Eigenvalues of Hamiltonian

  • Thread starter Krischi
  • Start date
  • #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: 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!
Krischi
 

Answers and Replies

  • #2
980
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.
 
  • #3
3
0
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)
 
  • #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
 

Related Threads on Eigenvalues of Hamiltonian

  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
5
Views
6K
  • Last Post
Replies
1
Views
663
  • Last Post
Replies
7
Views
2K
Replies
7
Views
7K
Replies
6
Views
5K
  • Last Post
Replies
1
Views
1K
Replies
0
Views
1K
Replies
4
Views
2K
Replies
5
Views
1K
Top