3-D harmonic oscillator expectation value

[dyn]

Homework Statement

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

Homework Equations

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

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

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

 

[vela]

Please show your work.

 

[nrqed]

Homework Statement

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

Homework Equations

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

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

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

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