Historical research guide · 1150 · 1984
Quantal phase factors accompanying adiabatic changes
M. V. Berry
In one minute
Take a quantum system in a definite energy level and slowly turn the knobs on its Hamiltonian, all the way round a closed loop, so that everything ends up exactly as it started. Michael Berry's paper shows that the system does not quite come back: it picks up an extra phase, over and above the familiar one that just counts energy times time, and that extra phase depends only on the shape of the loop in the space of knob settings. Nothing is dissipated and nothing has moved; the geometry of the circuit is remembered. Berry derives a general formula for it, shows it becomes very simple near a degeneracy, and then does two concrete things with it. A spin in a slowly rotated magnetic field acquires a phase equal to the solid angle the field direction sweeps out. And the Aharonov-Bohm effect — an electron registering a magnetic flux it never touches — turns out to be a special case of the same geometrical phase.
Connection
Chapter 10 is built on the vector potential doing real physical work where there is no field to point at, and this is the paper that says what kind of work it is: phase, set by geometry. It is also the discipline of §7.4 in the site's own hand — the potential controls phase, which is settled, and this is the derivation that makes the Aharonov-Bohm effect a corollary rather than a curiosity.
From the source
A quantum system in an eigenstate, slowly transported round a circuit by varying the parameters in its Hamiltonian, acquires a geometrical phase factor in addition to the familiar dynamical phase factor — and that geometrical phase is a circuit integral in parameter space, independent of how fast the circuit is traversed.Abstract, first sentence; Section 2, Eqs. (5) and (6), and the paragraph following (6)
The geometrical phase can be written as a surface integral over any surface spanning the circuit, of a quantity built only from the spectrum and the eigenstates of the Hamiltonian. Because the dependence on the choice of eigenstate phases drops out of that expression, the result does not require a globally single-valued choice of eigenvectors — which Berry calls a surprising conclusion.Section 2, Eqs. (9) and (10), and the paragraph beginning 'Equations (9) and (10) embody the central results'
For a particle of spin s in a slowly rotated magnetic field, the geometrical phase factor is the exponential of minus i times n times the solid angle the field direction subtends at the origin of field space — the phase of a monopole field in parameter space. It depends only on the spin component along the field, not on the strength of the spin, and Berry notes that the sign change of spinors under a full rotation and the sign change of wavefunctions round a degeneracy therefore have the same mathematical origin.Section 4, Eqs. (26) and (27), and the two paragraphs following (27)
The Aharonov-Bohm effect is a special case of the geometrical phase. For a box of charge q carried round a circuit threaded by a flux line, the geometrical phase is q times the flux divided by the reduced Planck constant — independent of the state and of the circuit, provided it winds once round the line.Section 5, Eqs. (30) to (35)
Berry's derivation of the Aharonov-Bohm phase uses only single-valued wavefunctions, which he offers as a third route past the standing objection to the elementary Dirac-phase-factor presentation — that the wavefunction it constructs is not single-valued. He also records the anticipation of the effect by Ehrenberg and Siday in 1949 and its observation by Chambers in 1960.Section 5, page 54, the paragraph beginning 'In elementary presentations of the Aharonov-Bohm effect'; Section 1, final paragraph
What to watch: Berry proposes the experiment that would test the spin result directly — split a polarized monoenergetic beam in two, keep the magnetic field constant along one path and slowly rotate its direction round a circuit of adjustable solid angle along the other, recombine, and measure the count rate against solid angle. He gives the predicted fringe contrast as a cosine-squared law in the solid angle, and stresses that this differs from the earlier neutron experiments of Rauch and of Werner, whose beams were unpolarized, not in an eigenstate, and whose phase changed dynamically rather than geometrically.Section 4, page 52, Eq. (28) and the two paragraphs around it
The way in
https://doi.org/10.1098/rspa.1984.0023How to cite it
M. V. Berry (1984) Quantal phase factors accompanying adiabatic changes doi:10.1098/rspa.1984.0023