Solution of the Radial Schrödinger Equation by the Fox-Goodwin Method

Abstract

Solving the Schrödinger equation is the central problem of non-relativistic quantum mechanics. A simple case is the study of the motion of a particle without spin in an external potential. The time-independent Schrödinger equation in this case reads

$$ H\psi \left( r \right) = E\psi \left( r \right), $$

(13.1a)

with the Hamilton operator

$$ H = - \frac{{{\hbar ^2}}}{{2m}}\Delta + V\left( r \right). $$

(13.1b)

If the particle is scattered by the potential, the energy E may be given any positive value. If the particle is bound by the potential, E becomes negative and can only take discrete values.