Thought experiment / paradox question

[NateTG]

Let's say I start out with a (spin) neutral electron beam. Now, I can place an up/down Stein-Gerlach spin filter/detector in the beam so that I have a running tally of the "up" electrons, and a spin-down exit beam.
Since the measuring device is accumulating spin-up electrons, it should also be accumulating net spin-up.
Now, let's say I have a terminal for the beam, which splits the beam into left and right spin electrons using a Stein-Gerlach device, and takes counts of both. Now, the terminal is interacting with (what I expect to be) a roughly equal number of spin-up and spin-down electrons, so there should be no net accumulation of spin in the terminal.
This seems problematic because the filter is accumulating net spin without conservation of spin. What am I missing?

 

[vanesch]

What am I missing?

The angular momentum in the magnetic field of the second apparatus.
For each individual |z+> -> |x+> transition, there is a transfer of Lz from the electron to the magnetic field ; and for each |z+> -> |x-> transition, there is an EQUAL transfer of Lz from the electron to the magnetic field.
Of course, for the |z+> -> |x+> transition, there was a positive contribution of Lx from the field to the electron, and for the |z+> -> |x-> transition, there was a negative contribution of Lx, so the Lx contribution averages out to 0, but not the Lz contribution.
There's a net flow of Lz momentum from the electron beam to the B-field (and hence to the magnet).
cheers,
Patrick.

 

[NateTG]

Sure, that makes sense, but doesn't that lead to a problem because you could know the spin state in two dimensions for a bunch of electrons?
Let's say we have an EPR-like set up with an entangled electron source, and an up-down split count target on one side, and a left-right split target on the other. And, using timing we manage to restrict our runs to getting 5 electrons at a time. Then there's a 1 in 1024 chane that all the hits on the up/down detector will be up, and all the hits on the left-right detector will be right. Thus we would 'know' that all of the electrons that hit the up/down detector would have to have been up/left electrons, but those measurements don't commute, and thus be able to measure (and predict) the net spin of the targets?