# Real Spherical Harmonics

1. Feb 28, 2004

### Cunicultor

Viva!

I wonder if anyone could explain me the difference between real spherical harmonics (SH) and complex SH.

What's the difference in doing an expansion in either situations?
And what are the orthogonality relations for each case?

Any help would be great...( websites, books..)

Cheers

2. Feb 28, 2004

### Dr Transport

The complex spherical harmonics are simply the product of the associated Legendre polynomials and a complex exponential, i.e.

$$Y_l^m(\theta ,\phi) = P_l^m(\theta)e^((im\phi))$$.

Look in any math methods text, they should be listed with all of their properties. The real shpeherical harmonics are defined as the real or imaginary parts of these. They have slightly different orthogonality properties.

Hope that this helps some

dt

3. Feb 29, 2004

### Cunicultor

Cheers!

But what I was really looking for were those orthogonality relations for the Real SH.

I've been looking in "Arfken" and also in "www.wolfram.com", but I didn't get much.

...

4. Feb 29, 2004

### Dr Transport

Look in Morse and Feshbach, they have them. The orthonormality relationships are the same, maybe a factor of two different, I can't remember, I used them in my dissertation a few years ago and have not used them since. It is simple to derive anyhow, just the product of the Associated Legendre polynomials and trigonometric functions, should only take a few minutes to do.

dt

5. Feb 29, 2004

### Cunicultor

I've got the Morse and Feshbach book just here. There's nothing there that can help. Anyway, I've just derived myself the orthogonality relations; it's true, they are diferent by a scale factor from those for the complex SH. I just wanted to know if it is correct.

Please take a look at this, and see the way he defines the real SH.

What do you think?

6. Feb 29, 2004

### Dr Transport

That .pdf file is correct. In general, the normalization is different by a factor of $$\frac{1}{\sqrt{2}}$$. The real functions are also normalized to 1 anc can be seen from the definition in the .pdf file.

dt

7. Sep 3, 2008

### doruforum

Hello everybody,

I bring this old topic to life, because I would like to convert a matrix expressed in complex spherical harmonics (Y_{lm}) into a matrix expressed in real spherical harmonics (S_{lm}). I have found here http://www1.elsevier.com/homepage/saa/eccc3/paper48/eccc3.html
the transformation matrix C^{l} between Y_{lm} and S_{lm} for a given l.
Is it just a similarity transformation that I have to do in order to get the new matrix? I mean:
$$M_{real}=C^{-1}M_{complex}C$$
$$M_{complex}=C^{-1}M_{real}C$$

I want to be sure that I do this transformation in the correct way, because the results are not the ones that I am expecting :)

Thank you!

Last edited: Sep 3, 2008
8. Jan 13, 2010