Density Matrix for Spin 1/2 particle in a magnetic field

[khfrekek92]

Hi everyone!

I am trying to create the density matrix for a spin-1/2 particle that is in thermal equilibrium at temperature T, and in a constant magnetic field oriented in the x-direction. This is a fairly straightforward process, but I'm getting stuck on one little part.

Before starting I need to find the energy eigenvalues (In order to find the partition function):

H=-μS⋅B=-μBσ_x

But since σ_x is an off-diagonal matrix (unlike σ_z), plugging this Hamiltonian into the Schrödinger Equation yields two equations (By letting |ψ>=(ψ1,ψ2))

Eψ1=-μBψ2
Eψ2=-μBψ1

And then solving these like normal for E gives us only one energy eigenstate for this system (with degeneracy 2):

E=μB

However, when the magnetic field was in the z direction, the z pauli spin matrix was diagonal and didn't switch the positions of ψ1 and ψ2, which gave me two energy eigenstates (±μB).

So my question is, why would a magnetic field in the x-direction NOT break the degeneracy of the energy eigenstates, while in the z-direction it does? These directions are completely arbitrary and should yield the same results, right?

Thanks

 

[Avodyne]

You have made a mistake. Your equations lead to $E^2=(\mu B)^2$, which has two solutions.

 

[khfrekek92]

Ah man such a small algebra error! That makes everything work out perfectly, duh! Thank you so much!