Functions, operator => eigenfunction, eigenvalue

[sundriedtomato]

[SOLVED] Functions, operator => eigenfunction, eigenvalue

Homework Statement

Show, that functions
f1 = A*sin($$\theta$$)exp[i$$\phi$$] and
f2 = B(3cos$$^{2}$$($$\theta$$) - 1) A,B - constants
are eigenfunctions of an operator
http://img358.imageshack.us/img358/3406/98211270ob1.jpg
and find eigenvalues

The Attempt at a Solution

This is what i got for the first function:
http://img391.imageshack.us/img391/4461/38657444ih2.jpg

The next step is to solve left part of the equation, and than compare it to the right part.

The question arises, how to solve that equation?

I tried simplifying left part of an equation in mathcad, and I got
http://img391.imageshack.us/img391/1897/28095157bm3.jpg

Next question from that part, is if I am doing it right, how to compare those parts, and answer a question - weather this function is an eigenfunction of an operator?

Thank You in advance, and I am constantly near computer and waiting for suggestions.

 

[Reshma]

Your eigen operator has partial derivatives wrt to $\theta$ and $\phi$. When you operate it on your given eigen function, you should get back your original function multiplied by a scaling factor which is your eigenvalue.

 

[sundriedtomato]

Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?

 

[nrqed]

Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?

Just go ahead and apply the derivatives! It's that simple. (btw, I don't know what you entered in mathcad but what it gave you is wrong).

all you have to do is to apply the derivatives

 

[nrqed]

Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?

For the first term, what you have to calculate is

$$\frac{1}{sin \theta} \frac{\partial}{\partial \theta} ( sin \theta ~\frac{\partial}{ \partial \theta} (A sin \theta} e^{i \phi}}) )$$

 

[sundriedtomato]

I will give it a try right know. Thank You.

 

[sundriedtomato]

So, after proper calculations, the result is
http://img76.imageshack.us/img76/3385/38806746cn9.jpg

Is this one correct?

I posted an image of what I am given, and as far as I know, differentiation sign usually is placed before the function?
I just don't get it - to what parts of an equation do underlined derivatives belong t?
http://img511.imageshack.us/img511/9895/22kn1.jpg

Thank You.

 

[rahuldandekar]

The part in brackets is an "operator"... every incomplete differentiation sign (ie, without anything to differentiate) operates on whatever is "multiplied" to it.

For example, the d/d(theta) is your image also operates on A*sin(theta)*exp(i*phi), because when you open the bracket, it gets to differentiate that term.

Eg. (the d's are partial)
[d/dx + d/dy]*x*y^2 = y^2 + 2*y*x

 

[Reshma]

Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?

Just operate it on the function. They are simple partial derivatives, I won't take much time to solve. Just a little patience for the initial steps. I tried your problem and many terms get canceled out and you get the solution correctly. You don't need mathcad to do it.

 

[sundriedtomato]

Thank You Reshma! I already managed to solve it correctly, and find eigenvalues as well.
For the case with the first function : a = 2h^2, and for the case with second function b = 6h^2. The problem was initially in understanding how to apply operator properly. Once example have been given, and operator properties reanalyzed - problem got solved in like 10 minutes (with putting it all on a paper as well). Thank You everybody who took part in this one!