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angular momentum commutation relations. The restriction of l to integers is a consequence of restricting m to integers. 

Let’s go back to equation (6), and write down its famous solutions which are the associated Legendre functions [1, 2]( N m l is a normalization constant): 

where P m l are given by 

<img src='content_image/336077.jpg'>

<img src='content_image/336078.jpg'>

where P$_{l}$ ( x ) are the the Legendre polynomials which are given by the Rodrigues formula [1, 2]: 

<img src='content_image/336076.jpg'>

The solutions | P m l (cos θ ) | 2 represent the probability of finding the particle at a certain angle θ , which means that they give the probability distribution of the particle about the z -axis. We will show below that we can view this distribution as being shaped by a polar potential. 

## III. THE POLAR POTENTIAL 

Now we can bring the polar equation (6) to the Schr¨ odinger form, just as we did with the radial equation. Defining: 

and multiplying by 1/2, equation (6) becomes 

where E = (1 / 2)( l ( l + 1) + 1 / 4). 

<img src='content_image/336074.jpg'>

<img src='content_image/336075.jpg'>

This equation can be thought of as a one dimensional Shr¨ odinger equation (with ℏ = m = 1 ) for a particle confined between 0 and π , and satisfying the boundary conditions y (0) = y ( π ) = 0. The term in brackets is a one dimensional potential, more precisely it is a family of potentials depending on the choice of the value of m . These potentials 

