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Prove the wave function for dxy orbital

  1. Jun 30, 2012 #1
    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.
     
  2. jcsd
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