Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  At the opposite end of the floor, where the machine gun had been placed before, was another gun. This was a small stubby affair, like a very small version of the cannons from which people are sometimes shot during circus performances. "What is that?" asked Alice.

  "Why, it's an electron gun, of course." As Alice looked more carefully, she could see a short flight of steps leading up to the mouth of the cannon and a line of electrons waiting to be fired from it. They seemed to be a great deal smaller than when she had last seen them. "But of course," she told herself, "these are only thought electrons."

  As she looked at them, she was surprised to see the electrons all turn and wave to her. "I wonder how they know me?" she asked herself. "But then I suppose that they are all the same electron that I met before!"

  "Commence firing!" commanded the Quantum Mechanic, and the electrons hurried up the steps into the gun and shot out in a steady stream. Alice could not make them out at all when they were in flight, but she saw a bright flash where each one hit the screen. As each flash died, it left a small glowing star which rose up the screen and remained behind to provide a marker for the position where the electron had landed.

  As had been the case for the machine gun before it, the electron gun continued to fire out its stream of electrons and the stacks of little glowing stars began to build up a recognizable distribution. At first Alice could not be too sure what she was seeing, but as the number of little stars displayed became larger it was clear that their distribution was quite different from that represented by the previous stacks of bullets.

  Instead of a slow, steady decrease from a maximum number in the center, the stars were now arranged in bands, with dark gaps between where there were few if any of the glowing markers. Alice realized that this was in a way like the case she had seen for the water waves, where there had been regions of high activity with calm areas in between. Now there were regions where many electrons had been detected, with very few in the areas between. It consequently came as no great surprise to her when Quantum Mechanic said, "There you see a clear interference effect. With the water waves you had regions of greater and lesser motion at the surface. Now each electron will be detected at one position only, but the probability of detecting an electron varies from one position to another. The distribution of different wave intensities which you saw before is replaced by a probability distribution. With one or two electrons such a distribution is not obvious, but when you use a lot of electrons you will find more of them in the regions of high probability. With one slit alone we would have seen that the distribution would decrease smoothly to either side, much as the bullets or the water waves did when there is only one slit. In this case we see that, when there are two slits open, the amplitudes from the two slits are interfering and are producing obvious peaks and troughs in the probability distribution. The behavior of the electrons is quite different from that of my friend's bullets."

  "I do not understand," said Alice. This seemed to her to be the only thing she ever said. "Do you mean that there are so many electrons going through that somehow the electrons which go through one hole are interfering with the ones which go through the other?"

  "No, that is not what I mean. Not at all. You shall now see what happens when there is only one electron in flight at any time." He clapped his hands and cried "OK! Let's do it again, but slowly this time." The electrons sprang into action or rather, to be strictly accurate, one climbed up into the cannon and shot off. The others continued to sit around where they were. A little later another electron climbed in and was fired on its way. This continued for some time, and Alice could see the same pattern of clumps and gaps appearing. These clumps and gaps were not so clear this time as they had been before because the slow rate at which the electrons were arriving meant that there were not very many in the clumps, but the pattern was clear enough. "There, you see that the interference effect works just as well even when there is only one electron present at any time. One electron on its own can show interference. It can go through both slits and interfere with itself, so to speak."

  "But that is silly!" cried Alice. "One electron cannot go through both slits. As the Classical Mechanic said, it just isn't sensible." She went up to the barrier and peered more closely, to try and see where the electrons went as they passed through the slits. Unfortunately the light was poor and the electrons moved by so quickly that she could never quite make out which slit any one had passed through. "This is ridiculous," thought Alice. "I need more light." She had forgotten that she was in the "thinking room" and was startled when an intense spotlight mounted on a stand appeared by her elbow. Quickly she directed the light toward the two slits and was pleased to find that now there was a visible flash near one hole or the other when the electron passed through. "I have done it!" she cried. "I can see the electrons as they go through the slits, and it is just as I said it must be. Each one does go through just one slit."

  "Aha!" replied the Quantum Mechanic meaningfully. "But have you looked to see what is happening to the interference pattern?" Alice looked back toward the far screen and was amazed to see that now the distribution of little stars fell smoothly from a central maximum, just like the distribution that she had seen for the classical bullets. It didn't seem fair somehow.

  "That is how it always happens; there is nothing that you can do about it," said the Quantum Mechanic soothingly. "If you don't have any observation to show which hole the electrons go through, then you get interference between the effects of the two holes. If you do observe the electrons, then you find that indeed they are in one place or the other, not both, but in that case they also act as you would expect if they had come through one hole only and you do not get any interference. The problem is that there is no way in which you can look at the electrons without disturbing them, as when you shone that light on them, and the very act of making the observation forces the electrons to choose one course of action. It doesn't matter whether or not you make a note of which hole the electron came through. It does not matter whether you are aware which hole it came through. Any observation which could tell you this will disturb the electron and stop the interference. The interference effects only happen when there is no way that you could know which slit the electron went through. Whether or not you do know does not matter.

