What does the sum of eigenfunctions represent?

1. Oct 13, 2016

Kara386

1. The problem statement, all variables and given/known data
I've been given the spherical harmonics $Y_{l,m}$ for the orbital quantum number $l=1$. Then told to calcute the sum of their squares over all values of m and explain the significance of the result.

2. Relevant equations
$Y_{1,1} = -\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{i\phi}$
$Y_{1,0} = -\sqrt{\frac{3}{4\pi}}\cos(\theta)$
$Y_{1,-1} = -\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{-i\phi}$
3. The attempt at a solution
A linear combination of eigenfunctions can be used to represent a wavefunction because they form an orthonormal basis but this is the sum of eigenfunctions squared, no coefficients. The result of the sum from m=l to m=-l is
$\frac{3}{4\pi}(\sin^2(\theta)\cos(2\phi)+cos^2(\theta))$
But I'm wondering if that's wrong because it doesn't seem especially significant to me...

Last edited: Oct 13, 2016
2. Oct 13, 2016

PeroK

My first thought is what is $\phi$ doing in the answer?

3. Oct 13, 2016

Kara386

Well if I could get rid of them I'd be sorted. But they don't cancel, I'll type up my workings. So the sum is given by each of the three spherical harmonics summed and squared:
$\frac{3}{8\pi}\sin^2(\theta)e^{2i\phi} + \frac{3}{4\pi}\cos^2(\theta) + \frac{3}{8\pi}\sin^2(\theta)e^{-2i\phi}$
$= \frac{3}{8\pi}\sin^2(\theta)(e^{2i\phi}+e^{-2i\phi})+\frac{3}{4\pi}\cos^2(\theta)$
$= \frac{3}{4\pi}\sin^2(\theta)\frac{1}{2}(e^{2i\phi}+e^{-2i\phi})+\frac{3}{4\pi}\cos^2(\theta)$
$= \frac{3}{4\pi}\sin^2(\theta)\cos(2\phi)+\frac{3}{4\pi}\cos^2(\theta)$
Which then becomes the equation in my first post. Any mistakes?

Oh wait, mistake is that they are complex so the magnitude means something different! Got it! so I get $\frac{3}{4\pi}$ as my answer, I just need the interpretation.

Last edited: Oct 13, 2016
4. Oct 14, 2016

Fred Wright

I suggest that you decompose the product of the spherical harmonics into a linear combination of lower and higher l order harmonics. For example,
$Y_{10}(\theta,\phi)Y_{10}(\theta,\phi)=c_{00}Y_{00}(\theta,\phi)+c_{20}Y_{20}(\theta,\phi)$You can easily find that$c_{00}=\frac{1} {\sqrt {4\pi}}, c_{20}=\frac{1} {\sqrt{5\pi}}$
Do the same for the other two products and see if this gives some insight.

Last edited: Oct 14, 2016
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