A particle is confined to an infinite square well of width with orthonormal energy…

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

The Structure of Quantum Mechanics, Ch Hilbert space and Dirac notation

Problem

A particle is confined to an infinite square well of width LL with orthonormal energy eigenfunctions φn(x)=⟨x∣φn⟩=2/Lsin⁡(nπx/L)\varphi_n(x)=\langle x|\varphi_n\rangle = \sqrt{2/L}\sin(n\pi x/L) on [0,L][0,L]. Let ∣ϕ⟩=∣φ1⟩|\phi\rangle = |\varphi_1\rangle and ∣ψ⟩=(∣φ1⟩+∣φ2⟩)/2|\psi\rangle = (|\varphi_1\rangle+|\varphi_2\rangle)/\sqrt2. ((a)) Using the resolution of the identity in the position representation, 1=∫0L∣x⟩⟨x∣ dx\mathbb{1}=\int_0^L |x\rangle\langle x|\,dx (used on credit, as ∣x⟩|x\rangle has been throughout this chapter), write the abstract overlap ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle as an explicit integral over xx involving φ1(x)\varphi_1(x) and ψ(x)=⟨x∣ψ⟩\psi(x)=\langle x|\psi\rangle. ((b)) Evaluate that integral using the orthonormality of φ1,φ2\varphi_1,\varphi_2, and give the numerical value of ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle.

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