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Spherical Harmonics within QM

  1. May 25, 2007 #1
    1. The problem statement, all variables and given/known data
    The spherical harmonics [tex]Y^m_l[/tex] with l=1 are given by
    [tex]Y^{-1}_1 = \sqrt{\frac{3}{8\pi}}\frac{x-iy}{r}, Y^0_1 = \sqrt{\frac{3}{4\pi}}\frac{z}{r}, Y^1_1 = -\sqrt{\frac{3}{8\pi}}\frac{x+iy}{r}[/tex]

    and they are functions of [tex]L^2[/tex] and [tex]L_z[/tex] where L is the angular momentum.

    i) From these functions find a new set of three functions [tex]X^m_1[/tex] which are now eigenfunctions of [tex]L^2[/tex] and [tex]L_x[/tex].

    2. Relevant equations

    3. The attempt at a solution
    I'm not 100% sure about this question. Is it asking me to give the spherical harmonics in terms of [tex]\theta, \phi[/tex]? I think I can do that, but if thats not the question, could someone please explain to me what is being asked of me.


  2. jcsd
  3. May 25, 2007 #2
    i think it would be better if you wrote your spherical harmonics in spherical coords ..

    but in any case
    if [itex] X_{lm} [/itex] is an eignefunction of L^2 and Lz then
    [tex] \hat{L_{z}} X_{lm} = mX_{lm} [/tex]
    [tex] \hat{L^2} X_{lm} = l(l+1) X_{lm} [/tex]
    and i think Xlm would b acquired by finding a superposition of the three given eigenfunctions.
  4. May 25, 2007 #3
    Thats what the question gives, cartesian coords.
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