Quantum Mechanics | Part IB, 2004

Consider a Hamiltonian of the form

H=12m(p+if(x))(pif(x)),<x<,H=\frac{1}{2 m}(p+i f(x))(p-i f(x)), \quad-\infty<x<\infty,

where f(x)f(x) is a real function. Show that this can be written in the form H=p2/(2m)+V(x)H=p^{2} /(2 m)+V(x), for some real V(x)V(x) to be determined. Show that there is a wave function ψ0(x)\psi_{0}(x), satisfying a first-order equation, such that Hψ0=0H \psi_{0}=0. If ff is a polynomial of degree nn, show that nn must be odd in order for ψ0\psi_{0} to be normalisable. By considering dxψHψ\int \mathrm{d} x \psi^{*} H \psi show that all energy eigenvalues other than that for ψ0\psi_{0} must be positive.

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

Hoscillator =p22m+12mω2x2.H_{\text {oscillator }}=\frac{p^{2}}{2 m}+\frac{1}{2} m \omega^{2} x^{2} .

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