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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 Schrodinger Equation yields two equations (By letting |ψ>=(ψ1,ψ2))

Eψ1=-μBψ2

Eψ2=-μBψ1

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

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 metwoenergy 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

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

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