Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1804.06681 · Journal of Physics A Mathematical and Theoretical

Contact interactions and Kronig–Penney models in Hermitian and P T symmetric quantum mechanics

Foster Thompson, Katherine Brown, Harsh Mathur, Kristin McKee

Published Oct 16, 2018• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aae8af

Related papers

Analytic properties of energy levels in models with delta-function potentials

A.G. Ushveridze

The analytic properties of energy levels as the functions of coupling parameters in one-dimensional quantum mechanical models described by the potentials of delta-function type are investigated. The following models are considered: (i) the rectangular potential well with delta-function barrier, (ii) two delta-function wells in the unbounded space, (iii) the Kronig-Penney model with periodic delta-function potential, and (iv) the Bose gas with delta-function interaction between particles.

Journal of Physics A Mathematical and General•Feb 21, 1988•22 citations

Solvable simulation of a double-well problem in -symmetric quantum mechanics

Miloslav Znojil

Within the framework of 𝒫𝒯-symmetric quantum mechanics of bound states (which works with parity-pseudo-Hermitian Hamiltonians H = 𝒫 H †𝒫 and real spectra) we mimic some effects of the double-well structure of potentials by a pair of δ functions with mutually complex conjugate strengths. The model is solvable by the standard matching technique and exhibits several interesting features. We observe an amazingly robust absence of any 𝒫𝒯-symmetry breaking. A quasi-degeneracy which occurs in the high-energy domain is interpreted as a manifestation of certain ‘quantum beats’.

arXiv:quant-ph/0303122•Journal of Physics A Mathematical and General•Jun 24, 2003•25 citations

Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation

Hugh F Jones

We consider the interaction between the Hermitian world, represented by a real delta-function potential $\ensuremath{-}\ensuremath{\alpha}\ensuremath{\delta}(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\ensuremath{\beta}(\ensuremath{\delta}(x\ensuremath{-}L)\ensuremath{-}\ensuremath{\delta}(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave function is small for $L$ large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $\ensuremath{\eta}$, and consequently a new wave function $\ensuremath{\Psi}(x)$, defined in terms of the original wave function $\ensuremath{\psi}(x)$ by means of $\ensuremath{\eta}$. In this picture, working perturbatively in $\ensuremath{\beta}$, the bound-state wave function is actually unchanged from its unperturbed form for $|x|\ensuremath{\ll}L$. However, the scattering wave function, for $|x|\ensuremath{\gg}L$, is changed in a significant manner. In particular, there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.

arXiv:0805.1656•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 19, 2008•66 citations

Application of pseudo-Hermitian quantum mechanics to a complex scattering potential with point interactions

Hossein Mehri‐Dehnavi, Ali Mostafazadeh, Ahmet Batal

We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p 2 /2 m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h . η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ + ) = −ℑ(ζ − ). This in particular contains the -symmetric case: ζ + = ζ* − . We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H . We show that these physical quantities are not directly sensitive to the presence of the -symmetry.

arXiv:1002.1221•Journal of Physics A Mathematical and Theoretical•Mar 16, 2010•25 citations

Exact solutions of the Dirac equation with surface delta interactions

Francisco Domínguez-Adame

The Dirac equation with vector plus scalar surface delta interactions supported by a sphere is exactly solved for all partial waves and parities. Unlike the relativistic one-dimensional delta potentials, surface delta interactions with finite values of the vector and scalar coupling constants require a minimum strength to bind particles. Strong scalar couplings confine relativistic particles at high energies inside the sphere, while the confinement is no longer possible for strong vector potentials. It is also found that the potential causes no effects on the scattering phase shift in the zero-range limit. Therefore, the well known Dirac-Kronig-Penney model in one dimension cannot be generalised to three-dimensional crystal lattices.

