Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  Well, no-if you make many measurements the average result is accurately predictable. Bookmakers do not know which horse will win each race, but they confidently expect to make a profit at the end of the day. They do not anticipate large surprise losses even though they have to work with rather small numbers, so that the averaging is not too reliable. The number of gamblers will be a mere few thousand people rather than the 1,000,000,000,000,000,000,000,000 or more atoms you will get in even a tiny speck of matter. This looks less like a number than a repetitive wallpaper pattern, but it is undeniably large. The overall statistical fluctuations to be expected for measurements made on such a large number of atoms are negligible, even though the result for each individual atom may be quite random.

  Quantum-mechanical amplitudes may be calculated very accurately and compared with experiments. An often quoted result is for the magnetic moment of the electron. Electrons spin like little tops and they also have electrical properties: They behave rather like tiny bar magnets. The magnetic strength and the electron spin are related, and their ratio may be calculated using suitable units.

  A classical calculation gives the result 1 (with rather arbitrary assumptions about the distribution of the electric charge in an electron).

  The quantum calculation gives the result 2.0023193048 (±8) (the error is in the last decimal place).

  A measurement has given the result 2.0023193048 (±4).

  This is good agreement! The probability of getting by chance a value which is in such good agreement is similar to the probability of throwing a dart at random and hitting the bull's-eye on a dartboard-when the dartboard is as far away as the Moon. This particular result is often given as an example of the success of quantum theory. It is possible to calculate accurately the amplitudes for other processes just as accurately, but there are very few quantities which you can measure to this precision.

  lice entered the wood and made her way along a path which wound between the trees, until she came to a place where it forked. There was a signpost at the junction, but it did not appear very helpful. The arm pointing to the right bore the letter "A," that to the left the letter "B," nothing more. "Well, I declare," exclaimed Alice in exasperation. "That is really the most unhelpful signpost I have ever seen." She looked around to see if there were any clues as to where the paths might lead, when she was a little startled to see that Schrödinger's Cat was sitting on the bough of a tree a few yards off.

  "Oh Cat," she began rather timidly. "Would you tell me please which way I ought to go from here?"

  "That depends a good deal on where you want to get to," said the Cat.

  "I am not really sure where.... " began Alice.

  "Then it doesn't matter which way you go," interrupted the Cat.

  "But I have to decide between these two paths," said Alice.

  "Now that is where you are wrong," mused the Cat. "You do not have to decide, you can take all the paths. Surely you have learned that by now. Speaking for myself, I often do about nine different things at the same time. Cats can prowl around all over the place when they are not observed. Talking of observations," he said hurriedly, "I think that I am about to be obs..." At that point the Cat vanished abruptly.

  "What a strange cat," thought Alice, "and what a strange suggestion. He must have been referring to that superposition of states that the Mechanic was talking about. I think that it must be something like the time that I left the Bank. Somehow I managed to go in many different directions that time, so I suppose I shall just have to try and do it again."

  State: Alice (A1)

  Alice turned right at the sign and walked farther along the winding path, looking around her at the trees as she passed. She had not gone very far when she came to another fork in the path; this time the signpost had two arms, labeled "1" and "2." Alice turned to the right and continued on her way.

  As she walked along, the trees thinned out and she found herself trudging up a steep, rocky track. It became steeper and steeper as she went on, until she found herself climbing the side of a lonely mountain. The track brought her along a narrow ledge running along the side of a precipitous cliff. This finally ended at a little grassy-floored area with vertical sides. Before her eyes was a yawning mouth in the cliff face, from which a passage led in and down.

  The passage was very dark, but to her surprise Alice found herself creeping on down it. It had a smooth floor and sides and ran straight ahead, sloping gently downward toward a dimly visible distant glow. As she went, the light steadily became brighter and also redder, and the tunnel got hotter. Wisps of vapor floated past her, and she heard a sound like some vast animal snoring in its sleep.

  At the end of the tunnel Alice peeped out into a great cellar. Its dark vastness could only be guessed dimly, but rising from close below her feet was a great glow. There lay a vast reddish-gold dragon fast asleep with his huge tail coiled around him. Beneath him, forming his bed, was an enormous pile of gold and silver, jewels, and marvelously carved objects, all red-stained in the ruddy light.

  State: Alice (A2)

  Alice turned right at the sign and walked farther along the winding path, looking around her at the trees as she passed. She had not gone very far when she came to another fork in the path; this time the signpost had two arms, labeled "1" and "2." Alice turned to the left and continued on her way.

  As she was walking along, she looked down and found that the path she was on had changed from a forest track to a narrow road paved with yellow bricks. She followed this through the trees until the wood opened out into a wide meadow. It was very wide, extending as far as Alice could see, and the whole field was covered with bright poppies. The yellow brick road ran through the middle of the meadow up to the gates of a distant city. From where she stood, Alice could see that the high walls of the city were a brilliant green and the gates were studded with emeralds.

