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Journal of Mathematical Physics

Bound states for square well potentials extending to infinity in D≥2

H. Rupertsberger

Published Apr 1, 1993• 0 citations
Publisher

Abstract

It is well known that quantum mechanics allows the penetration into classically forbidden regions (tunneling). Less well known seems to be the fact that in some sense the converse is true also. Potentials with classically allowed regions where a particle can move freely to infinity can nevertheless lead to bound states in quantum mechanics due to the stringent requirements of the boundary conditions, thus forbidding an escape to infinity. This effect is demonstrated by using an obvious generalization of the well-known one-dimensional (D=1) square well potential to arbitrary space dimensions.

DOI
10.1063/1.530210

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It is well known that quantum mechanics allows the penetration into classically forbidden regions (tunneling). Less well known seems to be the fact that in some sense the converse is true also. Potentials with classically allowed regions where a particle can move freely to infinity can nevertheless lead to bound states in quantum mechanics due to the stringent requirements of the boundary conditions, thus forbidding an escape to infinity. This effect is demonstrated by using an obvious generalization of the well-known one-dimensional (D=1) square well potential to arbitrary space dimensions.
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.
For a quantum confinement model, the wave function of a particle is zero outside the confined region. Due to this, the negative energy states are, in fact, square integrable. As negative energy states are not physical, we need to impose some boundary conditions in order to avoid these states. For the case of the infinite square well model, we show the possible boundary conditions that avoid the existence of the negative energy states. The well-known boundary condition requiring the wave function be continuous on both sides of the well is one of the boundary conditions that avoids negative energy states. However, there are other types of boundary condition that can also avoid the negative energy states. This shows that there are the different branch of physical systems that can be established on the infinite square well model.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].