  "So you see, when there is interference it seems as if each electron is going through both slits. If you try and check on this, you will find that the electrons go through only one slit, but then the interference vanishes. You can't win!"

  Alice thought about this for a bit. "That is utterly ridiculous!" she decided.

  "Certainly it is," replied the Mechanic with a rather smug smile. "Quite ridiculous I agree, but as it also happens to be how Nature works we have to go along with it. Complementarity, that's what I say!"

  "Would you please tell me what you mean by complementarity?" asked Alice.

  "Why of course. By complementarity I mean that there are certain things you cannot know, not all at the same time anyhow."

  "Complementarity doesn't mean that," protested Alice.

  "It does when I use it," replied the Mechanic. "Words mean what I choose. It is a question of who is to be master, that is all. Complementarity, that's what I say."

  "You said that before," pointed out Alice, who was not entirely convinced by his last assertion.

  "No, I didn't," said the Mechanic. "This time it means that there are questions you cannot ask of a particle, such as where it is and, at the same time, how fast it is going. In fact it may not be really meaningful to talk about an electron having an exact position."

  "That is a great deal for one word to mean!" said Alice tartly.

  "Why, to be sure," answered the Mechanic, "but when I make a word do extra work like that I always pay it more. I am afraid that I cannot really explain what is happening to the electrons. An explanation is usually required to make sense in terms of things you already know about and quantum physics doesn't do that. It seems to make nonsen
se but it works. It is probably safe to say that no one really understands quantum mechanics, so I cannot explain, but I can tell you how we describe what goes on. Come into the back room and I will do my best."

  See end-of-chapter note 2

  They left the gedanken room, whose floor had returned to its original shimmering aspect, and walked down the corridor to another room furnished with scattered armchairs. When they had seated themselves, the Quantum Mechanic continued. "When we talk about a situation like the electrons passing through the slits, we describe it with an amplitude. This is something like the waves that you looked at, and indeed it is often called a wave function instead. The amplitude can pass through both the slits, and it is not always positive, like a probability. The lowest probability that you can have is zero, but the amplitude may be negative or positive, so the parts from different paths can cancel or add and give interference, again just like the water wave."

  "So where are the particles?" asked Alice. "Which slit do they actually go through?"

  "The amplitude doesn't really tell you about that. However if you square the amplitude, that is multiply it by itself so that it gives something that is always positive, then it gives you a probability distribution. If you choose any position this will tell you the probability that, when you observe a particle, you will find it at that position."

  "Is that all it can tell you?" exclaimed Alice. "I must say that it sounds very unsatisfactory. You would never know where anything is going to be."

  "Yes, that is true enough. For one particle you cannot tell where it will be found, except that it will not be at a position where there is zero probability of course. If you have a large number of particles, though, then you can be fairly sure that you will find more where the probability is high and far fewer where it is low. If you have a very large number of particles, then you can say quite accurately how many will end up where. That was the case with those builders you were telling us about. They knew what they would get because they used a large number of bricks. For really large numbers the overall reliability is very good."

  See end-of-chapter note 3

  "And there is no way you can say what each particle is doing until it is observed?" repeated Alice, just to get this clear.

  "No, no way at all. When the thing that you actually observe could have come about in several different ways, then you have an amplitude for each possible way, and the overall amplitude is given by adding all of these together. You have a superposition of states. In some sense the particle is doing all the things which it could possibly be doing. It is not just that you do not know what the particle is doing. The interference shows that the different possibilities are all present and affect one another. In some way they are all equally real. Everything that is not forbidden is compulsory."

  "Oh, I saw that on a notice in the Bank. It looked very stern."

  "You had better believe it! It is one of the main rules here. Where there are several things which might happen, they all do. Look at the Cat, for example."

  "What cat?," asked Alice, looking around her in confusion.

  "Why Schrödinger's Cat over there. He left it with us to look after." Alice looked over in the corner where the Mechanic was pointing and saw a large tabby cat sleeping in a basket in the corner. As if awakened by hearing its name the cat stood up and stretched. Or rather, it did and it didn't. Alice could see that, as well as the slightly hazy figure of the cat standing with back arched in the basket, there appeared to be another identical cat which was still lying on the bottom. It was very stiff and motionless and lay in a rather unnatural position. From the look of it, Alice would have sworn that it was dead.

