[Physics] Summation to Integration in Statistical Mechanics

statistical mechanics

In Statistical Mechanics, what is the procedure of replacing this summation by the integration given by $$\sum_{\vec k} \rightarrow \frac{V}{(2\pi)^3} \int_{0}^{\infty} 4\pi k^2 dk$$

Best Answer

The answer to your question is already contained inside this answer. Without reiterating the same answer again, essentially there is a conversion from Cartesian to spherical coordinates:

\begin{align} \int\int\int\rho(k)dk_xdk_ydk_z &= \int^k_0\int^{2\pi}_{0}\int^{\pi}_{0}\rho(k)k^2\sin\theta d\theta d\phi dk\\ &= \frac{V}{(2\pi)^3}\int^k_04\pi k^2dk \end{align}

where $k$ is the "radius" of the spherical coordinate system.

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