The Mechanic's Institute
Alice examined the building in front of her. It was unremarkable, a plain brick structure now rather the worse for wear. In front of her was the board which stated that this was "The Mechanics Institute." Beside this was a wooden door on which someone had pinned a note: "Don't knock. Just come in." Alice tried the door and found it was not locked, so she opened it and walked through.
Inside she found herself in a large, dark room. In the middle of the room there was an area of light and clarity. Within this limited area it was possible to make out a reasonable amount of detail. Beyond this there was a seemingly limitless expanse of darkness within which nothing meaningful could be discerned. In the pool of light was a billiards table, with two figures moving around it. Alice walked toward them, and as she approached they turned to look at her. They were an oddly assorted couple. One was tall and angular. He wore a starched white shirt with a tall stiff collar, a narrow tie, and, rather to Alice's surprise, a boiler suit. His face was aquiline, with bushy side whiskers. He regarded her with a gaze of such piercing intensity that Alice felt he could clearly distinguish every tiniest detail in whatever he saw. His companion was smaller and younger. He had a round face decorated with large, round metal-rimmed glasses. Behind the glasses his eyes were strangely hard to see; it was difficult to say where he was looking, or even exactly where his eyes were. He was wearing a white laboratory coat, which was open to display beneath it a T-shirt with a picture of something vaguely atomic on the front. It was not easy to say exactly what it was meant to be as the colors appeared to have run in the wash.
"Excuse me, is this the Mechanics Institute please?" asked Alice, mostly for the sake of making conversation. She knew from the notice outside that it must be.
"Yes, my dear girl," said the taller and more impressive looking of the two. "I myself am a Classical Mechanic from ClassicWorld, and I am visiting my colleague here, who is a Quantum Mechanic. Whatever your problem is, I am sure that between us we will be able to assist you, if you would just wait a moment while we finish our shots."
Both men turned back to the billiards table. The Classical Mechanic took careful aim, clearly judging all the angles involved to within a tiny fraction of a degree. At last, he very deliberately played his shot. The ball bounced to and fro in a remarkable series of ricochets, ending in a collision with the red ball and knocking it squarely into the center of a hole. "There you are," he exclaimed with satisfaction as he retrieved the ball from the pocket. "That is the way to do it, you know; careful and exact observation followed by precise action. If you do things that way you can produce any result you choose."
His companion did not respond, but took his place at the table and made a vague stab with his cue. After her previous recent experiences Alice was not really surprised to discover that the ball shot off in every direction at once, so that there was no part of the table where she could say definitely that the ball had not gone, though equally she could in no way say where it actually was. After a moment the player went over and peered into one of the pockets, then reached in and drew out a red ball.
"If you do not mind my saying so," said Alice, "you do seem to play the game very differently."
"Quite so," replied the Classical Mechanic. "I hate the way he plays his shots like that. I like everything to be done very carefully and precisely and to be planned in every detail in advance. However," he added, "I imagine that you did not come here to watch us play billiards, so tell us what you wanted to know."
Alice recounted all her experiences since she came into Quantumland and explained how confusing she found it and how everything seemed so strange and somehow indefinite. "And I do not even know how I came to find this building," she finished. "I was told that the interference would probably bring me to the right place, but I do not understand what happened at all."
"Well now," began the Classical Mechanic, who seemed to have appointed himself as the spokesman for the two. "I cannot say that I really understand all of it either. As I have said, I like things to be clear-cut, with cause following effect in a sensible fashion and everything clear and predictable. If truth be told, not a lot that goes on here makes much sense to me," he whispered confidentially to her. "I am just visiting from ClassicWorld. That is a splendid place where everything happens with mechanical precision. Cause follows effect in a wonderfully predictable fashion, so it all makes sense and you know what is going to happen. What is more, the trains all run on time," he added as an afterthought.
See end-of-chapter note 1
"That sounds very impressive," said Alice politely. "If it is so well organized, is everything run by computers?"
"Well, no," answered the Classical Mechanic. " We do not use computers at all. In fact electronics will not work in ClassicWorld. We are bet ter with steam engines. I do not really feel at home in Quantumland. My friend here is much more familiar with quantum conditions.
"However," he went on more confidently, "I can tell you what interference is. That happens in classical mechanics as well. Just follow me and I shall demonstrate how it works."
He led Alice out through a door, down a short corridor, and into another room. This one was well illuminated, with a clear light which was equally bright everywhere and did not seem to come from any particular source. They stood on a narrow wooden walkway which ran around the edges of the room. The floor in the center was covered with some sort of shimmering grayish material, which did not look solid. It was shot through with random flashes of light, rather like a television set when there is no picture being received.
Her guide explained, "This is the gedanken room, which means a 'thinking room.' You know that many gentlemen's clubs have a writing room and a reading room. Well, we have a thinking room. In here one's thoughts can take on substance, so that anyone can look at them. It allows us to do thought experiments. These allow us to work out what would happen in various physical cases, and they are much cheaper than real experiments of course."
