Are These Wavefunctions Orthonormal in Spherical Coordinates?

[MontavonM]

Given 2 wavefunctions with respect to (r,theta,phi)...

To prove that the functions are orthonormal, you would let the first w.function = Y(psi) and the 2nd = Y* (psi star), then you would integrate --> SY*Y dr dtheta dphi (integral of psi* times psi) dr dtheta dphi.

Correct?

 

[jfy4]

yeah, over a whole period, or the appropriate bounds.

 

[strangerep]

Given 2 wavefunctions with respect to (r,theta,phi)...

To prove that the functions are orthonormal, you would let the first w.function = Y(psi) and the 2nd = Y* (psi star), then you would integrate --> SY*Y dr dtheta dphi (integral of psi* times psi) dr dtheta dphi.

Correct?

That looks like the wrong measure for spherical polar coordinates.
I would have expected something like:
$$\int_0^{2\pi} d\phi \int_0^\pi \sin\theta \; d\theta \int_0^\infty \bar f g \, r^2 dr$$
where f,g are functions of $r,\theta,\phi$.

See also:

http://en.wikipedia.org/wiki/Volume_integral