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Commutators of angular momentum

  1. Jan 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Show the three components of angular momentum: L_x, L_y and L_z commute with nabla^2 and r^2 = x^2 + y^2 = z^2


    2. Relevant equations
    [A, B] = AB - BA
    For example:
    [itex]
    [L_x, \nabla^2] = L_x \nabla^2 - \nabla^2 L_x
    [/itex]

    3. The attempt at a solution
    [itex]
    L_x \nabla^2 = -i\hbar(y\frac{\partial}{\partial z} \nabla^2 - z \frac{\partial}{\partial y}\nabla^2)
    [/itex]

    [itex]
    \nabla^2 L_x = -i\hbar\nabla^2(y\frac{\partial}{\partial z} - z \frac{\partial}{\partial y})
    [/itex]

    How can I simplify these?
     
  2. jcsd
  3. Jan 24, 2009 #2

    Hurkyl

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    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Commutators

    If you have no better ideas, then why not rewrite [itex]\nabla^2[/itex], just like you did with [itex]L_x[/itex]?
     
  4. Jan 24, 2009 #3
    Re: Commutators

    after rewriting [itex]\nabla^2[/itex] as Hurkyl suggested, you should apply the commutator (it is also an operator) to a differentiable function [itex]f(x,y,z)[/itex] and see whether the differentiation rules (e.g.differentiation of a product) make the equation look simplier.
     
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