"How does it work?" asked Alice. "Do you just think of something and it appears?"
"That is correct; in essence that is all you have to do."
"Oh please, may I try?" asked Alice.
"Yes certainly, if you wish."
Alice thought very intensely at the shifting, flickering surface. To her surprise and delight, where there had before been a featureless area there was now a group of small furry rabbits hopping about.
"Yes, very pretty," said the Mechanic rather impatiently. "But this is not helping to explain interference." He made a gesture and all the rabbits vanished, all but one little one who remained unnoticed in a corner of the area.
"Interference," he began authoritatively, "is something which happens with waves. You can have all kinds of waves in physical systems, but it will be simplest to consider water waves." He stared hard at the floor, which turned before Alice's eyes into a sheet of water, with gentle ripples running over the surface. In the corner the rabbit vanished below the surface with a "plop" as the floor turned to water beneath it. It quickly struggled out again and glared at them. Then it shook itself, looked mournfully at its damp fur, and vanished.
"Now we want some waves," continued the Classical Mechanic, paying no attention to the unhappy rabbit. Alice obligingly thought at the floor and a long curling wave came sweeping across the surface and broke dramatically upon a beach at one end.
"No, that is not the sort of wave that we want. Those large breaking waves are too complicated. We want the sort of gentler ripple which spreads out when you throw a stone into water." As he spoke a series of circular ripples spread out from the middle of the water.
"But we need to think about what are called plane waves where they all move in the same direction." The circular ripples changed to a series of long, parallel furrows, like a wet plowed field, all moving across the floor from one side to the other.
"Now we put a barrier in the middle." A low fence sprang up across the center, dividing the floor in two. The waves flowed up to the barrier and lapped up and down against it, but there was no way for them to get through and the water beyond was now still and calm.
"Now we make a hole in the barrier, so that the waves can get through there." A neat little gap appeared just to the left of the fence's center point. Where the ripples struck this narrow gap they could pass through and spread out in circular ripples into the calm region beyond.
"And now, see what happens when we have two holes in the barrier," cried the Mechanic. Abruptly there were holes both to the right and to the left of the center. Circular ripples spread out from both of these. Where they crossed, Alice could see that in some places the water was surging up and down much more than it had when there was only one hole open, whereas in other places it hardly moved at all and was locally quite still.
"You can see what is happening if we freeze the motion. We can do that of course in a thought experiment." All motion on the water stopped, and the patterns of ripples were frozen into position, as if the whole area had turned abruptly to ice.
"Now we shall mark regions of maximum and minimum amplitude," continued the Classical Mechanic determinedly. "The amplitude is the amount by which the water moves from the surface level it had when calm." Two fluorescent arrows appeared, hanging in space above the surface. One was an apple green color and was pointing down at a point where the disturbance was greatest. The other was a pale red and pointed to a spot where the surface was almost undisturbed.
"You will be able to see what is happening if we now look at the effect of only one hole at a time," he said, with steadily increasing enthusiasm. One of the gaps in the fence vanished, and there were left only the circular ripples spreading out from the other one, though still frozen in position as if they were made from glass. "Now we will switch to the other hole." Alice could see very little difference when this happened. The position of the gap had moved and the pattern of circular ripples coming from it had moved very slightly, but overall it looked much the same. "I am afraid that I cannot understand what you are trying to show me," she said. "The two cases look just the same to me."
"It will help you to see the difference if we cut quickly from one case to the other." Now the gap in the fence leapt to and fro, first to the right, then to the left. As it moved, the pattern of ripples on the surface shifted slightly back and forth.
"Look at the wave patterns under the green arrow," cried the Mechanic, who seemed to Alice to have become quite unnecessarily excited about the subject. However, she did as requested and saw that at the point indicated there was a hump in the water in each case. "Each gap in the fence has produced a wave which is high at this particular point, so when both gaps are open the wave is twice as high here and the overall rise and fall of the water is much greater than it is for one gap alone. This is called constructive interference.
"Now look at the wave patterns under the red arrow." Here Alice saw that, while one gap gave rise to a hump at that point, the other produced a trough in the surface. "You can see that in this position the wave from one gap goes up and that from the other goes down, so when you have the two present together, they cancel one another out and you get no overall effect. This is called destructive interference.
"That is all there is to wave interference really. When two waves overlap and combine with one another, their amplitudes, the amounts by which they go up or down, combine with one another. In some places the contributing waves are all going in the same direction, so the disturbances add up and you get a large effect. At other positions they go in different directions and cancel one another out."
"Yes, I think that I follow that," said Alice. "So you are saying that the doors in the Bank acted rather like the gaps in the fence here and gave rise to some sort of large effect in the place where I needed to go and can - celed one another out in other positions. I do not see how that can apply to my case though. With your water wave you say that there is more of the wave in one place and less in another because of this interference, but the wave is spread out over the whole area, while I am always in just one place at any time."
