A free particle is confined to a one-dimensional ring of circumference with periodic…

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

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

Problem

A free particle is confined to a one-dimensional ring of circumference LL with periodic boundary conditions. Its normalised momentum eigenfunctions are ψn(x)=A eipnx/ℏ\psi_n(x)=A\,e^{ip_nx/\hbar}, 0≤x<L0\le x<L, with pn=2πℏn/Lp_n = 2\pi\hbar n/L for integer nn, and ∫0L∣ψn(x)∣2dx=1\int_0^L|\psi_n(x)|^2dx=1. ((a)) Find AA in terms of LL. ((b)) Evaluate AA for L1=1.0×10−6 mL_1=1.0\times10^{-6}\,\mathrm{m} and L2=1.0×10−3 mL_2=1.0\times10^{-3}\,\mathrm{m}. ((c)) State what happens to AA as L→∞L\to\infty, and explain why this is consistent with the continuum normalisation ⟨p∣p′⟩=δ(p−p′)\langle p|p'\rangle=\delta(p-p'), under which ∣p⟩|p\rangle has infinite norm even though ⟨x∣p⟩=(2πℏ)−1/2eipx/ℏ\langle x|p\rangle=(2\pi\hbar)^{-1/2}e^{ipx/\hbar} stays finite at every xx.

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