A particle of mass $\mu$ moves in two dimensions in an axisymmetric potential. Show that the time-independent Schrödinger equation can be separated in polar coordinates. Show that the angular part of the wave function has the form $e^{i m \phi}$, where $\phi$ is the angular coordinate and $m$ is an integer. Suppose that the potential is zero for $r<a$, where $r$ is the radial coordinate, and infinite otherwise. Show that the radial part of the wave function satisfies

$$\frac{d^{2} R}{d \rho^{2}}+\frac{1}{\rho} \frac{d R}{d \rho}+\left(1-\frac{m^{2}}{\rho^{2}}\right) R=0$$

where $\rho=r\left(2 \mu E / \hbar^{2}\right)^{1 / 2}$. What conditions must $R$ satisfy at $r=0$ and $R=a$ ?

Show that, when $m=0$, the equation has the solution $R(\rho)=\sum_{k=0}^{\infty} A_{k} \rho^{k}$, where $A_{k}=0$ if $k$ is odd and

$$A_{k+2}=-\frac{A_{k}}{(k+2)^{2}}$$

if $k$ is even

Deduce the coefficients $A_{2}$ and $A_{4}$ in terms of $A_{0}$. By truncating the series expansion at order $\rho^{4}$, estimate the smallest value of $\rho$ at which the $R$ is zero. Hence give an estimate of the ground state energy.

[You may use the fact that the Laplace operator is given in polar coordinates by the expression

$$\left.\nabla^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \phi^{2}}\right]$$