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

  1. Nov 26, 2014 #1
    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 Schrodinger 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
     
  2. jcsd
  3. Nov 26, 2014 #2

    Avodyne

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    Science Advisor

    You have made a mistake. Your equations lead to ##E^2=(\mu B)^2##, which has two solutions.
     
  4. Nov 26, 2014 #3
    Ah man such a small algebra error! That makes everything work out perfectly, duh! Thank you so much!
     
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