Pöschl–Teller potential
In mathematical physics, a Pöschl-Teller potential, named after the physicists Herta Pöschl and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions.Definition
In its symmetric form is explicitly given by
and the solutions of the time-independent Schrödinger equation
with this potential can be found by virtue of the substitution, which yields
Thus the solutions are just the Legendre functions with, and,. Moreover, eigenvalues and scattering data can be explicitly computed. In the special case of integer, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg-de Vries equation.
The more general form of the potential is given byRosen–Morse potential
A related potential is given by introducing an additional term:
