On the half line

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

On the half line x>0x>0, define p^=−iℏ d/dx\hat p=-i\hbar\,d/dx acting on wave functions with ψ(0)=0\psi(0)=0 and ∫0∞∣ψ∣2dx<∞\int_0^\infty|\psi|^2dx<\infty. Someone argues: "Integration by parts gives ⟨ϕ∣p^ψ⟩−⟨p^ϕ∣ψ⟩=−iℏ[ϕ∗ψ]0∞=0\langle\phi|\hat p\psi\rangle-\langle\hat p\phi|\psi\rangle=-i\hbar[\phi^*\psi]_0^\infty=0 for every ϕ,ψ\phi,\psi in this domain, because both vanish at x=0x=0 and decay at infinity. So p^\hat p is Hermitian, and Hermitian operators are self-adjoint, so p^\hat p is a genuine observable here." ((a)) Identify the false step in the argument. ((b)) The function ϕ(x)=e−x/λ\phi(x)=e^{-x/\lambda} with decay length λ=1.0 nm\lambda = 1.0\ \mathrm{nm} is normalisable on x>0x>0 and satisfies p^ϕ=p0ϕ\hat p\phi = p_0\phi for some p0p_0. Compute p0p_0 (as a pure imaginary number) and its magnitude in kg m s−1\mathrm{kg\,m\,s^{-1}}, and explain why ϕ\phi lies in the domain of p^†\hat p^\dagger but not in the domain of p^\hat p itself.

Related theory

Browse the problem library