As every schoolboy knows, the energy of a wave is proportional to the square of its amplitude. And when waves interfere, the amplitudes add.
This is a paradox. Consider:
Lasers produce coherent light. Pure waves.
If I take two lasers, and focus them on a wall, they will interfere.
If they interfere constructively, then their amplitude will double, so their energy will multiply by four.
If they interfere destructively, then their amplitude will be zero, and there will be no energy.
These energies are real and measurable things. I can buy two laser pointers from a store, connect them to mains power, and focus them on a thermometer.
I have found a method of heating my house that will either halve my heating bills or allow me to burn money without warming anything up.
When I was myself a schoolboy, I didn't know about lasers, but this question occurred to me about water waves, and I asked my physics teacher. He thought for quite a long time, and said that the answer must be that *whenever* waves interfere, they must interfere both constructively and destructively, so that whenever we win, we must also lose.
He was a clever man, my physics teacher. The answer shut me up, and I haven't thought about it since. But even at the time, I found the answer suspicious, because it seemed like the universe was conspiring to preserve the conservation of energy.
I didn't believe in a teleological universe then, and I don't believe in one now.
Water waves are made up of particles, and the particles do not calculate. It must be possible to explain why there are always both sorts of interference in terms of particles bouncing off each other. But I knew that I was not clever enough to do that, and my paradox had gone away, so I put the question on the back burner and forgot about it.
The reason that I have suddenly begun to worry about this problem again is that I am making one of my periodic attempts to understand quantum mechanics, and it occurred to me that the same mathematics, and the same paradox, occur in quantum mathematics.
If a particle has two routes to the same destination, then they must interfere. You can have the same sorts of interference.
Suppose you close one of the routes, but leave the other one open, and you leave your experiment running for a while, and count how many particles get to your detector.
And suppose you had got 1000 particles arriving in your detector. And it is the same for the second route.
And now you open both routes. Particles will start arriving at a rate that depends on how the apparatus is set up. You might get anything up to 4000 in an hour. You might get none.
What you will never find is that you can use these effects to create particles for free! You can't put 2000 particles in and get 4000 out!
The mathematics of it is exactly the same as the mathematics for the wave.
In quantum mechanics the conservation of particle number is explained by the calculating machinery that we believe nature must use to advance the state of the world.
The world proceeds through unitary operators. One way of looking at a unitary operator is that it is an operator that preserves lengths.
But since the lengths in quantum mechanics are just the probabilities of a particle arriving, we are not really saying much here except that 'the universe must play according to rules that conserve the numbers of particles'.
So I've been wondering how my laser pointer experiment really works. Why does the conservation of energy survive the superposition of waves?
My physics teacher thought that if there was interference, there must always be both types in equal measure. So perhaps the reason that the spot on the wall does not go away or shine four times as bright is that it is made up of some places where there is one sort of interference and other places where there is the other.
Perhaps it must always be that the spot made by two laser pointers has bright bits and dark bits?
Why would that be true for the laser pointers?
Well, I can't put the laser pointers on top of each other, but I can put them right next to each other. At that point, their waves will be coming into the spot at very similar angles.
But of course the wavelength of visible light is very small, so even the very small angles might be enough, if the spot is large enough. I would imagine that the spot will be made up of bright and dark stripes, like a zebra.
But surely, if I make the spot very small, and put the lasers very close together, and put the lasers far away from the wall, I can make it so that I just make a bright band, and not any of the dark ones?
If I shone the lasers at the moon, that might do it!
But perhaps the laser beams would spread out, and the spot would get too big, and the bands would still be there?
So perhaps I should use red light instead of blue, to make the bands spread out more?
But perhaps as I do that, the beams will spread out more, so that as the bands get wider, the spot gets wider too, and I still have bright red and black stripes. In exactly the right proportion.
I will begin to feel that the universe is conspiring against me.
I haven't done any of these experiments, or anything like them. I am just trying to work out what might happen from half-remembered school physics.
And it has occurred to me that another place where it can appear that the universe is conspiring to make troubling questions hard to ask is the uncertainty principle that prevents you working out which slit a particle has gone through in Feynman's two slit experiment.
And perhaps if I can work out why it must always be true for waves, which can be explained in terms of particles bouncing around, I might be able to work out why it is true for quantum mechanical probabilities.
Maybe. I have looked on the web for the answer, and I have not found one, and I have asked some physics students, and I have asked a friend who did a doctorate in light waves, and none of them has thought about it, so I have written this blog post and I am going to submit it to reddit and see what happens.