A particle is confined to with wave functions required to satisfy (the Dirichlet box)

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Function spaces and operator theory · Function spaces and operator theory

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

Problem

A particle is confined to 0≤x≤L0\le x\le L with wave functions required to satisfy ψ(0)=ψ(L)=0\psi(0)=\psi(L)=0 (the Dirichlet box). Someone argues: "The Hamiltonian H^=−ℏ22md2dx2\hat H=-\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} is self-adjoint on this domain, since its boundary term [ϕ∗ψ′−ϕ′∗ψ]0L[\phi^*\psi'-\phi'^*\psi]_0^L vanishes whenever ψ(0)=ψ(L)=0\psi(0)=\psi(L)=0. The momentum operator p^=−iℏ d/dx\hat p=-i\hbar\,d/dx is defined on the very same space of functions, so by the same reasoning it too must be self-adjoint here." ((a)) Decide whether p^\hat p with this Dirichlet domain is self-adjoint. Justify using p^\hat p's boundary term. ((b)) State the difference between H^\hat H's boundary term and p^\hat p's that explains why the argument fails for momentum but not for energy.

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