How Does Angular Dependence Arise in a Spherical Symmetric Potential?

[touqra]

For a spherical symmetric potential, the wavefunction can be expanded in terms of partial waves which is dependent on r and $$\theta$$. How would this be possible, when the potential only depends on distance from source? Classically, there's no quantity, that could have depend even on $$\theta$$.

 

[wdlang]

this is a good question

i also want to know the answer

 

[ZapperZ]

For a spherical symmetric potential, the wavefunction can be expanded in terms of partial waves which is dependent on r and $$\theta$$. How would this be possible, when the potential only depends on distance from source? Classically, there's no quantity, that could have depend even on $$\theta$$.

You are also forgetting $\phi$, the azimuthal angle.

Say you draw a circumference around the central potential. How many "standing waves" can you fit on that circumference, especially if you are allowed only certain wavelengths? Do you think this changes as you increase the radius of the circumference?

What you are solving even for a spherically symmetric potential is similar to that. Even though the potential only depends on r, the angular part of the wavefunction has angular dependence because of what I just mentioned. It is similar to Sommerfled quantization procedure, which also only had a central potential.

Zz.