- #1

- 3

- 0

## Homework Statement

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?

## 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: λ

_{1,2}=-((2h-K)/β)±sqrt[(2h-K)

^{2}/β

^{2})-((2h-K)

^{2}/β

^{2}-(2K/β)]

and: λ

_{1,2}=-((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!

Krischi