Chapter 2, Problem 2.52P

(a)

To determine

Graph the potential $V\left(x\right)=-\frac{{\hslash }^2{a}^2}{m}{\text{sech}}^2\left(ax\right)$

(b)

To determine

To check that the potential $V\left(x\right)=-\frac{{\hslash }^2{a}^2}{m}{\text{sech}}^2\left(ax\right)$ has ground state ${\psi }_0\left(x\right)=A\text{sech}\left(ax\right)$ and find its energy, normalize ${\psi }_0$, and sketch its graph.

(c)

To determine

Show that the function ${\psi }_k\left(x\right)=A\left(\frac{ik-a\tanh \left(ax\right)}{ik+a}\right)e^{ikx}$, solves the Schrodinger equation for any energy E. The asymptotic form of ${\psi }_k\left(x\right)$ at large positive x ? What are R and T, for this potential?