Consider a quantum mechanical particle of mass $m$ in a one-dimensional stepped potential well $U(x)$ given by:

$$U(x)= \begin{cases}\infty & \text { for } x<0 \text { and } x>a \\ 0 & \text { for } 0 \leqslant x \leqslant a / 2 \\ U_{0} & \text { for } a / 2<x \leqslant a\end{cases}$$

where $a>0$ and $U_{0} \geqslant 0$ are constants.

(i) Show that all energy levels $E$ of the particle are non-negative. Show that any level $E$ with $0<E<U_{0}$ satisfies

$$\frac{1}{k} \tan \frac{k a}{2}=-\frac{1}{l} \tanh \frac{l a}{2}$$

where

$$k=\sqrt{\frac{2 m E}{\hbar^{2}}}>0 \quad \text { and } \quad l=\sqrt{\frac{2 m\left(U_{0}-E\right)}{\hbar^{2}}}>0$$

(ii) Suppose that initially $U_{0}=0$ and the particle is in the ground state of the potential well. $U_{0}$ is then changed to a value $U_{0}>0$ (while the particle's wavefunction stays the same) and the energy of the particle is measured. For $0<E<U_{0}$, give an expression in terms of $E$ for prob $(E)$, the probability that the energy measurement will find the particle having energy $E$. The expression may be left in terms of integrals that you need not evaluate.