Introduction to Quantum Mechanics · Problem 2.4

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · harmonic_oscillator

Problem

A particle of mass mm in the harmonic oscillator potential (Equation 2.44) starts out in the state Ψ(x,0)=A(1−2mωℏ x)2e−mω2ℏx2,\Psi(x, 0) = A\left(1 - 2\sqrt{\frac{m\omega}{\hbar}}\,x\right)^2 e^{-\frac{m\omega}{2\hbar}x^2}, for some constant AA. (a) Determine AA and the coefficients cnc_n in the expansion of this state in terms of the stationary states of the harmonic oscillator. (b) In a measurement of the particle's energy, what results could you get, and what are their probabilities? What is the expectation value of the energy? (c) At a later time TT the wave function is Ψ(x,T)=B(1+2mωℏ x)2e−mω2ℏx2,\Psi(x, T) = B\left(1 + 2\sqrt{\frac{m\omega}{\hbar}}\,x\right)^2 e^{-\frac{m\omega}{2\hbar}x^2}, for some constant BB. What is the smallest possible value of TT?

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