Quantal phase factors accompanying adiabatic changes
08 Mar 1984-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 392, Iss: 1802, pp 45-57
TL;DR: In this article, it was shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor and a general formula for γ(C) was derived in terms of the spectrum and eigen states of the Hamiltonian over a surface spanning C.
Abstract: A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.
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Cites background from "Quantal phase factors accompanying ..."
...In 1984, Michael Berry wrote a paper that has generated immense interests throughout the different fields of physics including quantum chemistry (Berry, 1984)....
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...This conclusion remained unchallenged until Berry (1984) reconsidered the cyclic evolution of the system along a closed path C with R(T ) = R(0)....
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...Following Berry’s original paper (Berry, 1984), we first discuss how the Berry phase arises as a generic feature of the adiabatic evolution of a quantum state....
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References
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TL;DR: In this article, it was shown that there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish.
Abstract: In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials.
5,553 citations
"Quantal phase factors accompanying ..." refers background in this paper
...Finally, it is shown in § 5 th a t physical effects of magnetic vector potentials in the absence of fields, predicted by Aharonov & Bohm (1959) and observed by Chambers (i960), can be understood as special cases of the geometrical phase factor....
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...Aharonov & Bohm (1959) showed th a t in quantum mechanics such vector potentials have physical significance even though they correspond to zero field....
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Book•
27 Dec 2005
TL;DR: In this paper, the authors present a list of applications of the path integral formula in statistical mechanics, including the application of the Path Integral formula to Statistical Mechanics, asymptotic analysis, and the phase space path integral.
Abstract: Partial table of contents: Probabilities and Probability Amplitudes for Paths. Correspondence Limit for the Path Integral (Heuristic). Vector Potentials and Another Proof of the Path Integral Formula. Doing the Integral: Free Particle and Quadratic Lagrangians. Brownian Motion and the Wiener Integral Kac's Proof. Perturbation Theory and Feynman Diagrams. SELECTED APPLICATIONS OF THE PATH INTEGRAL. Asymptotic Analysis. The Calculus of Variations. WKB Near Caustics. The Phase of the Semiclassical Amplitude. Scattering Theory. Geometrical Optics. The Polaron. Spin and Related Matters. Quantum Mechanics on Curved Spaces. Relativistic Propagators and Black Holes. Applications to Statistical Mechanics. Critical Droplets. Alias Instantons, and Metastability. Phase Space Path Integral. Omissions, Miscellany, and Prejudices. Indexes.
2,079 citations
TL;DR: In this paper, it was shown that dislocations are to be expected whenever limited trains of waves, ultimately derived from the same oscillator, travel in different directions and interfere -for example in a scattering problem.
Abstract: When an ultrasonic pulse, containing, say, ten quasi-sinusoidal oscillations, is reflected in air from a rough surface, it is observed experimentally that the scattered wave train contains dislocations, which are closely analogous to those found in imperfect crystals. We show theoretically that such dislocations are to be expected whenever limited trains of waves, ultimately derived from the same oscillator, travel in different directions and interfere - for example in a scattering problem. Dispersion is not involved. Equations are given showing the detailed structure of edge, screw and mixed edge-screw dislocations, and also of parallel sets of such dislocations. Edge dislocations can glide relative to the wave train at any velocity; they can also climb, and screw dislocations can glide. Wavefront dislocations may be curved, and they may intersect; they may collide and rebound; they may annihilate each other or be created as loops or pairs. With dislocations in wave trains, unlike crystal dislocations, there is no breakdown of linearity near the centre. Mathematically they are lines along which the phase is indeterminate; this implies that the wave amplitude is zero.
1,984 citations
"Quantal phase factors accompanying ..." refers background in this paper
...A closer analogy is with wavefront dislocation lines, which are phase singularities of complex wavefunctions in three-dimensional position space (Nye & Berry 1974; Nye 1981; Berry 1981), tha t dominate the geometry of wavefronts without com pletely determining them....
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TL;DR: In this paper, an intrinsic and complete description of electromagnetism in a space-time region is formulated in terms of a nonintegrable phase factor, and connections on principal fiber bundles are identified.
Abstract: Through an examination of the Bohm-Aharonov experiment an intrinsic and complete description of electromagnetism in a space-time region is formulated in terms of a nonintegrable phase factor. This concept, in its global ramifications, is studied through an examination of Dirac's magnetic monopole field. Generalizations to non-Abelian groups are carried out, and result in identification with the mathematical concept of connections on principal fiber bundles.
1,121 citations
"Quantal phase factors accompanying ..." refers background in this paper
...The dependence on phase is of the following kind: if | ri)-> exp {ijn(R)} \n)then (n| Wn)-> + iV/£ (in another context the importance of such gauge transformations has been emphasized by Wu & Yang (1975))....
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856 citations
"Quantal phase factors accompanying ..." refers background in this paper
...This illustrates an old result of Von Neumann & Wigner (1929): for generic Hamiltonians (Hermitian matrices), it is necessary to vary three parameters in order to make a degeneracy occur accidentally, tha t is, not on account of symmetry....
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...Professor Barry Simon (1983), commenting on the original version of this paper, points out tha t the geometrical phase factor has a mathematical interpretation in terms of holonomy, with the phase two-form emerging naturally (in the form (76)) as the curvature (first Chern class) of a Hermitian line bundle....
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...The results obtained here are not restricted to quantum mechanics, but apply more generally, to the phase of eigenvectors of any Hermitian matrices under a natural continuation in parameter space....
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...I f the circuit is close to a degeneracy in the spectrum of H, y takes a particularly simple form which will be derived in § 3; this contains, as a special case, the sign change around a degeneracy of the eigenstates of a system whose Hamiltonian is real as well as Hermitian (Herzberg & Longuet-Higgins 1963; Longuet-Higgins 1975; Mead 1979; Mead & Truhlar 1979; Mead 1980a, b;Berry & Wilkinson 1984)....
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...H(R) can be represented by a 2 x 2 Hermitian matrix coupling the two states....
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