Troubleshooting the Partition Function for a System with 3 Spins

[annaphys]

Homework Statement

Consider 3 spins s1,s2,s3 on a solid lattice. The spins can have the values -1 or +1. Find the partition function.

Relevant Equations

H = J1*s1*s2 + J2*s2*s3. J1,J2 >0

I'm having problems solving the partition function. I've attached a photo of where I currently am. Am I on the right track? What should be my next step?

 

[HallsofIvy]

I'm not an expert in this field but just on symmetry considerations, I would expect there to be a "$j_3s_1s_3$" term in it.

 

[annaphys]

The hamiltonian is given.

 

[annaphys]

Solved it. Thought about it too hard. Just need to work out the sums and simplify. One gets then two cosh() functions.