Partition Function

QUESTION:

A particle moving in one dimensions has energy

E=\frac{p^{2}}{2m}+\lambda x^{4} 

where p and q denote the generalized coordinate and momentum.

Show that heat capacity of a gas comprising of N such particles is

C_{v}=\frac{3NK_{B}}{4}

SOLUTION:

The single particle partition function can be written as

Z=\frac{1}{h}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\left[-\beta\left(\frac{p^{2}}{2m}+\lambda x^{4}\right)\right]dx\cdot dp

=\frac{1}{h}\int_{-\infty}^{\infty}\exp\left[-\beta\frac{p^{2}}{2m}\right]dx\int_{-\infty}^{\infty}\exp\left[-\beta\lambda x^{4}\right]dx

Z=\frac{1}{h}\sqrt{\frac{2m\pi}{\beta}}\cdot2\cdot\frac{1}{4}\left(\lambda\beta\right)^{-\frac{1}{4}}\Gamma\left(\frac{1}{4}\right)

\left\langle E\right\rangle =-\frac{\partial\ln Z}{\partial\beta}=\frac{3}{4\beta}=\frac{3K_{B}T}{4}

\left\langle E\right\rangle _{N}=\frac{3K_{B}NT}{4}

Now,

C_{v}=\frac{\partial\left\langle E\right\rangle _{N}}{\partial T}=\frac{3NK_{B}}{4}

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Two Useful Integration:

\int_{0}^{\infty}e^{-ax^{b}}dx=\frac{1}{b}a^{-\frac{1}{b}}\Gamma\left(\frac{1}{b}\right)

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\int_{-\infty}^{\infty}e^{-ax^{2}}dx=\sqrt{\frac{\pi}{a}}, a>0