# 3-D harmonic oscillator expectation value

1. Aug 20, 2014

### dyn

1. The problem statement, all variables and given/known data
The Hamiltonian for the 3-D harmonic oscillator in spherical polar coordinates is given in the question.The question then asks : using the trial wavefunction $ψ=e^(-αr)$ show that

2. Relevant equations

$<ψ|H|ψ>/<ψ|ψ> = ($\hbar$α)^2/2m + 3mω^2/2α^2$

The following integral is also given$∫ x^nexp(-ax) = n!/a^(n+1)$for n≥0 with limits from 0 to ∞

3. The attempt at a solution

I applied the Hamiltonian to ψ. The $\theta$ and $\phi$ terms drop out. I then perform the integral of ∫ ψ^*Hψ but I have a term containing 1/r and the given integral doesn't apply for n<0
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 21, 2014

### vela

Staff Emeritus
What did you use for the measure of integration dV? It contains $r^2 dr$, did you forget that $r^2$?