AMATH 481/581 Autumn Quarter 2024

Homework 2: Quantum Harmonic Oscillator

DUE: Friday, October 18 at midnight

The probability density evolution in a one-dimensional harmonic trapping potential is governed by the partial differential equation:

where is the probability density and V(x) = kz2 /2 is the harmonic confining potential. A typical solution technique for this problem is to assume a solution of the form.

which is called an eigenfunction expansion solution (o=eigenfunction, E,=eigenvalue). Plugging in this solution ansatz to Eq. (1) gives the boundary value problem:

where we expect the solution (z)→0 as z→±oo and e, is the quantum energy. Note here that K=km/h2and en=E,m/h2. In what follows, take K=1 and always normalize so that nl2dr=1.

(a) Calculate the first five normalized eigenfunctions (фn) and eigenvalues (en) using a shooting scheme. For this calculation, use -L, L] with L4 and choose zspan-L: 0.1: L. Save the absolute value of the eigenfunctions in a 5-column matrix (column 1 is 1, column 2 is o2 etc.) and the eigenvalues in a 1x5 vector.