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 1


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.

  • $\begingroup$ yes, you are right. I should have noticed it. Thank you :)) $\endgroup$
    – Dimitri
    Commented Sep 16, 2019 at 21:26

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