What does the sum of eigenfunctions represent?

[Kara386]

Homework Statement

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.

Homework Equations

$Y_{1,1} = -\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{i\phi}$
$Y_{1,0} = -\sqrt{\frac{3}{4\pi}}\cos(\theta)$
$Y_{1,-1} = -\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{-i\phi}$

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...

 

[PeroK]

Homework Statement

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.

Homework Equations

$Y_{1,1} = -\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{i\phi}$
$Y_{1,0} = -\sqrt{\frac{3}{4\pi}}\cos(\theta)$
$Y_{1,-1} = -\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{-i\phi}$

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...

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

 

[Kara386]

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

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.

 

[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.