A particle of mass $m$ moves in a one-dimensional potential defined by

$$V(x)= \begin{cases}\infty & \text { for } x<0 \\ 0 & \text { for } 0 \leqslant x \leqslant a \\ V_{0} & \text { for } a<x\end{cases}$$

where $a$ and $V_{0}$ are positive constants. Defining $c=\left[2 m\left(V_{0}-E\right)\right]^{1 / 2} / \hbar$ and $k=$ $(2 m E)^{1 / 2} / \hbar$, show that for any allowed positive value $E$ of the energy with $E<V_{0}$ then

$$c+k \cot k a=0$$

Find the minimum value of $V_{0}$ for this equation to have a solution.

Find the normalized wave function for the particle. Write down an expression for the expectation value of $x$ in terms of two integrals, which you need not evaluate. Given that

$$\langle x\rangle=\frac{1}{2 k}(k a-\tan k a),$$

discuss briefly the possibility of $\langle x\rangle$ being greater than $a$. [Hint: consider the graph of - ka cot $k a$ against $k a .]$