  State: Alice (B1)

  Alice turned left at the sign and walked farther along the winding path. There was nothing very remarkable to see as yet. She turned a corner and came to another fork in the path; this time the signpost had two arms, labeled 1 and 2. Alice turned to the right and continued on her way.

  The undergrowth between the trees became thicker, and it was difficult to see anything that was at all far from the path, though the path itself was still quite clear as it wound between the closely packed trees. Alice turned a corner and came suddenly upon an open space. In the center of the clearing stood a little building with a steeply pitched roof and a small belfry at one end. The words "Copenhagen School" were carved deeply into the stone lintel over the door.

  "This must be the place I was told to go to," Alice remarked to herself. "I am not sure that I want to go to a school though! I spend quite enough time at school as it is. But maybe a school here will be quite different from the one that I am used to. I will go in and see!" Without knocking, she opened the door and went in.

  State: Alice (B2)

  Alice turned left at the sign and walked farther along the winding path. There was nothing very remarkable to see as yet. She turned a corner and came to another fork in the path; this time the signpost had two arms, labeled 1 and 2. Alice turned to the left and continued on her way.

  A little further along, the path began to rise, and Alice climbed up the side of a little hill. At the top of the hill she stood for some minutes looking out in all directions over the country-and a most curious country it was. There were a number of little brooks running across it from side to side, and the ground between was divided up into squares by a number of hedges, which reached from brook to brook.

  "I declare. It's marked out just like a large chessboard," Alice said at last.

  "Ah, come in, my dear," a voice called softly, and Alice realized that she had been observed. She stepped through the door and looked around the schoolroom. It was quite a large room with high windows all round. There were rows of desks down the middle of the room. At one end there was a blackboard and a large table behind which
stood the Master.

  "It does look very much like an ordinary school," Alice admitted to herself as she turned to look at the children in the class. She found that the desks were not occupied by children, however, but by a most remarkable selection of beings who clustered around the front of the room. There was a mermaid, with long flowing hair and a scaly fish's tail. There was a uniformed soldier who, on closer examination, appeared to be made out of tin and a ragged little girl with a tray full of matches. There was a dramatically ugly duckling and a haughty looking man of regal bearing who for some reason was clad only in his underclothes.

  Or is he?" Alice wondered to herself. As she looked again she fancied she could see him wearing rich embroidered garments and a thick flowing velvet robe. When she looked once more, however, all she could see was a rather portly man dressed in his undergarments.

  "Hello, my dear," said the Master, who was a kindly looking avuncular figure with bushy eyebrows. "Have you come to join our discussion?"

  "I am afraid that I do not know how I have come to be here," said Alice. "I seemed to be in several other places just a moment ago, and I am not at all sure why I have ended up here, rather than in one of the others."

  "That is because we have observed you to be here, of course. You were in a superposition of quantum states, but once you had been observed to be here, why you were here, naturally. Obviously you were not observed in any of the other places."

  "What would have happened if I had been?" asked Alice curiously.

  "Why then your set of states would have collapsed to that other one. "You would not be here, but would instead be at the position where you had been observed to be, of course."

  "I really do not see how that can be," replied Alice, who was once again feeling terribly confused. "What difference does it make whether I was observed or not? Surely I must be in one place or the other no matter who sees me."

  "Not at all! After all, you cannot say what is happening in any system if you do not observe it. There may be a whole range of things that it might be doing and you could give a probability that it is or is not doing any one of them as long as you do not look. In fact the system will be in a mixture of states corresponding to all the things that it might be doing. That will be the situation up to the point when you look to see what it is doing. At that point of course one possibility is selected and the system will then be doing only that."

  "Then what happens to all the other things it was doing?" asked Alice. "Do they just vanish?"

  "Well, there are more things that it may be doing than things it was doing, but yes," answered the Master, beaming at her. "You have got it exactly. All the others states just vanish. The land of maybe becomes the land that never was. At that point all the other states cease to be in any way real. They become, if you like, just dreams or fantasies, and the observed state is the real one. This is called reduction of the quantum states. You will soon get used to it."

  "Does that mean that when you look at something you can choose what you will see?" asked Alice in some disbelief.

  "Oh no, you do not get any choice in the matter. What you are likely to see is determined by the probabilities for the various quantum states. What you actually do see is a matter of random chance. You do not get to choose what will happen; the quantum amplitudes only give the probability of different results, but they do not fix what will happen. That is pure chance and only becomes fixed when an observation is made." The Master said this very earnestly, though so quietly that Alice had to strain to catch everything that he said.

