# Aharonov-Bohm effect

The

**Aharonov-Bohm effect** is a

quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded, first proposed by Aharonov and

Bohm in

1959. The most common case, sometimes called the

**Aharonov-Bohm solenoid effect**, is where a charged particle passing around a long

solenoid experiences a quantum phase shift as a result of the enclosed

magnetic field, despite the absence of any magnetic field in the region through which the particle passes. This phase shift has been observed experimentally by its effect on interference fringes. There are also magnetic Aharonov-Bohm effects on bound energies and scattering cross sections, as well as a proposed electric effect on charges moving through conducting cylinders, but these cases are more difficult to test. In general, the profound consequence of Aharonov-Bohm effects is that knowledge of the classical electromagnetic field acting

*locally* on a particle is not sufficient to predict the quantum-mechanical behavior.

The magnetic Aharonov-Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the vector potential **A**. This implies that a particle with charge *q* travelling along some path P in a region with zero magnetic field () must acquire a phase φ given in SI units by

with a phase difference Δφ between any two paths with the same endpoints therefore determined by the

magnetic flux Φ through the area between the paths (via

Stokes theorem and ), and given by:

This phase difference can be observed by placing a

solenoid between the slits of a double-slit experiment (or equivalent). A solenoid encloses a magnetic field

**B**, but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an

electron) passing outside experiences no classical effect. However, there is a (

curl-free) vector potential outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on. This corresponds to an observable shift of the interference fringes on the observation plane.

*Schematic of double-slit experiment in which Aharonov-Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field* **B** *is turned on in the cylindrical solenoid.*
The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization is due to the fact that the superconducting wave function must be single valued: its phase difference Δφ around a closed loop must be an integer multiple of 2π (with the charge *q*=2*e* for the electron Cooper pairs), and thus the flux Φ must be a multiple of *h*/2*e*. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by London in 1948 using a phenomenological model.

The magnetic Aharonov-Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole necessarily implies that both electric and magnetic charges are quantized. A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as an infinitely long *Dirac string* of infinitesimal diameter that contains the equivalent of all of the 4π*g* flux from a monopole "charge" *g*. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization: must be an integer (in cgs units) for any electric charge *q* and magnetic charge *g*.

- Y. Aharonov and D. Bohm, "Significance of electromagnetic potentials in quantum theory,"
*Phys. Rev.* **115**, 485–491 (1959).
- S. Olariu and I. Iovitzu Popèscu, "The quantum effects of electromagnetic fluxes,"
*Rev. Mod. Phys.* **57**, 339–436 (1985).
- M. Peshkin and A. Tonomura,
*The Aharonov-Bohm Effect* (Springer-Verlag: Berlin, 1989).
- F. London, "On the problem of the molecular theory of superconductivity,"
*Phys. Rev.* **74**, 562–573 (1948).