What is the meaning of the pauli matrices in the Hamiltonian summation?

[aaaa202]

For an exercise I am given the attached Hamiltonian, but I don't understand it completely. We sum over spin -½ and ½ and the paulimatrices seem to be dependent on this since they are labeled by σσ'. What does this mean? I mean the pauli matrices are just operators for the spin in the x,y,z-direction, so what does it exactly have to do with our summation of spin eigenvalues.

 

[tiny-tim]

hi aaaa202!

… We sum over spin -½ and ½ and the paulimatrices seem to be dependent on this since they are labeled by σσ'. What does this mean? I mean the pauli matrices are just operators for the spin in the x,y,z-direction, so what does it exactly have to do with our summation of spin eigenvalues.

i'm not sure what you're asking

each pauli matrix, here written as τ¹ τ² and τ³, is a 2x2 matrix, so it has four entries, labelled ½½ ½-½ -½½ and -½-½

each entry of τⁱ is written τⁱ_σσ'

 

[aaaa202]

So in each component the spin operator S=(Sx,Sy,Sz) I add the entries of the corresponding pauli matrix?

 

[tiny-tim]

i don't understand

you add over every index (k k' σ and σ') in every ∑

(x y and z are not relevant indices for the spinors: those are ½ and -½)

 

[aaaa202]

Okay I misunderstood. I understand now - thanks :)

 

[aaaa202]

Hi again tiny Tim, maybe you can help me further with a problem concerning this Hamiltonian. I am supposed to perform a thermal average with respect to H0 of HS1*HS2
Now to perform a thermal average one simply just sums over:
<Eil HS1HS2 lEi> , where Ei are eigenstates of H0.
But what is the action of the product HS1HS2 on an eigenstate of H0?
I can understand the electron operators, but how do I interpret ∑Si¹∑Si² on an eigenstate of H0?
S1 and S2 simply just represent 2 different spin vectors. How do I use this on a single electron gas eigenstate (H0 describes an electron gas).

 

[tiny-tim]

hi again aaaa202!

sorry, can you ask this in a separate thread?