Chapter 3, Problem 3.47P

(a)

To determine

The potentials  $V_1\left(x\right)$ and $V_2\left(x\right)$ in terms of the superpotential, $W\left(x\right)$

(b)

To determine

To show that if  ${\psi }_n^{\left(1\right)}$ is an eigenstate of ${\widehat{H}}_1$ with eigenvalue $E_n^{\left(1\right)}$, then $\widehat{A}{\psi }_n^{\left(1\right)}$ is an eigenstate of ${\widehat{H}}_2$ with same eigenvalue. Similarly, to show that if  ${\psi }_n^{\left(2\right)}\left(x\right)$ is an eigenstate of ${\widehat{H}}_2$ with eigenvalue $E_n^{\left(2\right)}$, then ${\widehat{A}}^{†}{\psi }_n^{\left(2\right)}$ is an eigenstate of ${\widehat{H}}_1$ with the same eigenvalue. The two Hamiltonians therefore have essentially identical spectra.

(c)

To determine

The superpotential $W\left(x\right)$ in terms of the ground state wave function, ${\psi }_0^{\left(1\right)}\left(x\right)$

(d)

To determine

Use the results of parts (a) and (c), and Problem 2.23(b), to determine the superpotential $W\left(x\right)$ and the partner potential $V_2\left(x\right)$