The chapter shows that a real wave function can solve the Schrödinger equation only…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation and the Born rule

Problem

The chapter shows that a real wave function can solve the Schrödinger equation only if it does not change in time. Test this claim in full on a real, product-form candidate: let f(x)f(x) be a real function of xx only, ω\omega a real constant, and try Ψ(x,t)=f(x)cos⁡(ωt)\Psi(x,t) = f(x)\cos(\omega t) as a solution of the free equation iℏ ∂Ψ/∂t=−(ℏ2/2m) ∂2Ψ/∂x2i\hbar\,\partial\Psi/\partial t = -(\hbar^2/2m)\,\partial^2\Psi/\partial x^2. ((a)) For ω≠0\omega\ne0, show that no non-zero choice of f(x)f(x) makes this work. ((b)) The one case part (a) does not cover is ω=0\omega=0, where Ψ(x,t)=f(x)\Psi(x,t)=f(x) is genuinely time-independent, as the chapter's claim allows. Substitute this into the free equation, find the condition it places on f(x)f(x), and show that the only solutions are unphysical: they cannot be normalised on the whole line. ((c)) The building block used throughout this chapter for a moving wave is Ψ(x,t)=eikxe−iωt\Psi(x,t)=e^{ikx}e^{-i\omega t}, of the same product form f(x)g(t)f(x)g(t), but with g(t)=e−iωtg(t)=e^{-i\omega t} in place of cos⁡(ωt)\cos(\omega t). Substitute this form and explain, in one or two sentences, why it escapes the argument of part (a).

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