Partition function for generic spin state

I am studying statistical mechanics starting with the Gibbs state and the postulate of the partition function. I learned that the partition function is a sum over all the possible states of a system and applied it in a number of simple systems like spin 1/2, and other two or three level systems. But now I am stucked with a problem of calculating the partition function for a system of a generic spin state:

given that a spin state can be an integer or half-integer number like $$S = 1/2 , 1, 3/2, 2$$ etc and that $$s_z = S, S - 1, ... , -S + 1, -S$$ and the energy eigenvalues of this system in a magnetic field are given by:

' $$E_s = -hs$$ '

Since the eigenvalues are equally spaced and the ground state is $$s_z = S$$, how can I calculate the partition function?

I started in just writing the definition:

' Z = $$\sum_{s_z}^n \exp(-\beta E_s) = exp(h\beta S) + exp(h\beta (S-1)) + ... + exp(h\beta (-S+1)) + exp(h\beta (-S))$$ '

Is this right? If yes, I understood the "sum over all states" thing. But I do not know how to proceed in calculating this sum. If this is not right, then I do not even know how to start. One way of, maybe, simplifying this is adding a constant, since the it does not modify the final result. I can say that the ground state is of energy zero, then $$s_z = 0, -1, ... , 1, 0$$ and I have a more symmetric system with the exponentials. Can I do this?

1 Answer

Your expression for $$Z$$ looks correct. Notice that each term is $$e^{-h\beta}$$ times the previous term, so it is a geometric series, which you can easily sum.

• yes, you are right. I should have noticed it. Thank you :)) Commented Sep 16, 2019 at 21:26