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3D Harmonic Oscillator Circular Orbit

  1. Mar 23, 2014 #1
    1. The problem statement, all variables and given/known data

    I found this in Binney's text, pg 154 where he described the radial probability density ##P_{(r)} \propto r^2 u_L##

    rwi5o8.png

    2. Relevant equations



    3. The attempt at a solution

    Isn't the radial probability density simply the square of the normalized wavefunction, |ψ(x)|2? Why is there an additional factor of r2?
     
  2. jcsd
  3. Mar 23, 2014 #2
    When you integrate a function over a volume in spherical co-ordinates you integrate over r2sin(θ)drdθdø . The sin(θ)dθdø goes into the angular function and the r2dr into the radial function. I believe this is where it comes from.
     
  4. Mar 23, 2014 #3

    Probability = ##<\psi|\psi> = \int \psi^*\psi d^3r = \int \psi^*\psi r^2 dr d\Omega##

    Where the wavefunction corresponding to the ket ##|\psi>## is ##u_L Y_l^m##

    I think that's right.
     
    Last edited: Mar 23, 2014
  5. Mar 23, 2014 #4
    Correct.
     
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