter, so Alice walked on up the street.
A little farther along she came to an empty plot by the side of the road, where a group of builders was clustered around a pile of bricks. Alice assumed they were builders, as they were unloading more bricks from a small cart. "Well at least these people seem to be behaving in a sensible manner," she thought to herself. Just then another group came running around a corner carrying what looked like a very large rolled-up carpet and proceeded to spread it out on the site. When it was unrolled Alice could see that it was some sort of building plan. It did seem to be rather a large plan since it covered most of the available space. "Why, I do believe it must be exactly the same size as the building they are going to put up," said Alice, "but how will they manage to build anything if the plan is already taking up all the room?"
The builders had finished easing the plan into position and had retreated to the pile of bricks. They all picked up bricks and began throwing them at the plan, apparently quite at random. All was confusionsome fell in one place, some in another-and Alice could see no purpose in it at all. "What are you doing?" she asked a person who was standing to one side. He appeared to be doing nothing, and she assumed him to be the foreman. "You are just making untidy piles of bricks. Aren't you supposed to be putting up a building?"
"Ah, sure, and we are, me darling," answered the foremen. "It's true so it is that the random fluctuations are still large enough to hide the pattern, but since we have laid down the probability distribution for the result we are after needing, we'll be getting there, never fear."
Alice felt that this display of optimism was not very convincing, but she kept her peace and watched as the shower of bricks continued to descend onto the site. Gradually, to her amazement, she noted that more bricks were falling in some regions than in others, and she could begin to make out the patterns of walls and doorways. She watched in fascination as the recognizable shape of rooms began to appear out of the initial chaos. "Why, that is amazing," she cried. "How have you managed to do that?"
"Well now, haven't I already told you," smiled the foreman. "You watched us lay down the probability distribution before we began. This specifies where there should be bricks and where there should be none. We must do this before we start bricklaying as we cannot tell where each brick will go when we throw it, you know," he continued.
"I do not see why!" Alice interrupted him. "I am used to seeing bricks being laid in place one after another in neat lines."
"Well now, that is not the Quantum way. Here we cannot control where each individual brick goes, only the probability that it will go one place or another. This means that when you have only a few bricks, they can go almost anywhere and seem to have no sort of pattern at all. As the number becomes large, however, you find that there are bricks only where there is some probability that they should be there, and where the probability is higher, there you get more bricks. When you have large numbers of bricks involved it all works out very nicely in the end, so it does."
Alice found this all very peculiar, although the foreman spoke so definitely that it sounded as if it might make some sort of strange sense. She did not ask any more questions at this time, as his answers only made her feel more confused than ever, so she thanked him for his information and went on down the road.
Before long she came to a window in which was displayed a large notice:
"That all sounds very enticing I am sure, but I have no idea what it is talking about, and if I were to ask someone I am sure the answer would leave me even worse off than I am now," exclaimed Alice in desperation. "I have not really understood anything that I have seen so far. I wish I could find someone to give me a good explanation of what is going on around me."
She had not realized that she had spoken aloud, until she was answered by a passerby. "If you want to understand Quantumland you will need to find someone to explain to you about quantum mechanics. For that you ought to go to the Mechanics Institute," she was advised.
"Oh, will they be able to help me understand what is happening here?" cried Alice in delight. "Will they be able to explain all the things I have seen, such as that notice in the window there, and to tell me what these eV' are?"
"I should think the Mechanics will be able to give you an explanation for most of it," answered her informant, "but as 'eV' are units of energy you had probably best start by asking about them at the Heisenberg Bank, particularly as it is just across the road there."
Alice looked across where he was pointing and saw a large building with a very formal frontage, obviously designed to impress. It had a tall portico with stone pillars and over the top, in large letters, was carved the name THE HEISENBERG BANK. Alice
crossed the road, climbed the long flight of stone steps which led up to the lofty doorway, and passed through.
hen 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 ħ 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.
Alice in Quantumland: An Allegory of Quantum Physics Page 2
