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European Journal of Physics

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

Michael Berry, R. G. Chambers, M D Large, Colin Upstill, John Charles Walmsley

Published Jul 1, 1980• 223 citations

Abstract

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.

DOI: 10.1088/0143-0807/1/3/008

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

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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.

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 .

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.

In interferometers where a wave is divided into two beams that propagate along separate branches before being recombined, the closed circuit formed by the two branches must be threaded by wave dislocation lines. For a large class of interferometers, it is shown that the (signed) dislocation number, defined in a suitable asymptotic sense, jumps by +1 as the phase difference between the beams increases by 2 π . The argument is based on the single-valuedness of the wave function in the branches and leaking between them. In some cases, the jumps occur when the phase difference is an odd multiple of π . The same result holds for the Aharonov–Bohm wave function, where the waves passing above and below a flux line experience different phase shifts; in this case, where the wave is not concentrated onto branches, the threading dislocations coincide with the flux line.

Abstract We study an ensemble of random waves subject to the Aharonov–Bohm effect. The introduction of a point with a magnetic flux of arbitrary strength into a random wave ensemble gives a family of wavefunctions whose distribution of vortices (complex zeros) is responsible for the topological phase associated with the Aharonov–Bohm effect. Analytical expressions are found for the vortex number and topological charge densities as functions of distance from the flux point. Comparison is made with the distribution of vortices in the isotropic random wave model. The results indicate that as the flux approaches half-integer values, a vortex with the same sign as the fractional part of the flux is attracted to the flux point, merging with it in the limit of half-integer flux. We construct a statistical model of the neighbourhood of the flux point to study how this vortex-flux merger occurs in more detail. Other features of the Aharonov–Bohm vortex distribution are also explored.

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.