- #1
Dazed&Confused
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Homework Statement
If the energy ##E## of a system behaves like ## E = \alpha |x|^n##, where ## n =1, 2, 3, \dots ## and ## \alpha > 0##, show that ## \langle E \rangle = \xi k_B T ##, where ##\xi## is a numerical constant.
Homework Equations
$$ \langle E \rangle = \frac{ \int_{- \infty}^{ \infty} Ee^{-\beta E}}{\int_{-\infty}^{ \infty} e^{-\beta E}},$$ where ## \beta = \frac{1}{k_BT}.##
The Attempt at a Solution
Since the integral is even, it can be written as $$\frac{ \int_{0}^{ \infty} \alpha x^n e^{-\beta \alpha x^n}}{\int_{0}^{ \infty} e^{-\beta \alpha x^n}},$$
It can also be written as
$$ \frac{ \frac{ \partial}{ \partial \beta} \left ( \int_{0}^{ \infty} - e^{-\beta \alpha x^n} \right ) } {\int_{0}^{ \infty} e^{-\beta \alpha x^n}}$$
where the partial derivative was taken outside the integral.
I have no idea how to solve the integral. Mathematica didn't draw up anything useful.