- #1
El Hombre Invisible
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Hi
We're given the operators Lx, Ly and Lz in matrix form and asked to show that they have the correct eigenvalues for l=1. Obviously no problem determining the values and Lz comes out right, however we've never actually seen the e.v.s for Lx and Ly.
I tried finding the eigenvalues in the wave formulation for Lx and Ly using the operators in polar spherical co-ords:
[tex] L_{x} = i\hbar(sin\phi\frac{d}{d\theta} + cot \theta cos \phi \frac{d}{d\phi}) [/tex]
[tex] L_{y} = i\hbar(-cos\phi\frac{d}{d\theta} + cot \theta sin \phi \frac{d}{d\phi}) [/tex]
Dealing with l = 1, so my spherical harmonics are in one of two forms:
[tex] Y_{1,0} = (\frac{3}{4\pi})^{1/2}cos\theta [/tex]
[tex] Y_{1,\pm1} = \mp(\frac{3}{8\pi})^{1/2}sin\theta exp(\pm i \phi) [/tex]
Well, applying the operators to the wfs gives me nothing like an eigenvalue. For instance:
[tex] L_{x}Y_{1,0} = -i \hbar sin \phi tan \theta Y_{1,0} [/tex]
Anyone see where I'm going wrong, or just happen to know offhand the eigenvalues for Lx and Ly?
Cheers,
El Hombre
Homework Statement
We're given the operators Lx, Ly and Lz in matrix form and asked to show that they have the correct eigenvalues for l=1. Obviously no problem determining the values and Lz comes out right, however we've never actually seen the e.v.s for Lx and Ly.
Homework Equations
I tried finding the eigenvalues in the wave formulation for Lx and Ly using the operators in polar spherical co-ords:
[tex] L_{x} = i\hbar(sin\phi\frac{d}{d\theta} + cot \theta cos \phi \frac{d}{d\phi}) [/tex]
[tex] L_{y} = i\hbar(-cos\phi\frac{d}{d\theta} + cot \theta sin \phi \frac{d}{d\phi}) [/tex]
Dealing with l = 1, so my spherical harmonics are in one of two forms:
[tex] Y_{1,0} = (\frac{3}{4\pi})^{1/2}cos\theta [/tex]
[tex] Y_{1,\pm1} = \mp(\frac{3}{8\pi})^{1/2}sin\theta exp(\pm i \phi) [/tex]
The Attempt at a Solution
Well, applying the operators to the wfs gives me nothing like an eigenvalue. For instance:
[tex] L_{x}Y_{1,0} = -i \hbar sin \phi tan \theta Y_{1,0} [/tex]
Anyone see where I'm going wrong, or just happen to know offhand the eigenvalues for Lx and Ly?
Cheers,
El Hombre
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