Reflectionless eigenstates of the sech2 potential

Abstract

Any energy when is a positive integer. We show that in this reflectionless case the solutions of Schrödinger's equation can be expressed in terms of elementary functions. Wave packets can be constructed from these energy eigenstates, and the propagation of such wave packets through the potential region can be studied analytically. We find that the group velocity of a particular packet can substantially exceed the group velocity of a free-space Gaussian packet. The bound states of the potential can also be expressed in terms of elementary functions when is an integer. The special properties of the integer potentials are associated with critical binding. © 2007 American Association of Physics Teachers.