The reflection operator satisfies

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Observables, commutators and the uncertainty relations

Problem

The reflection operator P^ψ(x)=ψ(−x)\hat P\psi(x) = \psi(-x) satisfies P^2=1^\hat P^2 = \hat 1, since reflecting twice returns the original function. ((a)) Show that P^\hat P is Hermitian. ((b)) Use P^2=1^\hat P^2 = \hat 1 to find the only two possible eigenvalues of P^\hat P. ((c)) A particle moves in the tilted oscillator potential V(x)=12mω2x2+ϵxV(x) = \tfrac12m\omega^2x^2 + \epsilon x, for some small constant ϵ≠0\epsilon \ne 0. Using the criterion this chapter built for when P^\hat P and H^\hat H commute, decide whether P^\hat P and H^\hat H are compatible in this potential, and contrast with the case ϵ=0\epsilon = 0. ((d)) This chapter's compatibility theorem carries one exception: a state can still be shared even when the operators do not commute, if that particular state is annihilated by the commutator. Using the expression for [H^,P^]ψ(x)[\hat H,\hat P]\psi(x) found in (c), decide whether any normalisable state of the tilted oscillator is saved by this exception when ϵ≠0\epsilon \ne 0.

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