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Eigenfunctions of the angular momentum operator

  1. Oct 17, 2015 #1
    Hi everyone,

    I tried to find the Eigenstate of the angular momentum operator myself, more specifically I tried to find a Function [itex]Y_{lm}(\theta,\phi)[/itex] with

    [tex]L_zY_{lm}=mħY_{lm}[/tex] and [tex]L^2Y_{lm}=l(l+1)ħ^2Y_{lm}[/tex]

    where [tex]L_z=-iħ\frac{\partial}{\partial \phi}[/tex]
    and [tex]L^2=-ħ^2(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta\frac{\partial}{\partial\theta})+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial^2\phi})[/tex]

    These representations can be found here.
    Now lets look at the simple case of [itex]m=l=0[/itex]. The standard solution here is [itex]Y_{00}(\theta,\phi)=\frac{1}{\sqrt{4\pi}}[/itex]. However, it seems like the function [itex]Y_{00}(\theta,\phi)=A\ln(\cot\theta+\csc\theta)[/itex] is as well a solution to the differential equations above, since [itex]\frac{\partial Y}{\partial \phi}=0=mħY[/itex] and [itex]\frac{\partial Y}{\partial \theta}=-A\csc\theta[/itex] and therefore [tex]L^2Y=-ħ^2(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(-A \sin\theta\csc\theta))=0=l(l+1)ħ^2Y[/tex]
    Since [tex]\int_0^\pi YY^*\sin\theta d\theta[/tex] converges (you can look at the graph here) it is possible to find an [itex]A[/itex] that normalizes [itex]Y_{00}[/itex].

    So where is the mistake? I did not find this solution anywhere.
    Thank you for your help :)
     
  2. jcsd
  3. Oct 17, 2015 #2

    fzero

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    That function is singular at the poles of the sphere ##\theta = 0,\pi##. To do QM, generally we require the wavefunction and at least the 1st derivative to exist everywhere (2nd is good too), so we would not admit this as a possible wavefunction.
     
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