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Curl and Angular Momentum

  1. Sep 17, 2011 #1
    1. The problem statement, all variables and given/known data

    I want to prove this: [itex] \vec{r} \nabla^2 - \nabla(1+r \frac{\partial}{\partial {r}})=i \nabla \times \vec{L} [/itex]

    2. Relevant equations

    BAC-CAB rule:

    [itex] \nabla \times (\vec{A} \times \vec{B}) = (\vec{B} . \nabla) \vec{A} - (\vec{A} . \nabla ) \vec{B} - \vec{B} (\nabla . \vec{A} ) + \vec{A} (\nabla . \vec{B}) [/itex]

    3. The attempt at a solution

    We know that [itex] i \nabla \times \vec{L}= \nabla \times (\vec{r} \times \nabla) [/itex].

    Now how can I continue with [itex] \nabla \times (\vec{r} \times\nabla) = [/itex]?

    Can I use BAC-CAB rule? When I use BAC-CAB, I will have this: [itex] \nabla \times (\vec{r} \times\nabla) = \vec{r} (\nabla . \nabla) - \nabla (\nabla . \vec{r}) + (\nabla . \nabla) \vec{r}- (\vec{r} . \nabla) \nabla [/itex]

    And as we know [itex] \nabla (\nabla . \vec{r}) = \vec{0} [/itex].

    Now how can I continue? If my method is wrong, please tell me why!
    Last edited: Sep 17, 2011
  2. jcsd
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