Undergrad Help with partition function calculation

[hammer123]

Hi, would very much appreciate it if I could get some help for something I was trying to calculate. I'm not very good at latex so I've just attached images of my attempt. I would very much appreciate it if you could look over the images I've taken and provide some feedback, thank you.

I've been trying to calculate the partition function for spin-1/2 particles in a magnetic field, however instead of summing over the eigenstates of the spin operator in the direction of the field, I've summed over all possible normalized states.

Which leads me to my first question, is this allowed? And if not, why not?

My second question becomes apparent if you take a quick look at the images I've attached (end of part 2). Essentially I found that the average spin derived from the partition function did converge to +1/2 and -1/2 as the temperature went to zero, however I got a non-zero value as the temperature went to infinity.

I'm sure this indicates that something went wrong with my calculations, but I'm not sure if it's because I made a mistake or that the entire idea was invalid.

 

[PeterDonis]

I'm not very good at latex so I've just attached images of my attempt.

Unfortunately, that is against the forum rules. You need to put your equations directly into your post, so that they can be quoted by people who want to respond to you. Help with LaTeX can be found here:

https://www.physicsforums.com/help/latexhelp/

I have closed this thread in view of the above. Please re-post your question in a new thread using LaTeX.