Tagged Questions

2
votes
1answer
87 views

Accessible microstates of harmonic oscillator in microcanonical enemble

While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function ...
2
votes
1answer
154 views

Canonical partition of a boson gas

I have a 1D gas made of $N$ particles placed in a harmonic potential well, so the Hamiltonian is: $$ \mathcal H = \sum_{j=1}^N \left ( \frac{p_j^2}{2m} + \frac{1}{2}m\omega^2 x_j^2 \right )$$ The ...
2
votes
1answer
387 views

Helmholtz Free Energy, Partition Function

I'm trying to develop some basic intuition here, so this comes mostly as a jumble of commentary/questions. Hope its acceptable. Helmholtz Free Energy: $A = -{\beta ^{-1}}lnZ$. I find this statement ...
1
vote
1answer
596 views

Partition function of bosons vs fermions

I have two atoms, both of which are either bosons or fermions, with four allowed energy states: $E_1 = 0$, $E_2 = E$, $E_3 = 2E$, with degeneracies 1, 1, 2 respectively. What's the difference between ...
0
votes
1answer
215 views

Expressions for canonical partition function and probabilities $p(E_i)$

Given an atom with 4 allowed states corresponding to the energy levels $E_1 = 0$, $E_2 = E$, and $E_3 = 2E$ with degeneracies 1, 1, and 2 respectively. How do I find the expressions for the ...
4
votes
1answer
129 views

How do I calculate the probability that the oscillator is in a certain state using partition function?

So let's say I have a single ($N=1$) quantum harmonic oscillator and the energy is determined by $E_n = (n + 1/2) \cdot \hbar \omega$ (where $n$ is the quantum number and $n$ = $0, 1, 2, \ldots$) ...
2
votes
1answer
368 views

Calculating partition function of ultra-relativistic 1D gas

This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria. In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) ...
4
votes
1answer
210 views

Integration of partition-function over many momentum variables

My integral looks like $$Z(\beta) = \frac{1}{h^3}\int d^3p\ \exp{\left(-\frac{\beta}{2m}\sum^{3N}_{i=1}p_i^2\right)}.$$ I'm confused about how to integrate over seemingly 3N variables in only a ...
2
votes
1answer
165 views

Partition function of an interacting gas

By reading an article, I found a partition function that, according to the author, describes an interacting with random variables as coupling constant. $$Z =\int \mathrm{d} \lambda_i ...
7
votes
0answers
99 views

Measure of Lee-Yang zeros

Consider a statistical mechanical system (say the 1D Ising model) on a finite lattice of size $N$, and call the corresponding partition function (as a function of, say, real temperature and real ...
1
vote
1answer
129 views

Quantum Stat-Mech Proof of an Inequality for the Partition Function

I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish. I know I can't get this since it's already been ...
2
votes
4answers
606 views

What does third law of thermodynamics tell us?

I just have a question concerning the third law of thermodynamics. The third law describes that the entropy should be a well defined constant if the system reaches the ground state which depends ...
3
votes
1answer
124 views

From spectrum/dispersion relation to the partition function

I know the spectrum/dispersion relation for a bosonic system. $$E \left( \mathbf{k} \right) = \cdots$$ Is there a general method for writing down the partition function when the spectrum of the ...