Consider a quantum mechanical particle in a one-dimensional potential $V(x)$, for which $V(x)=V(-x)$. Prove that when the energy eigenvalue $E$ is non-degenerate, the energy eigenfunction $\chi(x)$ has definite parity.

Now assume the particle is in the double potential well

$$V(x)= \begin{cases}U, & 0 \leqslant|x| \leqslant l_{1} \\ 0, & l_{1}<|x| \leqslant l_{2} \\ \infty, & l_{2}<|x|\end{cases}$$

where $0<l_{1}<l_{2}$ and $0<E<U$ (U being large and positive). Obtain general expressions for the even parity energy eigenfunctions $\chi^{+}(x)$ in terms of trigonometric and hyperbolic functions. Show that

$$-\tan \left[k\left(l_{2}-l_{1}\right)\right]=\frac{k}{\kappa} \operatorname{coth}\left(\kappa l_{1}\right)$$

where $k^{2}=\frac{2 m E}{\hbar^{2}}$ and $\kappa^{2}=\frac{2 m(U-E)}{\hbar^{2}}$.