Journal of Physics A Mathematical and General•Jun 7, 1990•31 citations

Contact Interactions and Gamma Convergence

Gianfausto Dell’Antonio

In Classical Mechanics constraints describe forces restricting the motion of two systems when they are in contact. In Quantum Mechanics in the Schrodinger representation the state of the system is described by a (probability) wave and the Schrodinger equation is dispersive (it does not preserve locality of the wave function): at any time the wave function is spread out over all space. It is difficult to define "contact". To avoid this difficulty it is convenient to use the Heisenberg representation and describe the system by means of self-adjoint operators on some function space. Each self-adjoint operator operator has a domain of definition. We consider first in some detail the dynamics in R^3 and later consider the case of dimension two and dimension one. In three dimension contact interactions gives the Efimov spectrum of trimers and quadrimers in low energy physics and the Bose-Einstein condensate, both in low density and in high density. The case of dimension one is particularly interesting because the system may represent three particles on a Y-shaped graph with contact interaction at the vertex. We consider both the case of Bose particles which satisfy the Schrodinger equation and the case of spin 1\2 fermions which satisfy the Pauli equation. They form respectively Bose crystals and Fermi crystals (not necessary periodic). In both cases there is only one bound state at each vertex. In an extended crystal, due to Fermi-Dirac statistics, all bound states may be occupied to form the Fermi sea. The particles on the surface have a Dirac spectrum. We prove that in the semiclassical limit the dynamics of particles on the surface of the Fermi sea is the (classical) motion studied in detail by Novikov and Maltsev. In this semiclassical description the Fermi surface can be deformed by a magnetic field and one may have topological resonances.

Frontiers in Physics•Apr 9, 2019•4 citations
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.
Within the framework of 𝒫𝒯-symmetric quantum mechanics of bound states (which works with parity-pseudo-Hermitian Hamiltonians H = 𝒫 H †𝒫 and real spectra) we mimic some effects of the double-well structure of potentials by a pair of δ functions with mutually complex conjugate strengths. The model is solvable by the standard matching technique and exhibits several interesting features. We observe an amazingly robust absence of any 𝒫𝒯-symmetry breaking. A quasi-degeneracy which occurs in the high-energy domain is interpreted as a manifestation of certain ‘quantum beats’.
We consider the interaction between the Hermitian world, represented by a real delta-function potential $\ensuremath{-}\ensuremath{\alpha}\ensuremath{\delta}(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\ensuremath{\beta}(\ensuremath{\delta}(x\ensuremath{-}L)\ensuremath{-}\ensuremath{\delta}(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave function is small for LLL large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $\ensuremath{\eta}$, and consequently a new wave function $\ensuremath{\Psi}(x)$, defined in terms of the original wave function $\ensuremath{\psi}(x)$ by means of $\ensuremath{\eta}$. In this picture, working perturbatively in $\ensuremath{\beta}$, the bound-state wave function is actually unchanged from its unperturbed form for $|x|\ensuremath{\ll}L$. However, the scattering wave function, for $|x|\ensuremath{\gg}L$, is changed in a significant manner. In particular, there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.
We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p 2 /2 m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h . η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ + ) = −ℑ(ζ − ). This in particular contains the -symmetric case: ζ + = ζ* − . We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H . We show that these physical quantities are not directly sensitive to the presence of the -symmetry.
The Dirac equation with vector plus scalar surface delta interactions supported by a sphere is exactly solved for all partial waves and parities. Unlike the relativistic one-dimensional delta potentials, surface delta interactions with finite values of the vector and scalar coupling constants require a minimum strength to bind particles. Strong scalar couplings confine relativistic particles at high energies inside the sphere, while the confinement is no longer possible for strong vector potentials. It is also found that the potential causes no effects on the scattering phase shift in the zero-range limit. Therefore, the well known Dirac-Kronig-Penney model in one dimension cannot be generalised to three-dimensional crystal lattices.
In Classical Mechanics constraints describe forces restricting the motion of two systems when they are in contact. In Quantum Mechanics in the Schrodinger representation the state of the system is described by a (probability) wave and the Schrodinger equation is dispersive (it does not preserve locality of the wave function): at any time the wave function is spread out over all space. It is difficult to define "contact". To avoid this difficulty it is convenient to use the Heisenberg representation and describe the system by means of self-adjoint operators on some function space. Each self-adjoint operator operator has a domain of definition. We consider first in some detail the dynamics in R^3 and later consider the case of dimension two and dimension one. In three dimension contact interactions gives the Efimov spectrum of trimers and quadrimers in low energy physics and the Bose-Einstein condensate, both in low density and in high density. The case of dimension one is particularly interesting because the system may represent three particles on a Y-shaped graph with contact interaction at the vertex. We consider both the case of Bose particles which satisfy the Schrodinger equation and the case of spin 1\2 fermions which satisfy the Pauli equation. They form respectively Bose crystals and Fermi crystals (not necessary periodic). In both cases there is only one bound state at each vertex. In an extended crystal, due to Fermi-Dirac statistics, all bound states may be occupied to form the Fermi sea. The particles on the surface have a Dirac spectrum. We prove that in the semiclassical limit the dynamics of particles on the surface of the Fermi sea is the (classical) motion studied in detail by Novikov and Maltsev. In this semiclassical description the Fermi surface can be deformed by a magnetic field and one may have topological resonances.