The eigenvalue equation for the Hamiltonian

reads, in polar coordinates

,

:

. In the quantum-mechanical position basis (

-representation), the momentum operator

("nabla" operator), so that the energy eigenvalue equation is transformed into a partial differential equation;

is Planck's constant

. The unique solutions regular at

satisfying the boundary condition

are Bessel functions

of integer order

:

, where

represent the

zero of the Bessel function

,

. The integers

give the

* * components of the angular momentum

. The wave-mechanical probability densities

are oscillating functions and show analogous behavior to those of corresponding rectangular potential

* *problems in one, two, and three dimensions, respectively.

[1] J. J. Sakurai,

*Modern Quantum Mechanics*, Reading, MA: Addison–Wesley Publishing Company, 1995.

[2] L. D. Landau and E. M. Lifschitz,

*Quantum Mechanics, *Reading, MA: Addison-Wesley Publishing Company,

* *1958

*.*