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American Journal of Physics

Reflectionless eigenstates of the sech2 potential

John Lekner

Published Nov 20, 2007• 134 citations
Publisher

Abstract

The one-dimensional potential well V(x)=−(ℏ2ν(ν+1)∕2ma2)sech2(x∕a) does not reflect waves of any energy when ν 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 ν is an integer. The special properties of the integer ν potentials are associated with critical binding.

DOI
10.1119/1.2787015

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In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x)V(x)V(x) exceeds the value of the total energy EEE, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x)V(x)V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a.{V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}V(x),=0forx<0andforx>a,=V0​>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0x=0x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=ax=ax=a at about the time at which the incident packet reaches the point x=0x=0x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.
We construct exact, positive-energy, normalizable wave-packet solutions of the Dirac equation in the axisymmetric potential V=− v0/ρV=-\,v_0/ρV=−v0​/ρ -- to our knowledge, the first such solutions in any external potential. Remarkably, one family comprises only elementary functions whose longitudinal profiles reproduce the free-Schrödinger Hermite--Gauss wave packets in the nonrelativistic limit. All packets share two striking features: (i) a probability density that is pointwise decoupled from spin orientation -- despite the inherent spin-orbit coupling of the Dirac equation -- and (ii) a complete freezing of their time evolution at the critical coupling v0→ℏc/2v_0\to\hbar c/2v0​→ℏc/2. We also present a simple scheme that maps solutions of the 2D Helmholtz equation to further exact Dirac wave packets.
The specific features of scattering of the Gaussian wave packet by the simplest of all reflectionless potentials, the Sech-squared well, are considered. The results of numerical computation show that a wave packet that has passed through this potential overtakes a free wave packet and is both higher and narrower than a free wave packet; in other words, the scattered wave packet accelerates inside the well and preserves its shape better. Comparisons with scattering by the square well and by a series of reflectionless potentials are also presented. Possible applications of the Sech-squared as well as other reflectionless potentials are discussed.