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

Flux String in Quantum Billiards with Two Particles

Taksu Cheon, Takaomi Shigehara

Published Jan 1, 1996• 2 citations
OpenAlex

Abstract

We examine the quantum motion of two particles interacting through a contact force which are confined in a rectangular domain in two and three dimensions. When there is a difference in the mass scale of two particles, adiabatic separation of the fast and slow variables can be performed. Appearance of the Berry phase and magnetic flux is pointed out. The system is reduced to a one-particle Aharonov-Bohm billiard in two-dimensional case. In three dimension, the problem effectively becomes the motion of a particle in the presence of closed flux string in a box billiard. 3.65.Bz, 5.45.+b, 11.27.+d Typeset using REVTEX 1 The quantum billiard problem has been instrumental in revealing non-trivial features of seemingly simple systems with minimal degrees of freedom. As a consequence, unlike real-world billiards, quantum physicists ’ version of the billiard problem has only one particle moving within a walled boundary. The dynamics is controlled by the shape of the boundary,

Related papers

Dynamics of a particle confined in a two-dimensional dilating and deforming domain

Fabio Anzà, Sara Di Martino, ANTONINO MESSINA +1 more

Some recent results concerning a particle confined in a one-dimensional box with moving walls are briefly reviewed. By exploiting the same techniques used for the 1D problem, we investigate the behavior of a quantum particle confined in a two-dimensional box (a 2D billiard) whose walls are moving by recasting the relevant mathematical problem with moving boundaries in the form of a problem with fixed boundaries and time-dependent Hamiltonian. Changes of the shape of the box are shown to be important, as it clearly emerges from the comparison between the ‘pantographic’ case (same shape of the box through the entire process) and the case with deformation.

arXiv:1405.7195•Physica Scripta•Jun 1, 2015•7 citations

Berry phase , Aharonov-Bohm effect and topology

Donald H. Kobe

The Berry phase for an electron in a one-dimensional box rotated around a magnetic flux line has contributions from the geometry and the magnetic flux, which gives an Aharonov-Bohm effect. For a circular box enclosing the magnetic flux, the Berry phase depends on the boundary conditions.

Journal of Physics A Mathematical and General•Aug 7, 1991•6 citations

Information Theory and Semi-Quantum Maxent

Angel Ricardo Plastino, Claudia M. Sarris

This books presents a detailed treatment of the physics of the interaction between quantum and classical variables. Semiquantum Hamiltonians represent mixed physical systems for which one part is quantal and the other classical. These parts are coupled in nonlinear fashion. The coupling term contains both quantum and classical degrees of freedom. Physical systems that are appropriately described in this way are are to be found in molecular and solid state physics. Of course, so described is the interaction between a classical measurement's instrument and the quantum system being measured. Semiquantum systems have a technological counterpart as well, in connection with nano-technology (molecular transistors, quantum dots and SQUIDS, for instance). These special systems can be successfully modelled by quantum billiards, dynamic system in which a particle alternates between motion in a straight line and elastic reflections from a boundary. The interest focuses in vibrating quantum billiards, in which a particle of mass m is subjected to elastic interactions confined in a potential well V(q), generated by a classical mass M.

•Mar 21, 2014•0 citations

Quantum states without time-reversal symmetry: wavefront dislocations in a non-integrable Aharonov-Bohm billiard

Michael Berry, Marko Robnik

The authors display complex wavefunctions psi j for the jth eigenstate of a particle moving freely in a domain D (whose reflecting boundary gives classically chaotic motion) threaded by a single line of magnetic flux whose strength (in quantum units) is alpha . The wavefronts (phase contours of psi j ) show the expected dislocation singularities both away from the flux line (where their strength is +or-1) and at the flux line (where the strength is the integer closest to alpha ). psi 5 shows two dislocations away from the flux line for all alpha , and the birth of a dislocation at the flux line as alpha passes the value 1/2. psi 50 shows 43 dislocations, in reasonable agreement with a semiclassical theory based on regarding psi N as a complex Gaussian random function, which predicts N dislocations in the asymptotic limit N to infinity .

Journal of Physics A Mathematical and General•Jun 1, 1986•37 citations

The Aharonov–Bohm effect and fractional statistics of distinguishable particles

Stefan V. Mashkevich

In two-dimensional space, the topological coupling arising between charged particles with attached magnetic fluxes, via the Aharonov–Bohm effect, causes distinguishable particles to effectively acquire quantum statistics: a nontrivial quantum phase is generated when a particle of one species encircles one of a different species. We discuss a number of exact and numerical results concerning distinguishable particles with fractional statistics. Just like for anyons, the spectrum of such particles confined to the lowest Landau level of a strong magnetic field can be found exactly. Then a system of particles in the presence of static magnetic fluxes (equivalent to particles with infinite mass) is considered, and the low-lying states of one particle in the presence of two fluxes and two particles in the presence of one flux are analyzed.

