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.

 

[paranormal]

I would like to clarify that the radial probability density for a 3D harmonic oscillator in a circular orbit is not just the square of the normalized wavefunction, but also includes an additional factor of r^2. This is due to the fact that the wavefunction for a 3D harmonic oscillator is a product of three 1D harmonic oscillator wavefunctions, each of which has an additional factor of r^2 in its radial component. Therefore, when combining all three dimensions, the radial probability density also includes this factor. This is important to note, as it affects the overall shape and behavior of the probability density in a 3D harmonic oscillator system.