Aharonov-Bohm effect

Introduction

The Aharonov-Bohm effect is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential, despite being confined to a region in which both the magnetic field B and electric field E are zero. The underlying mechanism of this phenomenon is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, and the effect is observed in the interference pattern of a two-slit experiment. The Aharonov-Bohm effect illustrates the physical reality of electromagnetic potentials in quantum mechanics, which is not present in classical mechanics 1(https://www.nature.com/articles/nphys1170).

Photograph of a double-slit experiment setup

Historical Background

The Aharonov-Bohm effect is named after the physicists Yakir Aharonov and David Bohm who predicted the effect in 1959 2(https://journals.aps.org/pr/abstract/10.1103/PhysRev.115.485). However, the essence of this effect was first discussed by Werner Ehrenberg and Raymond E. Siday in 1949, in the context of electron optics 3(https://royalsocietypublishing.org/doi/10.1098/rspa.1949.0097).

Theoretical Description

The Aharonov-Bohm effect is fundamentally a quantum mechanical effect, and cannot be explained by classical physics. It arises from the fact that the quantum mechanical wave function of a charged particle is not only influenced by the electromagnetic fields, but also by the electromagnetic potentials. This is in stark contrast to classical physics, where the electromagnetic potentials are merely mathematical tools used to calculate the electromagnetic fields, and have no physical significance.

The quantum mechanical description of the Aharonov-Bohm effect is given by the Schrödinger equation, which describes the evolution of the wave function of a quantum system. In the presence of an electromagnetic potential, the Schrödinger equation takes the form:

HΨ = iħ ