# 412: Particle in an Infinite Spherical Potential Well

- Page ID
- 137741

Reduced mass: \( \mu\) = 1

Angular momentum: L = 2

Integration limit: r_{max} = 1

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

Given

\[ \frac{-1}{2 \mu} \frac{d^2}{dr^2} \psi (r) - \frac{1}{r \mu} \frac{d}{dr} \psi (r) + \bigg[ \frac{L (L + 1)}{2 \mu r^2} \bigg] \psi (r) = E \psi (r)~~~ \psi (.0001) = .1~~~ \psi '(.0001) = 0\]

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

Normalize the wavefunction:

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

Energy guess: E = 16.51

r = 0, .001 .. r_{max}