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European Journal of Physics

Piecewise zero-curvature energy eigenfunctions in one dimension

L. P. Gilbert, Mario Belloni, M. A. Doncheski, Richard W. Robinett

Published Sep 6, 2006• 7 citations
Publisher

Abstract

We discuss the general mathematical condition in one-dimensional time-independent quantum mechanics for an energy eigenfunction to have zero curvature over an extended region of space and still be valid. This condition and its solution are not often discussed as part of quantum mechanics texts at any level, yet have interesting consequences for experimental, pedagogical and theoretical investigations of systems with piecewise-constant potential energy functions. We present the solutions for several standard cases in which zero-curvature energy eigenfunctions are allowed as bound states, scattering states and threshold states. These states are of interest to supersymmetric quantum mechanics and the procedures we outline can be used to 'design' quantum wells for studies of wave packet dynamics.

DOI
10.1088/0143-0807/27/6/007

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We discuss the general mathematical condition in one-dimensional time-independent quantum mechanics for an energy eigenfunction to have zero curvature over an extended region of space and still be valid. This condition and its solution are not often discussed as part of quantum mechanics texts at any level, yet have interesting consequences for experimental, pedagogical and theoretical investigations of systems with piecewise-constant potential energy functions. We present the solutions for several standard cases in which zero-curvature energy eigenfunctions are allowed as bound states, scattering states and threshold states. These states are of interest to supersymmetric quantum mechanics and the procedures we outline can be used to 'design' quantum wells for studies of wave packet dynamics.
We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wavefunction to have zero curvature over an extended region of space and still be a valid wavefunction. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space.
In an earlier paper appropriate limiting procedure is discussed in a general way for quantum mechanical energy eigenfunctions (one-dimensional bound states) — a single interpretational postulate leading smoothly to entire compatible classical objective description without facing any contradiction. The method is consistently extended to two- and three-dimensional cases and it is interesting to note that results of earlier study on the classical limit of the radial distribution function of hydrogen atom are easily obtained as special cases of our analysis.
Sawtooth potentials-piecewise-linear potentials defined by alternating rising and falling slopes-have garnered interest for their roles in classical Brownian ratchets, quantum transport, and flat bands in ultracold-atom lattices.However, a general solution of the time-independent Schrödinger equation for an arbitrary asymmetric sawtooth potential and a study of its band structure remain absent from the literature.We derive the exact analytical eigenstates in terms of Airy functions, but due to numerical difficulty, we solved for the eigenstates using a piecewise constant "staircase approximation."We find localized low-energy states confined within individual wells and plane waves for high energies.We apply the obtained solutions to investigate the relationship between bandwidth and parameters of the sawtooth potentials, which could offer design principles for realizing flatbands in condensed-matter and ultracold-atom research.We find that the bandwidth of the lowest energy band is entirely independent of the sawtooth's asymmetry, while it remains inversely correlated with potential height and cell size.The methodologies presented here provide a toolkit for further exploration of sawtooth potentials.