The effect of the magnet in a Stern-Gerlach experiment

[Fredrik]

A beam of silver atoms (which are electrically neutral spin-1/2 particles) enters an inhomogeneous magnetic field, and is split in two.

The state of an atom that has passed the magnet is often described as |U>|↓>+|L>|↑>, where |U> and |L> are states that are localized to the upper and lower paths respectively, and |↓> and |↑> are the "spin down" and "spin up" states respectively. I have realized that I don't really understand how to justify this. Can we prove that each atom will end up in a |U>|↓>+|L>|↑> state?

 

[Fredrik]

Bump.

One idea that occurs to me is to take the wavefunction to be of the exp(-a(x-vt)²) form for x<0, and the potential to be 0 for x<0 and x>1, and $$=\vec\mu\cdot\vec B$$ when 0<x<1, but then I don't see how we can impose a boundary condition at x=1. Do we need one? Is this problem worked out in any of the standard textbooks?

The detail I'm the most interested in is why the wavefunction changes from having one peak to having two peaks, instead of just spreading out. Is this a result of some sort of decoherence in which the magnet serves as an "environment", or can it be derived from the Schrödinger equation alone?

 

[zenith8]

Hi Fredrik,

If you search through the archives, you'll find a thread - not too long ago - where you said you were going to order Peter Holland's "Quantum Theory of Motion" textbook - which is about the deBB perspective of these things. Did it ever arrive, I wonder? Anyway, there's a whole bunch of stuff about the SG experiment in there - though if you haven't got the physical book the crucial pages on Google Books are blocked, sadly.

Hey, it's even a demonstration on the Wolfram site:

http://demonstrations.wolfram.com/TheCausalInterpretationOfTheSternGerlachExperiment/"

Since the Schroedinger equation is about spinless particles, you do need to go the level of the Pauli equation or whatever, obviously.

Cheers,
Zenith

 

[Fredrik]

Thank you. It's been on my bookshelf for a long time, but I haven't read it yet. Not even a single page I'm afraid. I'm so dumb that it's taking me a very long time to learn functional analysis and a few other things that I've been giving a higher priority. I'll check it out.