1. Orbital angular momentum
Consider the angular wave function:
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The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
The normalized spherical harmonics are:
The wavefunction is:
Using Euler's identity:
The solution shows how to write the probability density function of a superposition of spherical harmonics as a product of normalized independent angular distribution functions