Stern-Gerlach, polarized atom beam.

[Telemachus]

Homework Statement

In a Stern-Gerlach device, an atom beam with angular impulse J, travels through a magnetic field applied normally to the trajectory. The beam is separated in 2j+1 beams in general. Find the relative intensities for these beams if J=1 and if the beam is polarized with jθ=1 in a direction that forms an angle θ with the direction of the magnetic field.

Homework Equations

The operator
J \cdot n_{\theta}=J_{\theta}=J_x\sin\theta \cos\phi+J_y \sin\theta\sin\phi+J_z\cos\phi

The Attempt at a Solution

I'm not sure what I'm supposed to do. I think I should find the eigenvalues for the given operator, and then determine the probability of measuring each eigenvalue, projecting over each possible eigenstate. I need some guidance with this.

 

[TSny]

I think you have the right idea. Since you are dealing with spin 1 particles ($j = 1$), the possible eigenvalues for a measurement of the component of spin along any chosen direction in space are m\hbar where $m$ is an integer.

What are the possible values of $m$?

The corresponding eigenstates may be written $|j,m>$ where $j = 1$ and $m$ can be any of the possible values of $m$.

If you use primes to denote that you are considering spin components along the specific direction n_θ given in the problem, what is the value of $m'$ for the particles before they enter the B field? That is, what is the value of $m'$ for the initial state $|j,m'>$ ?

 

[Telemachus]

The possible values for m are $m=-1,0,1$.

I think that the value of $m′$ should be the eigenvalue for the projection of the eigenstates in the n_θ direction. So I think I should apply the given operator over the $|j,m>$, and those eigenvalues would give the $m′$, am I right?

Thank you very much TSny.

 

[TSny]

If I'm understanding the setup, the initial state of the particles is $j = 1$ and $m' = 1$ along n_θ: $|j, m'=1>$

If you let $m$ without a prime denote possible eigenstates having definite components of spin along the magnetic field direction, then the three output beams will correspond to the three states $|j, m>$ where $m$ = 1, 0, and -1. You can think of these three states as basis states for expressing any spin state. In particular, it should be possible to expand the input state $|j, m'=1>$ ("along n_θ") as a superposition of the states $|j, m>$ ("along B").

The easiest way to find the coefficients of the expansion is to consult a standard QM text that discusses rotations of quantum states. The $m'$ states are related to the $m$ states by a rotation.

 

[Telemachus]

You get the $m´=1$ from the given value of j_θ=1?

I think I get the setup, I saw it in Cohen, where this kind of configuration is discussed. There are two Stern Garlech apparatus, one giving the atoms in a given state, and the second one measuring the intensity I think. But I don't know how to work it out.

I'm sorry to insist with this, I'm having some trouble with QM.

 

[TSny]

I’m not sure of the meaning of the notation j_θ = 1. I assume that this means that the initial state is $|j = 1, m' = 1> \equiv |1, 1'>$. If not, then I don’t understand the setup.

You can expand $|1, 1'>$ in terms of the three eigenstates of spin along the B-field direction $|j = 1, m> \equiv |1,m>$ for m = 1, 0, -1. Thus, there exists constants $C_1, C_2, C_3$ such that

$|1, 1'>\; = \;C_1|1, 1> +\; C_2|1, 0> +\;C_3|1, -1>$

You can find tables containing the values of the coefficients. For example you can find these constants along with an outline of their derivation on the last 3 pages of

http://www.hep.phy.cam.ac.uk/~thomson/lectures/partIIIparticles/Handout4_2009.pdf \; [Don't panic! Skip quickly to the last three pages. Note that $|\psi>$ is used for what we have called $|1, 1'>\;$. See in particular the last equation on page 146.]

From the values of the $C$ constants you can calculate the relative intensities of the three beams exiting the apparatus.

 

[Telemachus]

Thanks, I think I got it.