Now, we show how to relate the behavior of the potential to that of the eigenenergies, eigenstates and the original associate Legendre functions. Let’s look again at the graphs of the potential for several m in Fig.(1). Here we note that as m increases the bottom of the potential is shifted up, and this is responsible for increasing the ground state energy with increasing m . The second observation here is that as one goes for higher values of m , the potential tends to confine the particle closer to the mid point of the potential $^{(}$θ = π 2 $^{)}$, which means confining the particle closer to the xy -plane. This is confirmed by looking at the behavior of the ground states for several m . In Fig. (3) we plot the ground states for various values of m just below the corresponding potential. It is clear that as m increases the ground state gets squeezed closer to middle of the box, giving rise to more confinement of the particle. The last line in Fig.(3) shows the corresponding associated Legendre functions which are related to the eigenfunctions through the transformation (11). It is quite obvious that the particle is more likely to be found near the xy − plane as m increases. Since higher m means higher angular momentum, this squeezing of the states towards the xy − plane can be explained semi-classically in terms of the increase of the moment of inertia of the particle about the z − axis. Finally, while the examples we have considered above are for the ground states only, the same arguments apply equally well to the excited states. Fig. (4) shows the first excited states; n = 1, for three m -values. It is clear that as one goes up in m , the particle gets more squeezed towards the xy -plane. In conclusion, we see that we can view the angular distribution in the wave functions of a particle in a central potential as being a result of confinement by an m − dependent ”effective” polar potential. [1] David Griffiths. Introduction to Quantum Mechanics . Pearson Prentice Hall, second edition, 2005. [2] Mary Boas. Mathematical Methods in the Physical Sciences . Wiley, third edition, 2006. [3] G. P¨ oschl and E. Teller. The factorization method. Z. Phys. , 83:143–151, 1933. [4] S.Flugge. Practical Quantum Mwchanics I . Springer-Verlag, 1970. [5] L. Infeld and T. E. Hull. The factorization method. Rev. Mod. Phys. , 23(1):21–68, Jan 1951. [6] Ranabir Dutt, Avinash Khare, and Uday P. Sukhatme. Am.J. Phys. , 56(2):163–168, 1988. 7