What does the sum of eigenfunctions represent?

In summary, the significance of the homework result is that it shows that the sum of the squares of the spherical harmonics is a rational function.
  • #1
Kara386
208
2

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...
 
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  • #2
Kara386 said:

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?
 
  • #3
PeroK said:
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.
 
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  • #4
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.
 
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FAQ: What does the sum of eigenfunctions represent?

1. What are eigenfunctions?

Eigenfunctions are specific functions in mathematics that, when multiplied by a constant factor, do not change their shape. They are often used in the study of linear transformations and differential equations.

2. What does the sum of eigenfunctions represent?

The sum of eigenfunctions represents a decomposition of a more complex function into simpler components. Each eigenfunction represents a different mode or pattern of the original function.

3. How does the sum of eigenfunctions relate to the original function?

The sum of eigenfunctions can be used to approximate the original function. As more eigenfunctions are included in the sum, the approximation becomes more accurate.

4. Can the sum of eigenfunctions be used to solve differential equations?

Yes, the sum of eigenfunctions can be used to solve certain types of differential equations. This is known as the method of separation of variables, where the solution is expressed as a sum of eigenfunctions.

5. Are eigenfunctions unique?

Yes, eigenfunctions are unique in that they are orthogonal to each other. This means that they are perpendicular in function space and have no overlap, making them distinct and identifiable.

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