414: Numerical Solutions for the Three-Dimensional Harmonic Oscillator

Reduced mass: \( \mu\) = 1

Angular momentum: L = 0

Integration limit: r_max = 6

Force constant: k = 1

Solve Schrödinger's equation numerically. Use Mathcad's ODE solve block:

Given

\[ \frac{-1}{2 \mu} \frac{d^2}{dr^2} \psi (r) - \frac{1}{r \mu} \frac{d}{dr} \psi (r) + \bigg[ \frac{L(L +1)}{2 \mu r^2} + \frac{1}{2} kr^2 \bigg] \psi (r) = E \psi (r)~~~ \psi (.001) = 1~~~ \psi '(.001) = 0.1\]

\[ \psi = Odesolve (r, r_{max})\]

\[ \psi (r) = \left( \int_{0}^{r_{max}} \psi (r)^2 4 \pi r^2 dr \right) ^{ \frac{-1}{2}} \psi (r)\]

Energy guess: E = 7.5