3D Harmonic Oscillator Circular Orbit

  • #1
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Homework Statement



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


Homework Equations





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?
 

Answers and Replies

  • #2
198
18
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.
 
  • #3
1,728
13
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.


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:
  • #4
198
18
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.

Correct.
 

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