Potential and Kinetic energy commutator

[582153236]

Homework Statement

[T,V]=[TV-VT]ψ

Homework Equations

T=(-ħ²/2μ)∂²/∂x²
V=(1/2)kx²[/B]

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''][/B]

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.

 

[vela]

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.