Radial part of wave function in respect to spherical harmonic

[Zaknife]

Homework Statement

Consider a Wavefunction:
[tex]\psi(x,y,z)=K(x+y+x^2-y^2)e^{-r/a}[/tex]
Find expectation value of [tex] L^{2} , L_{z}^{2}, L_{x}^{2}[/tex].

Homework Equations

The Attempt at a Solution

The first step would be a rewriting a wavefunction in terms of spherical coordinates:
[tex]\psi=Kr(\cos\phi \sin \theta + 2 \sin \phi \cos \theta +r(\cos^{2} \phi \sin^{2} \theta - \sin^{2} \phi \sin^{2} \theta )) [/tex]

My Question is : is it fair to skip the radial part and just forget about it. Normalize the Wavefunction for just the angular part , and then consider a mean values of Angular Momentum Operators ? Or should i normalize the wavefunction including r ? It bothers me because of the r squared in the equation.

 

[vela]

You can't neglect the radial functions.

 

[Zaknife]

Just to make it clear - i need to do all next steps with the radial part ?

 

[vela]

It depends what your next steps are. You eventually need to normalize the wave function, so you need to take into account the radial function somewhere along the way.

By the way, you seem to have dropped the exponential factor when you rewrote the function in terms of spherical coordinates.

 

[paranormal]

It is not fair to skip the radial part and just consider the angular part for calculating the mean values of Angular Momentum Operators. The radial part of the wavefunction is an essential component and cannot be ignored. It is important to normalize the entire wavefunction, including the radial part, before calculating the mean values of the Angular Momentum Operators. This is because the radial part affects the overall shape and magnitude of the wavefunction, which in turn affects the expectation values of the Angular Momentum Operators. Furthermore, the radial part also contributes to the probability density of the wavefunction, which is crucial in calculating the expectation values. Therefore, it is important to include the radial part in the normalization and calculation process.