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 ... (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.


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