Let be a unitary operator on a Hilbert space

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

The Structure of Quantum Mechanics, Ch Observables, spectra and the spectral theorem

Problem

Let U^\hat U be a unitary operator on a Hilbert space H\mathcal H, so that U^†U^=U^U^†=1\hat U^\dagger\hat U=\hat U\hat U^\dagger=\mathbb1, and let ∣ϕ⟩,∣ψ⟩|\phi\rangle,|\psi\rangle be arbitrary vectors in H\mathcal H. ((a)) Starting from the defining relation of the adjoint, ⟨A^ϕ∣χ⟩=⟨ϕ∣A^†χ⟩\langle\hat A\phi|\chi\rangle=\langle\phi|\hat A^\dagger\chi\rangle for any operator A^\hat A and any ∣ϕ⟩,∣χ⟩|\phi\rangle,|\chi\rangle, prove that ⟨U^ϕ∣U^ψ⟩=⟨ϕ∣ψ⟩\langle\hat U\phi|\hat U\psi\rangle=\langle\phi|\psi\rangle. ((b)) Deduce that if ∣ψ⟩|\psi\rangle is normalised then so is U^∣ψ⟩\hat U|\psi\rangle. ((c)) An observable A^\hat A is carried to A^′=U^A^U^†\hat A'=\hat U\hat A\hat U^\dagger, and ∣ψ⟩|\psi\rangle is carried to ∣ψ′⟩=U^∣ψ⟩|\psi'\rangle=\hat U|\psi\rangle. Show that ⟨ψ′∣A^′∣ψ′⟩=⟨ψ∣A^∣ψ⟩\langle\psi'|\hat A'|\psi'\rangle=\langle\psi|\hat A|\psi\rangle. ((d)) In one sentence, explain why part (c) means a unitary change of description cannot alter any prediction of the theory.

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