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Eigenvalues of Hamiltonian

by Krischi
Tags: eigenvalues, hamiltonian
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Krischi
#1
Sep16-08, 10:17 AM
P: 3
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 ħ1.
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)ħsqrt[(2h-K)22)-((2h-K)22-(2K/β)]

and: λ1,2=-((2h-K)/β)ħsqrt[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
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genneth
#2
Sep18-08, 10:24 AM
P: 981
You should use a 4x4 matrix for the Hamiltonian --- the system has 4 basis states (which you listed). Find the eigenvalues of that matrix.
Krischi
#3
Sep18-08, 10:35 AM
P: 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)

yaychemistry
#4
Sep18-08, 11:54 AM
P: 35
Eigenvalues of Hamiltonian

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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