Quantum Mechanics: Concepts and Applications · Problem 3.6

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · infinite-square-well-probability-current

Problem

A particle of mass mm, in an infinite potential well of length aa, has the following initial wave function at t=0t = 0: ψ(x,0)=35asin⁡(3πxa)+15asin⁡(5πxa),\psi(x, 0) = \sqrt{\frac{3}{5a}} \sin\left(\frac{3\pi x}{a}\right) + \frac{1}{\sqrt{5a}} \sin\left(\frac{5\pi x}{a}\right), and an energy spectrum En=−ℏ2π2n2/(2ma2)E_n = -\hbar^2\pi^2 n^2/(2ma^2). Find ψ(x,t)\psi(x, t) at any later time tt, then calculate ∂ρ∂t\frac{\partial\rho}{\partial t} and the probability current density vector J⃗(x,t)\vec{J}(x, t) and verify that ∂ρ∂t+∇⃗⋅J⃗(x,t)=0\frac{\partial\rho}{\partial t} + \vec{\nabla} \cdot \vec{J}(x, t) = 0. Recall that ρ=ψ∗(x,t)ψ(x,t)\rho = \psi^*(x,t)\psi(x,t) and J⃗(x,t)=iℏ2m(ψ(x,t)∇⃗ψ∗(x,t)−ψ∗(x,t)∇⃗ψ(x,t))\vec{J}(x,t) = \frac{i\hbar}{2m}\left(\psi(x,t)\vec{\nabla}\psi^*(x,t) - \psi^*(x,t)\vec{\nabla}\psi(x,t)\right).

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