# What Is the Correct Partition Function for a Spin System?

• happyparticle
In summary, the conversation discusses a mathematical equation involving sums and a geometric series. The final equation involves two finite geometric series and the individual parts are discussed. A helpful tip is provided to assist in solving the equation.
happyparticle
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.

Note that you've included ##n=0## twice!

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})##.

vanhees71
Thank you! I didn't see that I had included n=0 twice.
I spent hours trying to figure out what was wrong.

## 1. What is the partition function for a spin i?

The partition function for a spin i is a mathematical function that describes the statistical behavior of a system of particles with spin i. It is used to calculate the thermodynamic properties of the system, such as energy, entropy, and free energy.

## 2. How is the partition function for a spin i calculated?

The partition function for a spin i is calculated by summing over all possible states of the system, each weighted by the Boltzmann factor e^(-E/kT), where E is the energy of the state, k is the Boltzmann constant, and T is the temperature of the system.

## 3. What is the significance of the partition function for a spin i?

The partition function for a spin i is significant because it allows us to calculate the thermodynamic properties of a system with spin i. It is a fundamental concept in statistical mechanics and is used in various areas of physics, chemistry, and biology.

## 4. Can the partition function for a spin i be used for systems with multiple spins?

Yes, the partition function can be extended to systems with multiple spins by taking into account the spin states of each particle in the system. This is known as the multi-spin partition function and is used to study more complex systems.

## 5. How does the partition function for a spin i relate to the spin statistics of a system?

The partition function for a spin i is closely related to the spin statistics of a system. In quantum mechanics, particles with half-integer spin follow Fermi-Dirac statistics, while particles with integer spin follow Bose-Einstein statistics. The partition function takes into account these statistics when calculating the thermodynamic properties of the system.

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