  "Making an observation seems to be very important then," Alice mused, half to herself. "But then who can make an observation? Obviously the electrons are not able to observe themselves as they go through the slits in an interference experiment, as they seem to go through both slits. Or should I say that amplitudes for both slits are present?" she corrected herself, copying the way of speaking which she had heard so much recently. "Apparently I did not observe myself properly when I was in a superposition of states just now.

  "In fact," Alice said abruptly, struck by a sudden thought, "if quantum mechanics says that you must do everything you can do, then surely you must observe all the possible results of any measurement you make. If your quantum superposition principle is to work everywhere, then it is not possible to make measurements at all! Any measurement you tried to make could have several possible results. You might observe any of these results and, according to your rules, if you could observe any of them you would have to observe all of them. The results of your measurement would all be present in a new version of this superposition of states you talk about. You could never actually observe anything, or rather there would never be anything you could fail to observe."

  Alice paused for breath, quite carried away with this new thought, and noticed that everyone in the room was staring at her intently. When she stopped they all stirred a little uneasily.

  "Of course you have a very important point there," said the Master kindly. "It is known as the measurement problem and is the very subject that we have been considering in the class here."

  See end-of-chapter note 1

  The Master continued: "It is important to remember that it is a real problem. There must exist a mixture of amplitudes such as we describe for systems of one or two electrons, as in the two slit interference experiment you saw, because there is interference between the amplitudes. This is not just a way of saying that the electron may be in one state, but that you do not happen to know what state that is. That situation could not give any interference, so we are forced to accept that, in some sense, each electron is in all the states. I believe that it is not a proper question when you ask what the electron is really doing because there is no way you can ever find out. If you try to check you will alter the system, so that you are examining something different.

  "As you point out, there seems to be a problem here. Atoms, and systems containing a small number of particles, always do everything they possibly can, and they never make any decisions. We, on the other hand, always do one thing or the other and do not observe more than one outcome from any given situation. The students have each prepared a short talk about the measurement problem. They consider at what point, if any, the quantum behavior which allows all states to be present at the same time ceases to operate, so that unique observations may be made. You might like to sit and listen to their presentations." This seemed to Alice to be a good opportunity, so she sat at one of the desks and leaned forward expectantly.

  "The first talk," announced the Master, his quiet voice managing to quell the expectant buzz of comment from the students, "is from the Emperor." The portly gentleman in the tasteful purple underwear, whom Alice had noticed when she first entered the classroom, got up and walked to the front of the class.

  See end-of-chapter note 2

  The Emperor’s Theory (Mind Over Matter)

  "Our hypothesis," he began, with a haughty glance around the room, "is that it is all in the mind.

  "The laws that are obeyed by quantum systems," he continued, "the description of physical states by amplitudes, and the superposition of these amplitudes when there is more than one possible condition-these laws apply to every material thing in the world. We say 'every material thing,"' he repeated, "as Our contention is that such a superposition is not experienced by the conscious mind. The physical world is governed at every stage by quantum behavior, and any purely material system, large or small, will always be in a combination of states, with an amplitude present for everything which might be or might have been. It is only when the situation comes to the attention of the sovereign will of a conscious mind that a choice is made.

  "For the mind is a thing outside, or in Our case above, the laws of the quantum world. We are not tied by the need to do everything that might possibly be done; instead We are free to make selections. When We observe something, then that thing is observed; it knows that We have observed it, the Universe knows that We have observed it, and it remai
ns thereafter in the condition in which We have observed it. It is Our very act of observation which imposes a unique and definite form upon the world. We may not have the choice to select what We will see, but whatever We do observe has become uniquely real at that point."

  He paused and looked commandingly around the room once more. Alice found herself strangely impressed by his authoritative delivery, despite his purple underwear. "For example, when We look at Our magnificent new imperial clothing We observe that We are of course exquisitely attired." He looked down at himself and abruptly he was clad from head to foot in rich garments. His coat and waistcoat were smothered in fine embroidery and he wore a flowing velvet robe trimmed with ermine. "Now it is conceivable that, when Our attention was diverted from Our garments they might have been less tangibly real than they are now seen to be, but if that had been so, now that We have observed them they are seen by all to be of the finest cut and that is in reality what they are."

  The Emperor raised his head again and looked out at the class. Alice was intrigued to note that, although his observation of the clothes had fully established their rich aspect, as soon as he had looked away they gradually became hazy-looking once more and his tastefully monogrammed underclothes began to show through.

  "That then is Our thesis. The whole material world is indeed governed by the laws of quantum mechanics, but the human mind is outside the material world and not so restricted. We have the ability to see things uniquely. We cannot choose what We will see, but what We do see becomes reality in the world, at least for the time We observe it. When We have finished Our observation, then of course the world can once again begin to enter its customary set of mixed states."