Principles of Quantum Mechanics · Problem 7.4.9

The Path Integral Formulation of Quantum Theory — R. Shankar

Physics — Quantum Mechanics · unconventional-momentum-operator

Problem

*(Important). Consider the unconventional (but fully acceptable) operator choice X→xX \to x P→−iℏddx+f(x)P \to -i\hbar\frac{d}{dx} + f(x) in the XX basis. (1) Verify that the canonical commutation relation is satisfied. (2) It is possible to interpret the change in the operator assignment as a result of a unitary change of the XX basis: ∣x⟩→∣x~⟩=eig(X)/ℏ∣x⟩=eig(x)/ℏ∣x⟩|x\rangle \to |\tilde{x}\rangle = e^{ig(X)/\hbar}|x\rangle = e^{ig(x)/\hbar}|x\rangle where g(x)=∫xf(x′) dx′g(x) = \int^x f(x')\, dx' First verify that ⟨x~∣X∣x~′⟩=xδ(x−x′)\langle \tilde{x}|X|\tilde{x}'\rangle = x\delta(x-x') i.e., X→new X basisxX \xrightarrow{\text{new } X \text{ basis}} x Next verify that ⟨x~∣P∣x~′⟩=[−iℏddx+f(x)]δ(x−x′)\langle \tilde{x}|P|\tilde{x}'\rangle = \left[-i\hbar\frac{d}{dx} + f(x)\right]\delta(x-x')

Browse the problem library