Solutions of Schrödinger Equation for a Particle in a Finite Spherical Well
This Demonstration considers a particle bound to a finite spherical well in three dimensions. The potential energy is given by[more]
The Schrödinger equation is given by
For selected values of and the angular-momentum quantum number , the bound-state eigenvalues and eigenfunctions are determined. You may choose to display the energy diagram, the radial functions or contour plots of the eigenfunctions. The potential energy includes the centrifugal contribution. For either or 1, radial functions for the first three states are plotted, with the vertical blue line marking the radius . In the contour plots, the wavefunctions are positive in the blue regions, negative in the white regions.[less]
For simplicity, set (atomic units) and represent the energy by . Clearly the Schrödinger equation is separable in spherical polar coordinates and we can write
where is a spherical harmonic. The radial functions then obey the equations:
The solutions are spherical Bessel and Hankel functions:
The eigenfunctions are determined by the continuity conditions on the wavefunctions and their derivative at : and . This can be simplified to
For , this leads to a transcendental equation
For , the corresponding relation is
Solutions for real correspond to one of the finite number of bound states (with ).
 J. Branson. "Bound States in a Spherical Potential Well." (Jul 31, 2019) quantummechanics.ucsd.edu/ph130a/130_notes/node227.html.
 "Finite Spherical Square Well Bound States." (Jul 31, 2019) www.physics.csbsju.edu/QM/square.16.html.