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Homework Help: Eigenvalues of Hamiltonian

  1. Sep 16, 2008 #1
    1. The problem statement, all variables and given/known data Consider two Ising spins coupled together
    −βH = h(σ1 + σ2) + Kσ1σ2,
    where σ1 and σ2 commute and each independently takes on the values ±1.
    What are the eigenvalues of this Hamiltonian? What are the degeneracies of the states?

    3. The attempt at a solution Four possible combinations for (σ1,σ2): (1,1), (1,-1), (-1,1) and (-1,-1).
    Therefore H=(-h/β)*(σ1 + σ2) + K/β*σ1σ2 can be written in a 2×2 matrix. And the eigenvalues λ are obtained by det(H-Eλ)=0.

    it follows: [(-2h/β)-(K/β)-λ)][(-2h/β)-(K/β)-λ)]-(2K/β)=0

    and so: λ1,2=-((2h-K)/β)±sqrt[(2h-K)22)-((2h-K)22-(2K/β)]

    and: λ1,2=-((2h-K)/β)±sqrt[2k/β]

    Are these really the eigenvalues of the hamiltonian? I dont gain any physical insight by this solution and therefore I doubt my calculation. I dont know how to go on and clculate the degeneracies of the states.

    Thanks in advance!
  2. jcsd
  3. Sep 18, 2008 #2
    You should use a 4x4 matrix for the Hamiltonian --- the system has 4 basis states (which you listed). Find the eigenvalues of that matrix.
  4. Sep 18, 2008 #3
    Really, a 4[tex]\times[/tex]4 matrix? If there are 4 base states, why can't I use a 2[tex]\times[/tex]2 matrix? 4 states fit into a 2[tex]\times[/tex]2 matrix, right? I tried this and calculated 2 eigenvalues, [tex]\lambda[/tex], but I am not sure, if the result is correct, since it "looks" to complicated (see my 1st reply)
  5. Sep 18, 2008 #4
    genneth is right, for 4 base states you need a 4x4 matrix.
    Consider for a moment a Hamiltonian for a single spin that can be +1 or -1. We need to know how the Hamiltonian operator acts on the particle if its spin is +1 and also how the Hamiltonian acts on the particle if its spin is -1. Thus, we need a basis state for each state of the particle.

    In your case you have two particles. So you have two particles with two states each 2*2 = 4 basis states. You need to know how the Hamiltonian acts on each individual configuration of spins and there are 4 possible configurations.

    Hope that helps
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