Alice looked more closely at the building. On a faded board by the door she could make out the words "Mechanics Institute." This was indeed where she had intended to come!
lice 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 on
e 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 dist
ribution 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."
Alice in Quantumland: An Allegory of Quantum Physics Page 4