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

Journal of Physics A Mathematical and General

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

Michael Berry, Marko Robnik

Published Jun 1, 1986• 37 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/19/8/018

Related papers

Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue

Michael Berry, R. G. Chambers, M D Large +2 more

The authors study the wavefronts (i.e. the surfaces of constant phase) of the wave discussed by Aharonov and Bohm, representing a beam of particles with charge q scattered by an impenetrable cylinder of radius R containing magnetic flux Phi . Defining the quantum flux parameter by alpha =q Phi /h, they show that for the case R=0 the wave psi AB possesses a wavefront dislocation on the flux line, whose strength (i.e. the number of wave crests ending on the dislocation) equals the nearest integer to alpha . When alpha passes through half-integer values, the strength changes, by wavefronts unlinking and reconnecting along a nodal surface. In quantum mechanics this phase structure is unobservable, but they devise an analogue where surface waves on water encounter an irrotational 'bathtub' vortex; in this case alpha depends on the frequency of the waves and the circulation of the vortex. Experiments show dislocation structures agreeing with those predicted. psi AB is an unusual function in which incident and scattered waves cannot be clearly separated in all asymptotic directions; they discuss its properties using a new asymptotic method.

European Journal of Physics•Jul 1, 1980•223 citations

Circular quantum billiard with a singular magnetic flux line

S. M. Reimann, M. Brack, A. G. Magner +2 more

We discuss the application of Gutzwiller's semiclassical theory to a circular billiard with a singular magnetic flux line added at its center. The Aharonov-Bohm effect manifests itself through the cancellation of periodic orbits for particular flux strengths. Diffraction phenomena affect the gross-shell structure of the level density and require corrections of higher order in \ensuremath{\Elzxh}. The full quantization of the level spectrum, however, is much less affected. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Jan 1, 1996•43 citations

Semi-classical mechanics in phase space: A study of Wigner’s function

Michael Berry

Abstract We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.

Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences•Oct 4, 1977•575 citations

Regular and irregular semiclassical wavefunctions

Michael Berry

The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.

Journal of Physics A Mathematical and General•Dec 1, 1977•1,236 citations

Gauge-invariant Aharonov–Bohm streamlines

Michael Berry

Abstract The phase gradient of the wave describing the Aharonov–Bohm effect (AB) is proportional to the local canonical momentum. This vector field contains vortices (phase singularities), whose strengths cannot be detected in quantum mechanics because they increase (discontinuously) with the magnetic flux, violating gauge invariance. The analogous quantity which is gauge-invariant is the kinetic momentum field, proportional to the local electron velocity. Investigation of the streamlines (integral curves) of this velocity field reveals that as the flux increases from 0 to 1/2 (in quantum units), a vortex V is generated at the flux line, accompanied by a stagnation point (saddle) S that emerges from V and then collapses back into V. The VS pair is always small: the maximum distance between V and S is approximately 0.0209 de Broglie wavelengths. The VS phenomenon survives generalization to a superposition of AB waves. If the flux is confined within an impenetrable tube of radius R , S persists if R < 0.004 de Broglie wavelengths, and is swallowed by the tube for larger R . An experiment is envisaged.

Journal of Physics A Mathematical and Theoretical•Sep 8, 2017•9 citations

Aharonov - Bohm geometric phases for rotated rotators

Michael Berry

On a ring threaded by a flux line moves a charged quantum particle whose rotation is hindered by an angle-dependent potential. When the potential is rigidly rotated through , the particle (in the n th eigenstate of the potential on the ring) acquires a geometric phase . A general formula is , where is the dimensionless quantum flux and is a Schrödinger current associated with the state. Properties of are obtained in terms of the transmission coefficient round the ring. vanishes when the box is impenetrable or when is integer or half-integer. Energy levels form bands, with playing the role of Bloch pseudomomentum. For unhindered semiclassical states above the barrier, a WKB theory gives the geometric phase and hence the (previously calculated) classical Hannay angle . This does not vanish when is integer or half-integer, but these cases correspond to band edges where the semiclassical states are degenerate and differ greatly from the true asymptotic states. The theory is illustrated by the exact calculation of for a model where the potential is a delta function.

Journal of Physics A Mathematical and General•Dec 7, 1997•3 citations
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 .
The authors study the wavefronts (i.e. the surfaces of constant phase) of the wave discussed by Aharonov and Bohm, representing a beam of particles with charge q scattered by an impenetrable cylinder of radius R containing magnetic flux Phi . Defining the quantum flux parameter by alpha =q Phi /h, they show that for the case R=0 the wave psi AB possesses a wavefront dislocation on the flux line, whose strength (i.e. the number of wave crests ending on the dislocation) equals the nearest integer to alpha . When alpha passes through half-integer values, the strength changes, by wavefronts unlinking and reconnecting along a nodal surface. In quantum mechanics this phase structure is unobservable, but they devise an analogue where surface waves on water encounter an irrotational 'bathtub' vortex; in this case alpha depends on the frequency of the waves and the circulation of the vortex. Experiments show dislocation structures agreeing with those predicted. psi AB is an unusual function in which incident and scattered waves cannot be clearly separated in all asymptotic directions; they discuss its properties using a new asymptotic method.
We discuss the application of Gutzwiller's semiclassical theory to a circular billiard with a singular magnetic flux line added at its center. The Aharonov-Bohm effect manifests itself through the cancellation of periodic orbits for particular flux strengths. Diffraction phenomena affect the gross-shell structure of the level density and require corrections of higher order in \ensuremath{\Elzxh}. The full quantization of the level spectrum, however, is much less affected. \textcopyright{} 1996 The American Physical Society.
Abstract We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.
The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.
Abstract The phase gradient of the wave describing the Aharonov–Bohm effect (AB) is proportional to the local canonical momentum. This vector field contains vortices (phase singularities), whose strengths cannot be detected in quantum mechanics because they increase (discontinuously) with the magnetic flux, violating gauge invariance. The analogous quantity which is gauge-invariant is the kinetic momentum field, proportional to the local electron velocity. Investigation of the streamlines (integral curves) of this velocity field reveals that as the flux increases from 0 to 1/2 (in quantum units), a vortex V is generated at the flux line, accompanied by a stagnation point (saddle) S that emerges from V and then collapses back into V. The VS pair is always small: the maximum distance between V and S is approximately 0.0209 de Broglie wavelengths. The VS phenomenon survives generalization to a superposition of AB waves. If the flux is confined within an impenetrable tube of radius R , S persists if R < 0.004 de Broglie wavelengths, and is swallowed by the tube for larger R . An experiment is envisaged.
On a ring threaded by a flux line moves a charged quantum particle whose rotation is hindered by an angle-dependent potential. When the potential is rigidly rotated through , the particle (in the n th eigenstate of the potential on the ring) acquires a geometric phase . A general formula is , where is the dimensionless quantum flux and is a Schrödinger current associated with the state. Properties of are obtained in terms of the transmission coefficient round the ring. vanishes when the box is impenetrable or when is integer or half-integer. Energy levels form bands, with playing the role of Bloch pseudomomentum. For unhindered semiclassical states above the barrier, a WKB theory gives the geometric phase and hence the (previously calculated) classical Hannay angle . This does not vanish when is integer or half-integer, but these cases correspond to band edges where the semiclassical states are degenerate and differ greatly from the true asymptotic states. The theory is illustrated by the exact calculation of for a model where the potential is a delta function.