What Is the Correct Partition Function for a Spin System?

AI Thread Summary
The discussion centers on the correct formulation of the partition function for a spin system, with participants analyzing different summation approaches. One participant presents a formula involving finite geometric series, while another points out an error in including the term for n=0 twice. There is confusion regarding the derivation of the hyperbolic sine function from the proposed equations. A helpful mathematical identity is shared to clarify the situation. The conversation highlights the complexities involved in accurately calculating the partition function in statistical mechanics.
happyparticle
Messages
490
Reaction score
24
Homework Statement
Consider a spin i, integer or half integer with the states n = -i, -(i-1),..(i-1), i
Z component of the spin is ##S_z = n\hbar## and the energy eigenvalues of this system in a magnetic field are given by: ##E_n = nh##
Find the partition function in term of 2 sinh ratio
Relevant Equations
##Z = \sum_{-i}^{i} = e^{-E_n \beta}##
##Z = \sum_{-i}^{i} = e^{-E_n \beta}##

##Z = \sum_{0}^j e^{nh\beta} + \sum_{0}^j e^{-nh\beta}##
Those sums are 2 finites geometric series
##Z = \frac{1- e^{h\beta(i+1)}}{1-e^{h\beta}} + \frac{1-e^{-h\beta(i+1)}}{1-e^{-h\beta}}##
I don't think this is ring since from that I can't get 2 sinh. However, I'm not sure where is my error.
 
Physics news on Phys.org
Note that you've included ##n=0## twice!
 
  • Like
Likes happyparticle and Orodruin
happyparticle said:
I don't think this is ring since from that I can't get 2 sinh. However, I'm not sure where is my error.
This might help: ##1-e^x = e^{x/2}(e^{-x/2}-e^{x/2})##.
 
Thank you! I didn't see that I had included n=0 twice.
I spent hours trying to figure out what was wrong.
 
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'
(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...
Back
Top