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Prove the wave function for dxy orbital

  1. Jun 29, 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(2[itex]\phi[/itex]). 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([itex]\phi[/itex])=y/x

    trig identity:
    sin(2x)=2sinxcosx

    normalization:
    N^2∫ψ*ψdτ

    dτ=r^2sinθdrdθd[itex]\phi[/itex]

    0≤r≤∞
    0≤θ≤∏
    0≤[itex]\phi[/itex]≤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[itex]\phi[/itex]) as follows:

    xy=rsincos[itex]\phi[/itex]rsinθsin[itex]\phi[/itex]
    xy=rsin^2(θ)cos[itex]\phi[/itex]sin[itex]\phi[/itex]
    xy=rsin^2(θ)sin(2[itex]\phi[/itex])/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[itex]\phi[/itex] which equal zero.
     
  2. jcsd
  3. Jul 1, 2012 #2

    vela

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    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?
     
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