Spatial Quantization and the Measurement Postulate

One of the most baffling aspects of quantum mechanics is the notion that spin must be spatially quantized: that an electron can have its spin axis pointing up, or down, but nothing in between. This goes back to the Stern-Gerlach experiment: a beam of silver atoms with random spins is passed through a magnetic field, and instead of being spread out smoothly like you would expect “classically”, the beam splits in two. Half the atoms are “spin-up”, and half are “spin-down”.

Copenhagen explains this as an example of the Measurement Postulate. We have a silver atom intially in a random state: that is, in a superposition of up and down states. The Stern Gerlach apparatus is designed to clearly identify atoms in either of those two pure states. Therefore, when the atom enters the apparatus, it makes a decision: spin up, or spin down. The Born Postulate tells us that the probability of this decision is given by the amplitudes of the respective states.

Copenhagen is a bit sketchy on the question of just when and where the silver atom makes that decision. Some people think it happens when the atom passes between the magnets. Others defer the moment of truth to the point of impact on the screen. Perhaps I’m being unfair when I say “Copenhagen” is sketchy on this point; it is probably more accurate to say that the followers of the Copenhagen Interpretation are not, on the whole, especially clear on what they are supposed to believe about this question.

It is hard to believe that with all the nonsense written about the quantum leap and the collapse of the wave function, that nowhere will you find the straightforward explanation of the Stern Gerlach experiment that I am going to give you here, based on the the simple premise of matter waves as originally conceived by De Broglie and put into mathematical form by Schroedinger. From this perspective there is no issue of spatial quantization or quantum leaps. Everything happens through a natural time evolution of the wave function.

The critical step in demystifying the physics is to start by realizing that the beam of silver atoms is not a geometrical ray, but rather a spread-out beam which we can think of as more like a pencil than a thread. When we treat it as a wave function, it is obvious that the portion of the beam closer to the pointy magnet – that is, in the strongest part of the field - must have a different phase velocity than the portion farther away. And anyone who has analyzed wavefronts passing through different media, such as glass with a variable index of refraction, knows what this means: the beam must curve. The baffling aspect of this is, of course: why does the curvature of the path follow exactly two trajectories? Why is it not infinitely variable between the two extremes of spin alignment? Specifically, why does an atom whose spin is aligned perpedicular to the field axis not pass through undisturbed?

All of these questions are answered when we understand that for electrons, any arbitrary spin can be represented as the superposition of two spins, which we call “up” and “down”. The method is straightforward application of wave mechanics: we represent an arbitrary spin as the superposition of our two eigenstates, and analyze each eigenstate separately according to the straightforward method of wavefronts and phase velocities. It is crucial to represent the beam as a pencil and not a geometrical ray, because only then do we see clearly how the beams curve. Each of the two cases has its own characteristic path. After analyzing them separately, we then combine them and calculate the superposition of those two paths. The result gives us the trajectory of the beam, and we will show that the beam is simply split in two.

Where exactly is the deep mystery in all this? Where does an atom, initially in a superposition of states, decide to arbitrarily make that quantum leap into one or the other final state? I will have more to say on this in a future post, but first I want to point out that the exact same thing happens with light, and nobody goes around saying that a photon which is polarized at 45 degrees suddenly decides that it must be either vertically or horizontally polarized. Which is exactly what they do say for the silver atom.

Consider the well-known case of iceland spar, the prototypical “birefringent crystal”. A beam of light shone through the crystal is split in two. The vertically polarized component is refracted to a different extent than the horizontally polarized component. The mechanism whereby this happens is well understood. The crystal structure is not cubically symmetric, so it is easier to polarize in one direction than the other. Light passing through this crystal will travel at different speeds whether it is polarized along the easy axis or the stiffer axis. Ordinary light, which is polarized randomly, will naturally divide into two paths.

This is exactly the same thing that happens to silver atoms in the Stern Gerlach experiment, but no one points to iceland spar and says that it proves there is a spatial quantization of polarization: that light can be either vertically or horizontally polarized, but nothing in between.  That is just typical of the nonsense that is spouted everywhere you turn with regard to quantum mechanics.