Homework StatementConsider 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?
The Attempt at a SolutionFour 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)2/β2)-((2h-K)2/β2-(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!