Prove the wave function for dxy orbital

1. Jun 29, 2012

chem1309

1. The problem statement, all variables and given/known data
"The angular part of the wave function for the dxy orbital is (√(15/∏)/4)sin^2(θ)sin(2$\phi$). Show that this expression corresponds to the dxy orbital"

2. Relevant equations

conversion of Cartesian to spherical coordinates:
r=√(x^2+y^2+z^2)
cosθ=z/r
tan($\phi$)=y/x

trig identity:
sin(2x)=2sinxcosx

normalization:
N^2∫ψ*ψdτ

dτ=r^2sinθdrdθd$\phi$

0≤r≤∞
0≤θ≤∏
0≤$\phi$≤2∏

3. The attempt at a solution

in Cartesian coordinates dxy is represented as simply xy. I converted xy to spherical coordinates and manipulated the equation the relevant equations to get xy=(r/2)sin^2(θ)sin(2$\phi$) as follows:

xy=rsincos$\phi$rsinθsin$\phi$
xy=rsin^2(θ)cos$\phi$sin$\phi$
xy=rsin^2(θ)sin(2$\phi$)/2

Then I tried to normalize the equation, but I ended up with

∫r^3 from 0 to ∞, which goes to ∞/does not converge

and ∫sin2$\phi$ which equal zero.

2. Jul 1, 2012

vela

Staff Emeritus
I'm not sure why you think you need to normalize the wave function to show it corresponds to the dxy orbital.

In any case, the total wave function is of the form $\psi(\vec{r}) = R_{nl}(r)Y_l^m(\theta,\phi)$. Normalization requires that
$$\int \psi^*(\vec{r})\psi(\vec{r})\,d^3\vec{r} = \int R^*_{nl}(r)R_{nl}(r)\,dr \int {Y_l^m}^*(\theta,\phi)Y_l^m(\theta,\phi)\,d\Omega = 1.$$ Does seeing this clear up your two questions?