(i) Consider a particle of mass $m$ confined to a one-dimensional potential well of depth $U>0$ and potential

$$V(x)=\left\{\begin{array}{cl} -U, & |x|<l \\ 0, & |x|>l \end{array}\right.$$

If the particle has energy $E$ where $-U \leqslant E<0$, show that for even states

$$\alpha \tan \alpha l=\beta$$

where $\alpha=\left[\frac{2 m}{\hbar^{2}}(U+E)\right]^{1 / 2}$ and $\beta=\left[-\frac{2 m}{\hbar^{2}} E\right]^{1 / 2}$.

(ii) A particle of mass $m$ that is incident from the left scatters off a one-dimensional potential given by

$$V(x)=k \delta(x)$$

where $\delta(x)$ is the Dirac delta. If the particle has energy $E>0$ and $k>0$, obtain the reflection and transmission coefficients $R$ and $T$, respectively. Confirm that $R+T=1$.

For the case $k<0$ and $E<0$ show that the energy of the only even parity bound state of the system is given by

$$E=-\frac{m k^{2}}{2 \hbar^{2}}$$

Use part (i) to verify this result by taking the limit $U \rightarrow \infty, l \rightarrow 0$ with $U l$ fixed.