(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hi! I'm studying a computational physics course and have been ok so far but the newest problem relates to quantum mechanics (which I haven't studied) and I am struggling to understand... I would really appreciate it if someone could help me understand the physics side of things. The question relates to the energy spectrum of quantum spin chains.

The problem:

Write a program which sets up the Hamiltonian matrix for the Heisenberg model for

arbitrary chain size, N, and solve the matrix eigenvalue problem. Calculate the low

lying levels of the energy spectrum for various values of N. Compare your results for

the ground state energy density E0/N with the infinite chain result, E0/N = ?ln2+

1/4 (N ?infinity).

2. Relevant equations

The Hamiltonian of the Heisenberg model is given as:

H=[tex]\sum^{k=0}_{N-1}[H_{z}(k)+H_{f}(k)][/tex]

where

[tex]H_{z}(k)=S^{z}(k)S^{z}(k+1)[/tex]

[tex]H_{f}(k)=1/2[S^{+}(k)S^{-}(k+1)+S^{-}(k)S^{+}(k+1)][/tex]

The operation of the diagonal operator, [tex]H_{z}(k)[/tex], and the flipping operator, [tex]H_{f}(k)[/tex]on a

basis state gives:

[tex]H_{z}(k)|s^{z}(1)...s^{z}(k)s^{z}(k+1)...s^{z}(N)\right>\rangle =

s^{z}(k)s^{z}(k+1)|s^{z}(1)...s^{z}(k)s^{z}(k+1)...s^{z}(N)[/tex]

and

[tex]H_{f}(k)|s^{z}(1)...s^{z}(k)s^{z}(k+1)...s^{z}(N)\right>\rangle =

1/2|s^{z}(1)...-s^{z}(k)-s^{z}(k+1)...s^{z}(N) (for s^{z}(k)s^{z}(k+1)=-1)[/tex]

or

[tex]=0 (for s^{z}(k)s^{z}(k+1)=+1)[/tex]

3. The attempt at a solution

I'm completely lost. I understand what a Hamiltonian is, but no matter how much I read about the Heisenberg model I remain confused. Nothing outlines it simply and I have no idea how to set up a matrix with the equations above. Once I have the matrix I can easily program the row reduction but I really need some help to get my head around the concepts! Thankyou.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Understanding Heisenberg, Hamiltonians and Spin Chains

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