How Does Spin Orientation Affect Beam Intensity in a Stern-Gerlach Experiment?

[Rory9]

Homework Statement

Consider a beam of spin-(1/2) particles passed through a Stern-Gerlach apparatus (+ve z orientation).

The particles are in the ground-state of the Hamiltonian

$$H = \frac{g\mu_{B}\hbar}{2} \left( ^{B_{z}}_{ B_{x} + iB_{y} }^{ B_{x} - iB_{y}}_{ - B_{z} } \right)$$

and therefore aligned with the field B. For an incident beam intensity of 1, calculate the intensity of the transmitted beam(s).

The Attempt at a Solution

I'm honestly not sure what to do with this. I can compute the eigenvectors of the Hamiltonian (Bz + B,Bx + iBy) & (Bz - B,Bx + iBy), and deduce the ground state from the one whose eigenvalues gives the lowest energy (I presume), but what then?

I'd very much appreciate some helpful pointers! Thanks in advance.

(Also, perhaps there are some relevant examples somewhere online for this sort of problem?)

 

[Jhero]

Hello,

Thank you for your post. It seems like you are on the right track with computing the eigenvectors of the Hamiltonian. Once you have determined the ground state, you can use the intensity formula for a beam of particles to calculate the intensity of the transmitted beam(s). This formula is given by:

I = N * P * A

Where:
I = Intensity
N = Number of particles
P = Probability of particles being transmitted
A = Area of the beam

In this case, N = 1, since the incident beam has an intensity of 1. P can be calculated by taking the square of the absolute value of the projection of the ground state onto the eigenvector corresponding to the transmitted beam. Finally, A can be calculated using the dimensions of the Stern-Gerlach apparatus.

I hope this helps! Let me know if you have any further questions. Good luck with your calculations.