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Potential and Kinetic energy commutator

  1. Nov 9, 2014 #1
    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. jcsd
  3. Nov 9, 2014 #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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