Into Quantumland
alice was bored. All her friends were on holiday or visiting relations and it was raining, so that she was marooned indoors watching television. So far that afternoon she had watched part five of a series on introductory Esperanto, a program on gardening, and a paid political broadcast. Alice was really bored.
She looked down at the book lying on the floor beside her chair. It was a copy of Alice in Wonderland, which she had been reading earlier and had dropped there when she finished it. "I do not know why there cannot be more cartoons and interesting programs on the television," she wondered idly to herself. "I wish I could be like that other Alice. She was feeling bored and then she found her way to a land full of interesting creatures and strange happenings. If I could shrink down somehow and float through the television screen perhaps I might find all sorts of fascinating things."
She stared in frustration at the screen, which at that moment carried a picture of the Prime Minister telling her how, all things considered, everything was really far better than it had been three years ago, even if it didn't always seem that way. As she watched she was mildly surprised to see the picture of the Prime Minister's face slowly break apart into a mist of bright dancing speckles which all seemed to be rushing inward, as if they were beckoning her. "Why," said Alice, "I do believe that they want me to follow them in!" She leapt to her feet and started toward the television, but tripped on the book which she had discarded so untidily on the floor and fell headlong.
As she fell forward she was amazed to see the screen grow enormously, and she found herself in among the swirling speckles, rushing with them down into the picture. "I cannot see anything with these dots swirling all around me," thought Alice. "It is just like being lost in a snowstorm; why I cannot even see my feet. I wish I could see just a little. I could be anywhere."
At that moment Alice felt her feet strike something solid and she found herself standing on a hard, flat surface. All around her the swirling dots were fading away and she found that she was surrounded by a number of vague shapes.
She looked more closely at the one nearest to her and observed a small figure, coming roughly up to her waist. It was exceedingly difficult to make out, as all the time it kept hopping rapidly to-and-fro, moving so fast that it was very difficult to see at all clearly. The figure seemed to be carrying some sort of stick, or possibly a rolled umbrella, which was pointing straight up in the air. "Hello," Alice introduced herself politely. "I am Alice. May I ask who you are?"
"I am an electron," said the figure. "I am a spin-up electron. You can readily tell me apart from my friend there who is a spin-down electron, so, of course, she is quite different." Under his breath he added something which sounded rather like "Vive la difference!" As far as Alice could see, the other electron looked very much the same, except that her umbrella, or whatever it was, was pointing down toward the ground. It was very difficult to tell for sure, as this figure also was jigging to-and-fro as rapidly as the first.
"Oh please," said Alice to her first acquaintance. "Would you be good enough to stand still for a moment, as I really cannot see you at all clearly?"
"I am good enough," said the electron, "but I am afraid there is not room enough. However I will try." So saying he slowed his rate of jiggling. But as he moved more slowly, he began to expand sideways and become more and more diffuse. Now, although he was no longer moving at all quickly, he looked so fuzzy and quite out of focus that Alice could no more see what he looked like than she had been able to before. "That is the best I can do," he panted. "I am afraid that the more slowly I move, the more spread out I become. That is the way things are here in Quantumland: The smaller the space you occupy, the faster you have to move. It is one of the rules, and there is nothing I can do about it."
"There isn't really room to slow down here," continued Alice's companion as he began once more to leap rapidly around. "The platform is becoming so crowded that I have to be more compact." Sure enough, the space in which Alice stood had now become very crowded indeed, being closely packed with the small figures, each dancing feverishly to-and-fro.
"What strange beings," thought Alice. "I do not think I shall ever be able to see quite what they look like if they will not stand still for a minute, and there does not seem to be much chance of that." Since it did not look as if she could get them to slow down she tried another topic. "Would you tell me please what sort of platform we are on?" she asked.
"Why a railway platform, of course," replied one of the electrons cheerfully (it was very hard for Alice to say which had spoken; they really did all look very much the same). "We are going to take the wave train to the screen you see. You will change there to the photon express I expect, if you want to go any farther."
"Do you mean the television screen?" asked Alice.
"Why of course I do," cried one of the electrons. Alice could have sworn that it was not the same one which had just spoken, but it was very difficult to be certain. "Come on! The train is here and we have to get on."
Sure enough, Alice could see a line of small compartments drawn up at the platform. They were very small. Some were empty, some had one electron in, and some two. All of the empty compartments were filling rapidly-in fact there did not seem to be any left-but Alice noticed that not one of the compartments held more than two electrons. As they passed by any of these compartments, the two occupants would cry out "No room! No room!"
