# Prove the wave function for dxy orbital

1. Jun 30, 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(2ϕ). 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(ϕ)=y/x

trig identity:
sin(2x)=2sinxcosx

normalization:
N^2∫ψ*ψdτ=1

dτ=r^2sinθdrdθdϕ

0≤r≤∞
0≤θ≤∏
0≤ϕ≤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(2ϕ) as follows:

xy=rsincosϕrsinθsinϕ
xy=rsin^2(θ)cosϕsinϕ
xy=rsin^2(θ)sin(2ϕ)/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 ∫sin2ϕ which equal zero.