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 where P$_{l}$ ( x ) are the the Legendre polynomials which are given by the Rodrigues formula [1, 2]: 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. 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). 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 − 1 2 d $^{2}$y ( θ ) dθ 2 +$^{{}$ m 2 − 1 4 2 sin $^{2}$( θ )$^{}}$ y ( θ ) = Ey ( θ ) (12) Θ( θ ) = sin − 1 2 ( θ ) y ( θ ) (11) P$_{l}$ ( x ) = 1 2 $^{l}$l !$^{(}$ d dx$^{)}$$^{l}$ ( x 2 − 1) l (10) P m l ( x ) = (1 − x $^{2}$) | m | 2 ( d dx$^{)}$$^{|}$$^{m}$$^{|}$ P$_{l}$ ( x ) (9) Θ( θ ) = N m l P m l (cos θ ) (8) III. THE POLAR POTENTIAL 4