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I Measurement Values for z-component of Angular Momentum

  1. Nov 18, 2017 #1
    Given a wave function $$\Psi(r,\theta,\phi)=f(r)\sin^2(\theta)(2\cos^2(\phi)-1-2i*\sin(\phi)\cos(\phi))$$ we are trying to find what a measurement of angular momentum of a particle in such wave function would yield.
    Attempts were made using the integral formula for the Expectation Value over a spherical volume: $$<\hat L_z> = \iiint\Psi^*(\hat L_z)\Psi dV$$ where $$dV=r^2\sin(\theta)drd\theta d\phi$$ and $$\hat L_z=-i\hbar (\frac{\partial}{\partial \phi} ).$$ The Integral seemed really difficult to clear up and get a valid expression, which caused a doubt about whether the approach is incorrect in the first place. Any suggestions on an efficient or smarter way of approaching those types of problems?
    Thank you!
  2. jcsd
  3. Nov 18, 2017 #2


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    It's more useful if you try to think the wavefunction you are given with as a superposition of eigenfunctions of ##L_z##. At the same time try to get familiar with some low order spherical harmonics (e.g. in https://en.wikipedia.org/wiki/Table_of_spherical_harmonics) and recognize how they typically depend on the angles ##\theta## and ##\phi## for a given pair of ##l## and ##m##.
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