Potential and Kinetic energy commutator

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  • #1
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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.
 

Answers and Replies

  • #2
vela
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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.
 

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