"Exactly!" cried the Classical Mechanic triumphantly. "That is the problem. As you say, you are in one place. You are more like a particle than a wave, and particles behave quite differently in a sensible classical world. A wave is spread over a wide area and you look at only a small portion of it at any position. Because of interference you may get more or less of it at different positions, but it is only a small part of the whole wave wherever you look. A particle, on the other hand, is located at some point. If you look in various positions you will either find the whole particle or it is simply not there. In classical mechanics there is no question of particles showing interference effects, as we can show."
He turned to the floor of the gedanken room and stared firmly at it. The surface turned from water to a smooth area of steel armor, with armored barriers around the edges, high enough for them to hide behind. Across the middle of the floor, where the low fence had stretched across the water, was now a tall armored wall, with a narrow slit slightly to the left of center. "Now we can look at the same setup, but I have changed it so that we can look at fast particles. These are something like bullets from a gun, so that is what we will use."
He gestured toward one end of the room where there appeared an unpleasant-looking machine gun with many boxes of ammunition stacked beside it. "This gun has an unsteady mounting, so that it will not always shoot in the same direction. Some of the bullets will strike the gap in the wall and pass through, as part of the wave did in our last thought experiment. Most of them, of course, will hit the steel wall and bounce off. Oh that reminds me," he added abruptly. "We had better wear these in case we are struck by ricocheting bullets." He produced a pair of steel helmets and handed one to Alice.
"Do we really need these?" asked Alice. "If this is only a thought experiment, surely these are thought bullets, and can't do us any harm."
"Well, perhaps so. But you might still think that you had been hit by a bullet, and that would not be very nice you know."
Alice put the helmet on. She could not feel it on her head and did not think that it would be the least bit of use, but there did not seem to be much point in arguing any further. The Mechanic stood upright and gave an imperious wave of his hand, and the gun began firing very noisily. The bullets shot out in an unsteady stream; most hit the armored screen and whined off in all directions, but a few got through the slits in the barrier and hit the wall opposite. Alice was intrigued to note that when a bullet hit this wall, it immediately came to a stop and then rose slowly into the air to hang suspended in space, directly above the point where it had struck the wall.
"As you can see, whereas the water wave was spread out all over the far wall, a bullet will hit it in one position only. However, in this experiment there is a greater probability that the bullet will strike the far wall opposite the slit in the screen than there is that it will bounce off the slit edge and end up a long way off to the side. If we wait for a little we will see how the probability varies for different points along the wall." As time passed and the air became full of flying bullets, the number which were suspended above the wall grew steadily. As she watched, Alice could make out a distinct trend developing.
"There, you see how the bullets which have passed through the slit are distributed along the wall," remarked the Mechanic as the gun fell silent. "Most have ended up directly opposite the hole, and the number falls off steadily on either side. Now see what happens when the slit is offset to the right." With another wave of his hand the hovering bullets dropped to the ground, and the gun began to fire again. Though the demonstration was noisy and rather unsettling, as far as Alice could see the end result was just the same as last time. Frankly, it was disappointing.
"As you can see," said the Mechanic with misplaced confidence, "the distribution is similar to the previous one, but displaced slightly to the right because the center is now opposite the new position of the slit." Alice could not see any difference at all, but she was prepared to take his word for it.
"Now," continued the Mechanic dramatically, "see what happens when both slits are open." As far as Alice could see it did not make the slightest difference, except that, since two slits were now open, more bullets got through to hit the far wall. This time she decided to comment. "I am afraid that it looks just the same to me each time," she said rather apologetically.
"Exactly!" replied the Mechanic with satisfaction. "Except that, as you will of course have observed, the center of the distribution is now centered between the two slits. We had one distribution for the probability that bullets will pass through the left-hand slit and another distribution for the probability that bullets will pass through the right-hand slit. When we have both slits open, then bullets may pass through either slit, so the overall distribution is given by the sum of the probabilities that we got for the two slits on their own, since the bullets must have passed through one or the other. They cannot have passed through both you know," he added, addressing the Quantum Mechanic, who had just come into the room.
"You say that," replied his colleague, "but how can you be so sure? Just look what happens when we repeat your gedanken experiment with electrons."
In his turn, the Quantum Mechanic waved his hand at the floor of the room. His gestures were not so decisive as his companion's, but they seemed to work just as well. The gun and the armored walls all disappeared. The floor returned to the shimmering material which Alice had first seen, but the now-familiar wall with two slits near its middle was still there, stretching across the center of the floor. At the far end of the floor was a wide screen with a greenish glow. "That is a fluorescent screen," muttered the Mechanic in her ear. "It gives a flash of light when an electron hits it, so it can be used to detect where they are."
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 nonsense 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 Schr?dinger'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.
"Schr?dinger 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 Schr?dinger 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 quantum 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 ?, which features in most equations of quantum mechanics. The world is grainy on the scale defined by this constant, ?. 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, ? 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?
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.