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arXiv:quant-ph/9808034 · Birkhäuser Basel eBooks

Some Aspects of Generalized Contact Interaction in One-Dimensional Quantum Mechanics

Taksu Cheon, Takaomi Shigehara

Published Jan 1, 1999• 4 citations
arXivHTMLPDF

Abstract

A local construction is presented for a one-dimensional contact interaction (ε potential) which induces the discontinuity of the wave function while keeping its derivative continuous. By combining the ε potential and the Dirac’s δ function, we construct most general one-dimensional contact interactions allowable under the time reversal symmetry. We present some elementary results for the scattering problem which suggest a dual relation between δ and ε potentials.

DOI
10.1007/978-3-0348-8745-8_16

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A local construction is presented for a one-dimensional contact interaction (ε potential) which induces the discontinuity of the wave function while keeping its derivative continuous. By combining the ε potential and the Dirac’s δ function, we construct most general one-dimensional contact interactions allowable under the time reversal symmetry. We present some elementary results for the scattering problem which suggest a dual relation between δ and ε potentials.
Abstract The delta function potential is a simple model of zero-range contact interaction in non-relativistic quantum mechanics in one dimension. The Kronig–Penney model is a one-dimensional periodic array of delta functions and provides a simple illustration of energy bands in a crystal. Here we investigate contact interactions that generalize the delta function potential and corresponding generalizations of the Kronig–Penney model within conventional and symmetric quantum mechanics. In conventional Hermitian quantum mechanics we determine the most general contact interaction compatible with self-adjointness and in quantum mechanics we consider interactions that respect symmetry under the transformation where denotes parity and denotes time reversal. In both cases we find that the most general interaction has four independent real parameters and depending on the values of those parameters the contact interaction can support zero, one or two bound states. By contrast the conventional delta function can only support zero or one bound state. In the symmetric case moreover the two bound state energies can be both real or a complex conjugate pair. The transition from real to complex bound state energies corresponds to the spontaneous breaking of symmetry. The scattering states for the symmetric case are also found to exhibit spontaneous breaking of symmetry wherein the eigenvalues of the non-unitary S -matrix depart the unit circle in the complex plane. We also investigate the energy bands when the generalized contact interactions are repeated periodically in space in one dimension. In the Hermitian case we find that the two bound states result in two narrow bands generically separated by a gap. These bands intersect at a single point in the Brillouin zone as the interaction parameters are varied. Near the intersection the bands form a massless Dirac cone. In the symmetric case we find that as the parameters of the contact interaction are varied the two bound state bands undergo a symmetry breaking transition wherein the two band energies go from being real to being a complex conjugate pair. The symmetric Kronig–Penney model provides a simple soluble example of the transition which has the same form as in other models of symmetric crystals.
We propose a new method to construct a four parameter family of quantum-mechanical point interactions in one dimension, which is known as all possible self-adjoint extensions of the symmetric operator T=−Δ⌈C0∞(R\{0})T=-Δ\lceil C^{\infty}_{0}({\bf R} \backslash\{0\})T=−Δ⌈C0∞​(R\{0}). It is achieved in the small distance limit of equally spaced three neighboring Dirac's δδδ potentials. The strength for each δδδ is appropriately renormalized according to the distance and it diverges, in general, in the small distance limit. The validity of our method is ensured by numerical calculations. In general cases except for usual δδδ, the wave function discontinuity appears around the interaction and one can observe such a tendency even at a finite distance level.
A “one-point” approximation is proposed to investigate the transmission of electrons through the extra thin heterostructures composed of two parallel plane layers. The typical example is the bilayer for which the squeezed potential profile is the derivative of Dirac’s delta function. The Schr¨odinger equation with this singular one-dimensional profile produces a family of contact (point) interactions each of which (called a “distributional” б′-potential) depends on the way of regularization. The discrepancies widely discussed so far in the literature regarding the family of delta derivative potentials are eliminated using a two-scale power-connecting parametrization of the bilayer potential that enables one to extend the family of distributional б′-potentials to a whole class of “generalized” б′-potentials. In a squeezed limit of the bilayer structure to zero thickness, the resonant tunneling through this structure is shown to occur in the form of sharp peaks located on the sets of Lebesgue’s measure zero (called resonance sets). A four-dimensional parameter space is introduced for the representation of these sets. The transmission on the complement sets in the parameter space is shown to be completely opaque.