Why no smear in Stern-Gerlach experiment?

[Alfred Cann]

Why do the silver atoms not exist in a superposition of states with every possible mixture of spin-up and spin-down? Thermal photons do.

 

[Vanadium 50]

Because they are sent through a spin analyzer which forces them into eignestates of Jz.

 

[Alfred Cann]

What is the spin analyzer, the non-uniform magnetic field? How does that force a selection into eigenstates? If an atom has, for example, zero net spin because of superposition, wouldn't it go straight?

 

[Nugatory]

What is the spin analyzer, the non-uniform magnetic field? How does that force a selection into eigenstates? If an atom has, for example, zero net spin because of superposition, wouldn't it go straight?

The equal superposition of spin-up and spin-down is not a state of zero net spin (in fact the spin has no value at all until we perform a measurement - it's like asking whether a tossed coin is heads-up or tails-up before you've tossed it). Instead, it is a state in which an interaction that depends on the spin has a 50% chance of coming out as if the spin is up and a 50% chance of coming out as if the spin is down. We call these interactions "measurements" or "observations" Of the spin because when we look at the result we know what the spin is now that the interaction has happened. The interaction with the non-uniform magnetic field is such an interaction.

In the original formulation of quantum mechanics (the "Copenhagen interpretation") we would say that the wave function was a superposition of spin-up and spin-down until it encountered the non-uniform magnetic field; and the interaction with the magnetic field caused the wave function to "collapse" into one of the two non-superimposed possibilities.

One of the postulates of quantum mechanics is that the result of a measurement must be one of the eigenvalues of the observable we're measuring (spin-up and spin-down in this case), and that immediately after the measurement the system will be in the corresponding eigenstate.

 

[vanhees71]

The Stern-Gerlach apparatus does not make pure states of determined $J_z$ out of a beam of thermal (i.e., unpolarized) particles but it creates a state where $J_z$ is entangled with position. With an ideal Stern-Gerlach apparatus you can then make a beam of a (nearly) pure eigenstate with determined $J_z$ by blocking out all other partial beams with a different $J_z$. No collapse esoterics is needed just some material blocking the particles, you don't want to consider further ;-)).

 

[Alfred Cann]

I think I've got a partial understanding. The SG apparatus can only detect spin-up and spin-down. It is blind to all other states. If an atom comes in with, say, spin right, the apparatus will decompose that into its spin-up and -down components, each with probability amplitude of 0.707, and display those. This is quite counter-intuitive.
The photon case is easier to understand; a photon either gets thru a polarizer or not; you can't get 60% of a photon. If a photon goes thru a vertical polarizer, all we know is that it wasn't horizontally polarized. Its polarization could have been linear at any other angle, or elliptical of either handedness with any degree of ellipticity at any angle, which meant it had a vertical component, which, in turn meant it had some probability of going thru the polarizer. Oh, and after the polarizer we know it's vertically polarized.

 

[Nugatory]

The photon case is easier to understand; a photon either gets thru a polarizer or not

How so? The difference between the two cases is that either the photon goes through or it doesn't. We could produce the same effect with the S-G machine just by blocking one or the other paths out so that we can describe the situation as "either the atom gets through or not". You can't get 60% of a photon, but you also (even more forcefully) can't get 60% of an atom.

 

[TSny]

A birefringent crystal like calcite might be a good analog for photons of the SG apparatus for silver atoms.

http://fineartamerica.com/featured/birefringence-in-a-calcite-crystal-dirk-wiersma.html

https://en.wikipedia.org/wiki/Birefringence#/media/File:Positively_birefringent_material.svg

https://en.wikipedia.org/wiki/Birefringence