Statistical Mechanics: Canonical Partition Function & Anharmonic Oscillator

[binbagsss]

Homework Statement

With the Hamiltonian here:

Compute the cananonical ensemble partition function given by $\frac{1}{h} \int dq dp \exp^{-\beta(H(p,q)}$

for 1-d , where $h$ is planks constant

Homework Equations

The Attempt at a Solution

I am okay for the $p^2/2m$ term and the $aq^2$ term via a simple change of variables and using the gaussian integral result $\int e^{-x^2} dx = \sqrt{\pi}$

I am stuck on the $\int dq e^{\beta b q^{3}}$ and $\int dq e^{\beta c q^{3}}$ terms.

If these were of the form $\int dq e^{-\beta b q^{3}}$ I could evaluate via $\int dx e^{-x^n} = \frac{1}{n} \Gamma (1/n)$ where $\Gamma(1/n)$ is the gamma function;

however because it is a plus sign I have no idea how to integrate forms of $\int dq e^{x^n}$

Or should I be considering the integral over $q$ all together and there is another way to simply:

$\int dq e^{-\beta(aq^2-bq^3-cq^4)}$

Many thanks in advance

 

[SammyS]

Homework Statement

With the Hamiltonian here:

View attachment 204133

Compute the canonical ensemble partition function given by $\frac{1}{h} \int dq dp \exp^{-\beta(H(p,q))}$

for 1-d , where $h$ is planks constant

Homework Equations

The Attempt at a Solution

I am okay for the $p^2/2m$ term and the $aq^2$ term via a simple change of variables and using the gaussian integral result $\int e^{-x^2} dx = \sqrt{\pi}$

I am stuck on the $\int dq e^{\beta b q^{3}}$ and $\int dq e^{\beta c q^{3}}$ terms.

If these were of the form $\int dq e^{-\beta b q^{3}}$ I could evaluate via $\int dx e^{-x^n} = \frac{1}{n} \Gamma (1/n)$ where $\Gamma(1/n)$ is the gamma function;

however because it is a plus sign I have no idea how to integrate forms of $\int dq e^{x^n}$

Or should I be considering the integral over $q$ all together and there is another way to simply:

$\int dq e^{-\beta(aq^2-bq^3-cq^4)}$

Many thanks in advance

I suspect this thread would get better help in the "Advanced Physics Homework" section.

I'm no expert in this field, but it seems to me that you need to include distribution information ?

 

[stevendaryl]

I don't think that the integral can be evaluated exactly in terms of standard functions. Are you sure you aren't asked to come up with an approximate value?