  "Schrödinger devised a gedanken experiment in which an unfortunate cat was enclosed in a box, together with a flask of poison gas and a mechanism which would break the flask should a sample of radioactive material happen to decay. Now such a decay is definitely a quantum process. The material might or might not decay, so according to the rules of quantum physics you would have a superposition of states, in some of which the decay would have happened and in others it would not. Of course, for those states where a decay had happened the cat would have been killed, so you would have a superposition of cat-states, some dead and some alive. When the box was opened someone would observe the cat, and from that time on it would be either alive or dead. The question which Schrödinger posed was, 'What was the state of the cat before the box was opened?"'

  "And what did happen when the box was opened?" asked Alice.

  "Well actually, everyone was so engrossed in discussing the question that no one ever did open the box, which is why the Cat was left like that."

  Alice peered closely into the basket, where one aspect of the Cat was busily licking itself. "He looks pretty lively to me," she observed. No sooner were the words out of her mouth than the Cat became fully solid and the dead version vanished. With a satisfied purr the Cat leapt out of the box and began to stalk a mouse which had just popped out of the wall. Alice noted that there was no mouse hole visible-the mouse had simply come out of the solid wall. The Quantum Mechanic followed the direction of her gaze. "Ah, yes. That is an example of barrier penetration; we get it happening all the time. Where you have a region that a particle could not enter at all according to classical mechanics, the amplitude does not necessarily stop abruptly at the boundary, though it does die away rapidly inside the region. If the region is very narrow, then there is still some small amplitude left at the other side, and this gives a slight probability that the particle may appear there, having apparently tunneled through an impassible barrier. It happens quite often."

  Alice had been thinking through what she had seen and had noted a difficulty. "How is it that I was able to make an observation and fix the condition of the Cat if it was not able to do it for itself? What is it that decides when an observation is actually made and who is able to make one?"

  "There you have a good question," replied the Quantum Mechanic, "but we are only mechanics after all, so we do not worry too much about such things. We just get on with the job and use ways that we know will work in practice. If you want someone to discuss the measurement problem with you, you will need to go somewhere more academic. I suggest that you go to a class at the Copenhagen School."

  "And how do I get there?" asked Alice, resigned to being passed on somewhere else once again. In answer the Mechanic led her out into the corridor and opened yet another door. This did not lead into the alleyway from which she had entered, but into a wood.

  Notes

  1. Quantum mechanics is usually contrasted with classical or Newtonian mechanics. The latter covers the detailed description of moving objects which was developed before the early years of the twentieth century and was based on the original work of Galileo, Newton, and others both before and since. Newtonian mechanics works very well on a large scale. The motion of the planets can be predicted over long times and with great accuracy. It works almost as well for artificial planets and the various exploratory space missions: Their positions may be predicted years ahead. It also works pretty well for falling apples.

  In the case of a falling apple there will be significant resistance from the air that surrounds it. Classical mechanics describes this as the collision of vast numbers of air molecules bouncing off the apple. When you ask about air molecules you are told that they are small groups of atoms. When you ask about atoms there is an embarrassing silence.

  Classical mechanics had virtually no success in describing the nature of the world on the scale of atoms. Things must somehow be different for small objects from how they seem to be for large ones. If you argue in this way, then you must ask: large or small relative to what? There must be some dimension, some fundamental constant which fixes the size at which this new behavior becomes obvious. It is a definite change in the way things are observed to behave, and it is universal. Atoms in the sun and in distant stars emit light with a spectrum which is like that from a lamp on a table beside us. The onset of quant
um behavior is not something that just happens to take place locally; there is some fundamental property of Nature involved. This is given by the universal constant ħ, which features in most equations of quantum mechanics. The world is grainy on the scale defined by this constant, ħ. On this scale energy and time, position and momentum are blurred together. It need hardly be pointed out that, on the human scale of perception, ħ is very small indeed and most quantum effects are not at all obvious.

  2. What the Heisenberg uncertainty relations are telling us is that we are looking at things in the wrong way. We have a preconception that we ought to be able to measure the position and momentum of a particle at the same time, but we find that we cannot. It is not in the nature of particles for us to be able to make such a measurement on them, and the theory tells us that we are asking the wrong questions, questions for which there is no viable answer. Neils Bohr used the word complementarity to express the fact that there may be concepts which cannot be precisely defined at the same time: such pairs of concepts as justice and legality, emotion and rationality.

  There is, apparently, something fundamentally wrong with our belief that we should be able to talk about the position and momentum of a particle, or of its exact energy at a given time. It is not clear why it should be meaningful to talk simultaneously of two such different qualities, but it appears that it is not.

  3. Quantum mechanics is not really about definite particles in the traditional classical sense; instead you talk about states and amplitudes. If you square an amplitude (i.e., multiply it by itself), then you get a probability distribution which gives the probability of obtaining various results when you make an observation or measurement. The actual value that you get for any one measurement appears to be quite random and unpredictable. So it does look as if the suggestion made earlier that nature is uncertain and "anything goes" must, after all, be true, does it not?