"Surely you could squeeze more than two into a compartment, seeing as the train is so crowded?" Alice asked her companion.
"Oh no! Never more than two electrons together, that is the rule."
"I suppose we shall have to get into different compartments then," declared Alice regretfully, but the electron reassured her.
"There's no problem there for you, no problem at all! You can get into any compartment that you want, of course."
"I am sure that I do not see why that should be," Alice replied. "If a compartment is too full to hold you, then it must surely be too full for me as well."
"Not at all! The compartments are only allowed to hold two electrons, so almost all the places for electrons may be taken up, but you are not an electron! There is not a single other Alice on the train, so there is plenty of room for an Alice in any of the compartments."
This did not seem to follow so far as Alice could see, but she was afraid that the train would start to move off before they got seats, so she began looking for an empty space that could take another electron. "How about this one?" she asked her associate. "Here is a compartment with only one other electron already in it. Can you get in here?"
"Certainly not!" he snapped, sounding quite horrified. "That is another spin-up electron. I cannot share a compartment with another spin-up electron. What a suggestion! It is quite against my principle."
"Don't you mean against your principles?" Alice asked him.
"I mean what I say, against my principle, or rather Pauli's principle. It forbids any two of us electrons from doing exactly the same thing, which includes being in the same space and having the same spin," he responded crossly.
Alice did not really know why she had upset him, but she looked around hastily to find another compartment which might suit him better. She managed to find one that held a single electron who was of the spin-down variety, and Alice's companion leaped into this one readily enough. Alice was surprised to find that although the tiny compartment now seemed full there was somehow enough room for her to fit in quite easily.
No sooner were they settled in than the train moved off. The journey was uneventful and the scenery not very interesting, so Alice was rather glad when the train began to slow down. "This must be the screen, I suppose," thought Alice. "I wonder what will happen here."
As they alighted at the screen there was an enormous bustle everywhere. "Whatever is going on?" Alice wondered aloud. "Why does everyone seem to be so excited?" Her questions were answered by an announcement which appeared to come from the air all around her.
"The screen phosphor is presently being excited by the incoming electrons, and we shall be having photon emission soon. Stand by for the departure of the photon express." Alice looked around to see if she could see the express arriving, when there was a rush of bright shining shapes across the platform. Alice was caught up in the middle of the crowd and carried along with them as they all crowded into one compartment. "Well, they do not seem to be worried by any principle, Pauli or otherwise," thought Alice as they crowded in around her. "These ones are certainly not worried about all being in the same place. I suppose the express is going to start soon. I wonder where ...
"... we shall end up," she concluded as she stepped out onto the platform. "My, that was certainly a fast journey. Why, it seemed to take no time at all." (Alice was quite right about this. The journey did indeed take no time at all, as time is effectively frozen for anything which is traveling at the velocity of light.) Once again she found herself surrounded by a crowd of electrons, all rushing away from the platform.
"Come along!" one of them cried to her as it rushed off. "We must get out of the station now if we are to get anywhere."
"Excuse me," Alice asked it tentatively, "are you the same electron that I was talking to before?"
"Yes I am," answered the electron as it darted off down a side passage. Alice was swept along by the crowd of electrons and carried through the main gateway from the platform.
"I declare, this is really too irksome," said Alice. "Now I have lost the only person I know at all in this strange place and have no one to explain what is happening."
"Don't worry Alice," said a voice from about knee level. "I will show you where to go." It was one of the electrons.
"How do you know my name?" asked Alice in surprise.
"That's simple. I am the same electron that spoke to you before."
"You cannot be!" exclaimed Alice. "I saw that electron go off in a different direction. Perhaps he was not the same one I was talking to before?"
"Certainly he was."
"Then you cannot be the same one," said Alice reasonably. "You cannot both be the same one you know."
"Oh yes we can!" replied the electron. "He is the same. I am the same. We are all the same, you know, exactly the same!"
"That is ridiculous," argued Alice. "You are here beside me, while he has run off somewhere over there, so you cannot both be the same person. One of you must be different."
"Not at all," cried the electron, jumping up and down even faster in its excitement. "We are all identical; there is no way whatsoever that you can tell us apart, so you see that he must be the same and I am the same too."
At that point the crowd of electrons which surrounded Alice all began to cry out, "I am the same," "I am the same too," "I am just the same as you are," "I am too, just the same as you." The tumult was dreadful, and Alice closed her eyes and put her hands over her ears until the noise died down again.
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.