# Homework Help: Eigenfunction of Angular Momentum Squared Operator

1. Nov 29, 2011

### Silversonic

1. The problem statement, all variables and given/known data

The square of the angular momentum operator is (in SPC);

http://img6.imageshack.us/img6/67/54712598.png [Broken]

Show that Y($\theta$,$\phi$) = C$sin^{2}$$\theta$$e^{2i\phi}$

Is an eigenfunction. C is a constant.

2. Relevant equations

3. The attempt at a solution

I am not able to get it in the form of (A number) times Y($\theta$,$\phi$) = C$sin^{2}$$\theta$$e^{2i\phi}$. No matter what, I always end up with;

-2C$sin^{2}$$\theta$$e^{2i\phi}$ - 4C$e^{2i\phi}$ - and I don't know what to do with that.

This is clearly not what I want, there is an extra term that I just can't seem to eradicate. What am I doing wrong?

Last edited by a moderator: May 5, 2017
2. Nov 29, 2011

### fzero

You might want to show some of the earlier steps. You've made a mistake combining the first 2 terms.

3. Nov 29, 2011

### Silversonic

Hmm, well with

$\frac{\delta^{2}}{\delta\theta^{2}}$ of Y($\theta$,$\phi$) I get

(having taken the constant C out)

$-2e^{2i\phi}$($sin^{2} - cos^{2}$)

With $cot\theta \frac{\delta}{\delta\theta}$

I get

$-2e^{2i\phi}$$\frac{cos\theta}{sin\theta} sin\theta cos\theta$ = $-2e^{2i\phi}$$cos^{2}\theta$

and lastly with

$\frac{1}{sin^{2}\theta}\frac{\delta^{2}}{\delta {\phi}^{2}}$

I get $-4e^{2i\phi}$

Adding all the terms gives me;

$-2e^{2i\phi}$$sin^{2} \theta$ $-4e^{2i\phi}$

Have I done something wrong?

4. Nov 29, 2011

### fzero

You've factored out the $-\hbar^2$ so there's no minus sign in front of this term.

5. Nov 29, 2011

### Silversonic

$\frac{\delta}{\delta\theta}$ of

$sin^{2}$$\theta$$e^{2i\phi}$

is

$2sin\theta(-cos\theta)e^{2i\phi}$ = $-2sin\theta cos\theta e^{2i\phi}$

surely?

So times by $cot\theta$

is

$-2 cos^{2} \theta e^{2i\phi}$

?

6. Nov 29, 2011

### fzero

No,

$\frac{d}{d\theta}\sin\theta = \cos\theta,~~~\frac{d}{d\theta}\cos\theta = -\sin\theta.$

You'll want to doublecheck the first term as well then, but you had the overall sign correct there.

7. Nov 29, 2011

### Silversonic

Oh my good lord.

This is just embarrassing. It's not like I'm even tired or anything, I just continuously made an elementary mistake.

Thanks. I'll be handing in my University resignation form tomorrow ._.