Introduction to Quantum Mechanics · Problem 2.5

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · infinite_square_well

Problem

A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: Ψ(x,0)=A[ψ1(x)+ψ2(x)].\Psi(x, 0) = A [\psi_1(x) + \psi_2(x)]. (a) Normalize Ψ(x,0)\Psi(x, 0). (That is, find AA. This is very easy, if you exploit the orthonormality of ψ1\psi_1 and ψ2\psi_2. Recall that, having normalized Ψ\Psi at t=0t = 0, you can rest assured that it stays normalized—if you doubt this, check it explicitly after doing part (b).) (b) Find Ψ(x,t)\Psi(x, t) and ∣Ψ(x,t)∣2|\Psi(x, t)|^2. Express the latter as a sinusoidal function of time, as in Example 2.1. To simplify the result, let ω≡π2ℏ/2ma2\omega \equiv \pi^2 \hbar / 2ma^2. (c) Compute ⟨x⟩\langle x \rangle. Notice that it oscillates in time. What is the angular frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than a/2a/2, go directly to jail.) (d) Compute ⟨p⟩\langle p \rangle. (As Peter Lorre would say, "Do it ze kveek vay, Johnny!") (e) If you measured the energy of this particle, what values might you get, and what is the probability of getting each of them? Find the expectation value of HH. How does it compare with E1E_1 and E2E_2?
Intuitive explanation

The key insight is that orthonormality makes superpositions of stationary states especially clean to work with. Since ψ1\psi_1 and ψ2\psi_2 are orthonormal eigenstates, their inner products vanish except when identical—this lets you normalize the superposition almost trivially. Once you have Ψ(x,t)\Psi(x,t), each eigenstate picks up its own time-evolution phase e−iEnt/ℏe^{-iE_n t/\hbar}, creating interference terms in ∣Ψ∣2|\Psi|^2 that oscillate at the energy difference frequency. This oscillating probability density is the signature of a coherent quantum superposition.

Intuitive explanation

The key insight is that orthonormality makes superpositions of stationary states especially clean to work with. Since ψ1\psi_1 and ψ2\psi_2 are orthonormal eigenstates, their inner products vanish except when identical—this lets you normalize the superposition almost trivially. Once you have Ψ(x,t)\Psi(x,t), each eigenstate picks up its own time-evolution phase e−iEnt/ℏe^{-iE_n t/\hbar}, creating interference terms in ∣Ψ∣2|\Psi|^2 that oscillate at the energy difference frequency. This oscillating probability density is the signature of a coherent quantum superposition.

Related theory

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