Potential and Kinetic energy commutator

1. Nov 9, 2014

582153236

1. The problem statement, all variables and given/known data
[T,V]=[TV-VT]ψ

2. Relevant equations
T=(-ħ2/2μ)∂2/∂x2
V=(1/2)kx2

3. The attempt at a solution
[(-ħ2/2μ)∂2/∂x2((1/2)kx2ψ)]-[(1/2)kx2(-ħ2/2μ)∂2/∂x2(ψ)]

I think my problem is with executing the chain rule on the first term:

(-ħ2/2μ)[x2ψ''+2xψ'+2xψ'+2ψ-x2ψ'']

The x2ψ'' terms cancel out but I'm left with the +4xψ' term which I makes me suspect that I've made a mistake somewhere.

2. Nov 9, 2014

vela

Staff Emeritus
It's actually the product rule, not the chain rule. Your calculation is okay except for a few minor errors. (Where'd the k go?)

Neglecting the constant factors out front, you ended up with $(2x\frac{\partial}{\partial x} + 1)\psi$. You can show this equals $(x\frac{\partial}{\partial x} + \frac{\partial}{\partial x} x)\psi$, which probably looks more correct to you.