(a) Given $\int_{-\infty}^{+\infty} \exp \left(-\alpha x^2\right) d x=\sqrt{\pi / \alpha}$, show that
$$
\int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} d x=\frac{\sqrt{\pi}}{2} \alpha^{-3 / 2}, \quad \int_{-\infty}^{\infty} x^4 e^{-\alpha x^2} d x=\frac{3}{4} \sqrt{\pi} \alpha^{-5 / 2} .
$$
(b) Given that $\frac{\beta}{\sqrt{\alpha}} \ll 1$ and that $a$ is of the order of $\frac{1}{\sqrt{\alpha}}$, show
$$
\int_{-a}^a f(x) e^{-\alpha\left[x^2+\beta x^3 \mid\right.} d x \approx \int_{-a}^a f(x)\left(1-\alpha \beta x^3\right) e^{-\alpha x^2} d x .
$$
(c) Two atoms interact through a potential
$$
U(x)=U_0\left[\left(\frac{a}{x}\right)^{12}-2\left(\frac{a}{x}\right)^6\right],
$$
where $x$ is their separation. Sketch this potential. Calculate the value of $x$ for which $U(x)$ is minimum.
(d) Given a row of such atoms constrained to move only on the $x$ axis, each assumed to interact only with its nearest neighbors, use classical statistical mechanics to calculate the mean interatomic separation $\bar{x}(T)$.
To do this, expand $U$ about its minimum, keeping as many terms as necessary to obtain the lowest order temperature dependence of $\bar{x}(T)$. Assume that $k T \ll U_0$, and in the relevant integrals extend the limits of integration to $\pm \infty$ where appropriate. Explain clearly the justification for extending the limits. Also calculate
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