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Homework Help: Square of z-component of angular momentum eigenvalues

  1. Jul 23, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm trying to demonstrate that if:

    $$\hat{L}_z | l, m \rangle = m \hbar | l, m \rangle$$


    $$\hat{L}_z^2 | l, m \rangle = m^2 \hbar^2 | l, m \rangle$$

    2. Relevant equations

    $$\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2$$
    $$\hat{L}_z = -i\hbar \left [ x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right ]$$

    3. The attempt at a solution

    I'm not really sure where to start with this. I can apply the operator ##\hat{L}_z^2## to an arbitrary function ##f(x,y)##, but that gives me:

    $$\hat{L}_z^2 f(x,y) = \hbar^2\left(x^2 \frac{\partial^2}{\partial y^2} - xy \frac{\partial}{\partial y} \frac{\partial}{\partial x} -xy \frac{\partial}{\partial x} \frac{\partial}{\partial y} + y^2 \frac{\partial^2}{\partial x^2} \right)f(x,y) $$
    I've no idea if this demonstrates anything at all...
  2. jcsd
  3. Jul 23, 2013 #2


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    Don't think too complicated. Just apply [itex]\hat{L}_z[/itex] twice to the eigenstate. Generally one can say that almost any calculation concerning angular-momentum operators is easier in the representation-free Hilbert-space formulation than using the differential operators of the position representation.
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