3. A particle of mass $m$ moves in a one-dimensional potential well of the form $V(x) = -\mu \frac{\hbar^2 \alpha^2}{m} \text{sech}^2 \alpha x$, where $\mu$ and $\alpha$ are positive constants. The expectation value $<E>$ of the energy of the system is $\int \psi^* H \psi \, dx$ where the operator $H$ is defined by $-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x)$. Using trial wavefunctions of the form $y = A \text{sech}\beta x$, show the following:
a. for $\mu = 1$, there is an exact eigenfunction of $H$, with a corresponding $<E>$ of half of the maximum depth of the well; [6 points]
b. for $\mu = 6$ the 'binding energy' of the ground state is at least $10\hbar^2 \alpha^2/(3m)$. [6 points]
Hint: You will find it useful to note that for $u, v \ge 0$, $\text{sech}\, u \, \text{sech}\, v \ge \text{sech}(u+v)$.
