Eigenstates for Pöschl-Teller Potentials
It has been long known that the Schrödinger equation for a class of potentials of the form , usually referred to as Pöschl–Teller potentials, is exactly solvable. The eigenvalue problem[more]
(in units with has physically significant solutions for , for both bound and continuum states. For , we find the solution , , which follows simply from the derivative relation . More generally, the Schrödinger equation has the bound state solutions
, , , ,
where the are associated Legendre polynomials.
The Schrödinger equation has, in addition, continuum positive-energy eigenstates with . The trivial case gives a free particle . The first two nontrivial solutions are and . These represent waves traveling left to right. A remarkable property of Pöschl-Teller potentials is that they are "reflectionless", meaning that waves are 100% transmitted through the barrier with no reflected waves.[less]
G. Pöschl and E. Teller, "Bemerkungen zur Quantenmechanik des Anharmonischen Oszillators", Z. Phys., 83(3,4), 1933 pp. 143–151.
A recent discussion of reflectionless scattering is given by J. Lekner, "Reflectionless Eigenstates of the Potential," American Journal of Physics, 75(12), 2007 pp. 1151–1157.