Solutions For Potentials of Interest
Five special cases arise, of special importance:
- V(r) = 0, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
- (finite) for and 0 elsewhere, or a particle in the spherical equivalent of the square well, useful to describe scattering and bound states in a nucleus or quantum dot.
- As the previous case, but with an infinitely high jump in the potential on the surface of the sphere.
- V(r) ~ r2 for the three-dimensional isotropic harmonic oscillator.
- V(r) ~ 1/r to describe bound states of hydrogen-like atoms.
We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. This article relies heavily on Bessel functions and Laguerre polynomials.
Read more about this topic: Particle In A Spherically Symmetric Potential
Famous quotes containing the words solutions and/or interest:
“Science fiction writers foresee the inevitable, and although problems and catastrophes may be inevitable, solutions are not.”
—Isaac Asimov (19201992)
“I have an intense personal interest in making the use of American capital in the development of China an instrument for the promotion of the welfare of China, and an increase in her material prosperity without entanglements or creating embarrassment affecting the growth of her independent political power, and the preservation of her territorial integrity.”
—William Howard Taft (18571930)