Inner product of two spherical functions

[Niles]

Homework Statement

Hi all.

The inner product between two functions f(x) and g(x) is defined as:

$$<f | g> = \int f^*(x)g(x) dx,$$

where the * denotes the complex conjugate. Now if my functions f and g are functions of r, theta and phi (i.e. they are written in spherical coordinates), is the volume element then given as $drd\theta d\phi$ or $r^2\sin \theta drd\theta d\phi$?

The Attempt at a Solution

I personally think the last, because originally we have <f(x,y,z) | g(x,y,z)>, which we convert to spherical coordinates, so we need the Jacobian. Am I correct?

Thanks in advance.

 

[Unco]

Putting $$h(x,y,z) = f^\ast(x,y,z)g(x,y,z)$$, we have

$$\int\int\int h(x,y,z) dz \, dy\, dx = \int\int\int h(r\cos{\theta}\sin{\phi}, r\sin{\theta}\sin{\phi}, r\cos{\phi}) r^2{\underbrace{\sin{\phi}}_{\text{not \theta}} dr \, d\theta\, d\phi$$

with appropriate limits, as per usual. That is all your question appears to boil down to, Niles.

 

[Niles]

Then I was correct. Thanks for replying!