2.II.19D

Consider a Hamiltonian of the form

$H=\frac{1}{2 m}(p+i f(x))(p-i f(x)), \quad-\infty<x<\infty,$

where $f(x)$ is a real function. Show that this can be written in the form $H=p^{2} /(2 m)+V(x)$, for some real $V(x)$ to be determined. Show that there is a wave function $\psi_{0}(x)$, satisfying a first-order equation, such that $H \psi_{0}=0$. If $f$ is a polynomial of degree $n$, show that $n$ must be odd in order for $\psi_{0}$ to be normalisable. By considering $\int \mathrm{d} x \psi^{*} H \psi$ show that all energy eigenvalues other than that for $\psi_{0}$ must be positive.

For $f(x)=k x$, use these results to find the lowest energy and corresponding wave function for the harmonic oscillator Hamiltonian

$H_{\text {oscillator }}=\frac{p^{2}}{2 m}+\frac{1}{2} m \omega^{2} x^{2} .$

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