3D Harmonic Oscillator Circular Orbit

[unscientific]

Homework Statement

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

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?

 

[BOYLANATOR]

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.

 

[unscientific]

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.

 

[BOYLANATOR]

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.