Journal of Physics A Mathematical and Theoretical•Aug 12, 2010•0 citations

The quantum phase 2-form near degeneracies: two numerical studies

Raúl J. Mondragón, Michael Berry

Abstract The phase of a quantum state changes rapidly as parameters X = (X1, X2,...) are varied near a degeneracy X*, reflecting the monopole singularity of the underlying phase 2-form V(X) at X*. The singularities may be sources or sinks of V. We study them numerically and display them graphically for two families of hamiltonians whose degeneracy structure is typical. First is a particle moving along a line segment with kinetic energy quartic in the momentum (‘quartic-momentum square well’); the X are incorporated into the boundary conditions. Second is a charged particle moving in a domain D of the plane which is threaded by a magnetic flux line of strength α, with wavefunction vanishing on the boundary ∂D (‘Aharonov-Bohm billiards’); the X are α and parameters specifying ∂D; V is not invariant under gauge transformations of the vector potential generating the flux. For Aharonov-Bohm billiards we study how the spatial patterns of phase of wavefunctions change round circuits near degeneracies; these patterns also have singularities (wavefront dislocations) that appear and disappear by colliding with each other and with ∂D.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Aug 8, 1989•16 citations
We examine the quantum motion of two particles interacting through a contact force which are confined in a rectangular domain in two and three dimensions. When there is a difference in the mass scale of two particles, adiabatic separation of the fast and slow variables can be performed. Appearance of the Berry phase and magnetic flux is pointed out. The system is reduced to a one-particle Aharonov-Bohm billiard in two-dimensional case. In three dimension, the problem effectively becomes the motion of a particle in the presence of closed flux string in a box billiard. 3.65.Bz, 5.45.+b, 11.27.+d Typeset using REVTEX 1 The quantum billiard problem has been instrumental in revealing non-trivial features of seemingly simple systems with minimal degrees of freedom. As a consequence, unlike real-world billiards, quantum physicists ’ version of the billiard problem has only one particle moving within a walled boundary. The dynamics is controlled by the shape of the boundary,
Some recent results concerning a particle confined in a one-dimensional box with moving walls are briefly reviewed. By exploiting the same techniques used for the 1D problem, we investigate the behavior of a quantum particle confined in a two-dimensional box (a 2D billiard) whose walls are moving by recasting the relevant mathematical problem with moving boundaries in the form of a problem with fixed boundaries and time-dependent Hamiltonian. Changes of the shape of the box are shown to be important, as it clearly emerges from the comparison between the ‘pantographic’ case (same shape of the box through the entire process) and the case with deformation.
This books presents a detailed treatment of the physics of the interaction between quantum and classical variables. Semiquantum Hamiltonians represent mixed physical systems for which one part is quantal and the other classical. These parts are coupled in nonlinear fashion. The coupling term contains both quantum and classical degrees of freedom. Physical systems that are appropriately described in this way are are to be found in molecular and solid state physics. Of course, so described is the interaction between a classical measurement's instrument and the quantum system being measured. Semiquantum systems have a technological counterpart as well, in connection with nano-technology (molecular transistors, quantum dots and SQUIDS, for instance). These special systems can be successfully modelled by quantum billiards, dynamic system in which a particle alternates between motion in a straight line and elastic reflections from a boundary. The interest focuses in vibrating quantum billiards, in which a particle of mass m is subjected to elastic interactions confined in a potential well V(q), generated by a classical mass M.
The authors display complex wavefunctions psi j for the jth eigenstate of a particle moving freely in a domain D (whose reflecting boundary gives classically chaotic motion) threaded by a single line of magnetic flux whose strength (in quantum units) is alpha . The wavefronts (phase contours of psi j ) show the expected dislocation singularities both away from the flux line (where their strength is +or-1) and at the flux line (where the strength is the integer closest to alpha ). psi 5 shows two dislocations away from the flux line for all alpha , and the birth of a dislocation at the flux line as alpha passes the value 1/2. psi 50 shows 43 dislocations, in reasonable agreement with a semiclassical theory based on regarding psi N as a complex Gaussian random function, which predicts N dislocations in the asymptotic limit N to infinity .
In two-dimensional space, the topological coupling arising between charged particles with attached magnetic fluxes, via the Aharonov–Bohm effect, causes distinguishable particles to effectively acquire quantum statistics: a nontrivial quantum phase is generated when a particle of one species encircles one of a different species. We discuss a number of exact and numerical results concerning distinguishable particles with fractional statistics. Just like for anyons, the spectrum of such particles confined to the lowest Landau level of a strong magnetic field can be found exactly. Then a system of particles in the presence of static magnetic fluxes (equivalent to particles with infinite mass) is considered, and the low-lying states of one particle in the presence of two fluxes and two particles in the presence of one flux are analyzed.
Abstract The phase of a quantum state changes rapidly as parameters X = (X1, X2,...) are varied near a degeneracy X*, reflecting the monopole singularity of the underlying phase 2-form V(X) at X*. The singularities may be sources or sinks of V. We study them numerically and display them graphically for two families of hamiltonians whose degeneracy structure is typical. First is a particle moving along a line segment with kinetic energy quartic in the momentum (‘quartic-momentum square well’); the X are incorporated into the boundary conditions. Second is a charged particle moving in a domain D of the plane which is threaded by a magnetic flux line of strength α, with wavefunction vanishing on the boundary ∂D (‘Aharonov-Bohm billiards’); the X are α and parameters specifying ∂D; V is not invariant under gauge transformations of the vector potential generating the flux. For Aharonov-Bohm billiards we study how the spatial patterns of phase of wavefunctions change round circuits near degeneracies; these patterns also have singularities (wavefront dislocations) that appear and disappear by colliding with each other and with ∂D.