Listening to Feynman again, and I encountered yet another amazing idea. He already got me excited about the experiment with two holes. Now he describes an even simpler experiment, with only one hole.
Suppose a particle (electron, photon, it doesn’t matter) approaches and passes through a single hole in a wall from a great distance. The distance can be so great that we can say with arbitrary accuracy that the particle can have no up-down momentum. If it did, then the angle would cause the particle to miss the hole and in fact travel right out of the picture. In other words, this particle (or rather a whole group of them) must be traveling along a line perpendicular to the wall.
This realization means that we know precisely the value of the particle’s up-down momentum. It’s exactly zero.
The uncertainty principle says that if we know the momentum precisely, we can know nothing about the position of the particle. But that’s not right, because this particular particle, with zero up-down momentum, has a precise up-down position. We know this because this particular particle passes through the one hole in the wall. So now we have a precise up-down momentum and a precise up-down position. Have we beaten the uncertainty principle?
No! And this fact is demonstrated in a dramatic way, a way that makes the uncertainty principle as real and visible as a light beam passing through a keyhole. We know that light spreads out when it passes through a small opening, in a process called diffraction.
So what happens to this particle when it passes through the hole? It is diffracted! It is diffracted into some (unknown and unknowable) angle. And the smaller we make the hole (the better we know the up-down position, in other words), the greater the likely angle of diffraction.
Importantly, we can’t know the exact angle of diffraction. It might be anywhere from zero to some maximum. Only when we send lots and lots of particles through the same hole do we get a spread of angles that shows us the diffraction pattern.
Diffraction, then, is a direct result of the uncertainty principle. Why is the particle diffracted? Because if it weren’t, we’d know the precise up-down position of the particle (in the hole) and the precise up-down momentum of the particle (zero) at the same time. Since we know the first (the particle is in the hole) we can’t know the second, and so the particle goes flying off at some unknown (and unknowable) angle.
Feynman also makes the point that the uncertainty principle is strictly predictive. Yes, we know the momentum and position precisely the moment before the particle enters the hole. But that does us no good. We can’t use that information to predict anything. Only the forward-going information, where the particle is and what its momentum is right now, is any good for making predictions. And since that’s the “good” information, nature denies it from us, by making the particle diffract. You can’t beat the game.
But it’s very fun to try.