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arXiv:1610.09581

Robin boundary conditions are generic in quantum mechanics

Gwyneth Allwright, David M. Jacobs

Published Oct 29, 2016• 2 citations
arXivHTMLPDF

Abstract

Robin (or mixed) boundary conditions in quantum mechanics have received considerable attention in the last two decades, in particular, for applications to nanoscale systems. However, their utility has remained obscure to the larger physics community; in fact, one may find extant claims in the literature about the supposed naturalness of the specific Dirichlet (or vanishing) boundary condition. Here we demonstrate that not only are Dirichlet boundary conditions unnatural, but that Robin boundary conditions have utility in the long-wavelength approximation of short-ranged potentials. For illustration, we consider a non-relativistic particle in one dimension under the influence of the multi-step and Morse potential. We derive the scattering and bound states for these potentials and determine the parameters that describe those states in an effective system, namely one in which there is a free particle with a boundary. This method appears to be generically applicable and also generalizable to many other real systems.

DOI
10.48550/arxiv.1610.09581

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Robin (or mixed) boundary conditions in quantum mechanics have received considerable attention in the last two decades, in particular, for applications to nanoscale systems. However, their utility has remained obscure to the larger physics community; in fact, one may find extant claims in the literature about the supposed naturalness of the specific Dirichlet (or vanishing) boundary condition. Here we demonstrate that not only are Dirichlet boundary conditions unnatural, but that Robin boundary conditions have utility in the long-wavelength approximation of short-ranged potentials. For illustration, we consider a non-relativistic particle in one dimension under the influence of the multi-step and Morse potential. We derive the scattering and bound states for these potentials and determine the parameters that describe those states in an effective system, namely one in which there is a free particle with a boundary. This method appears to be generically applicable and also generalizable to many other real systems.
The physical origin of Robin boundary conditions is investigated for wavefunctions at an infinite reflecting wall. We consider both Schrödinger and phase-space quantum mechanics (also known as deformation quantization) for this simple example of a contact interaction. A non-relativistic particle moving freely on the half-line is treated as moving on the full line in the presence of an infinite potential wall, realized as a limit of a Morse potential. We show that the wavefunctions for the Morse states can become those for a free particle on the half-line with Robin boundary conditions. However, Dirichlet boundary conditions (standard walls) are obtained unless a mass-dependent fine tuning (to a reflection resonance) is imposed. This phenomenon was already observed for piece-wise flat potentials, so it is not removed by smoothing. We argue that it explains why standard quantum walls are standard. Next we consider the Wigner functions (the symbols of both diagonal and off-diagonal density operator elements) of phase-space quantum mechanics. Taking the (fine-tuned) limit, we show that our Wigner functions do reduce to the expected ones on the half-line. This confirms that the Wigner transform should indeed be unmodified for this contact interaction.
Abstract Real physical systems are only understood, experimentally or theoretically, to a finite resolution so in their analysis there is generally an ignorance of possible short-range phenomena. It is also well-known that the boundary conditions of wavefunctions and fields can be used to model short-range interactions when those interactions are expected, a priori . Here, a real-space approach is described wherein an artificial boundary of ignorance is imposed to explicitly exclude from analysis the region of a system wherein short-distance effects may be obscure. The (artificial) boundary conditions encode those short-distance effects by parameterizing the possible UV completions of the wavefunction. Since measurable quantities, such as spectra and cross sections, must be independent of the position of the artificial boundary, the boundary conditions must evolve with the radius of the boundary in a particular way. As examples of this approach, an analysis is performed of the non-relativistic free particle, harmonic oscillator, and Coulomb potential, and some known results for point-like (contact) interactions are recovered, however from a novel perspective. Generically, observables differ from their canonical values and symmetries are anomalously broken compared to those of idealized models. Connections are made to well-studied physical systems, such as the binding of light nuclei and cold atomic systems. This method is arguably more physically transparent and mathematically easier to use than other techniques that require the regularization and renormalization of delta-function potentials, and may offer further generalizations of practical use.
A proposal is given, of how to implement point interactions and boundary conditions in the path integral. The starting point is a path-integral formulation of a Dirac particle in one dimension. The implementation of a point interaction yields, by means of a perturbation expansion, the corresponding Green function for a relativistic particle. In the non-relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen. By a proper combination of the various possibilities of implementing the point interaction, the whole range of the four-parameter family of boundary conditions for point interactions can be exploited, in the relativistic case as well as in the non-relativistic limit. In addition, making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line. In particular, Dirichlet and Neumann boundary conditions emerge. The method is illustrated with some examples.
The problem of determining the Schr\"odinger wave function of a nonrelativistic particle that is either scattered by a potential of a finite range or that is bound to it is reformulated in a novel way. It is shown that in either case the wave function must satisfy a certain boundary condition on the surface that delimits the effective range of the potential. For scattering states the boundary condition is analogous to the mathematical formulation of the Ewald-Oseen extinction theorem of classical electromagnetic theory. The new formulation is illustrated by determining the scattering states and the bound states for a central potential. It is also shown that a boundary condition that is used in band-structure calculations in solids is an immediate consequence of our quantum-mechanical extinction theorem for bound states.
We consider the general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the “stargenvalue” equation and to the time evolution equation. These corrections can be cast in the form of a boundary potential contributing to the total Hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels. We show that a completely analogous formulation (in terms of boundary potentials) is also possible in standard operator quantum mechanics and that the Wigner and the operator formulations are also in one-to-one correspondence in the confined case. In particular, we extend Baker’s converse construction to bounded systems. Finally, we elaborate on the applications of the formalism to the subject of Wigner trajectories, namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit.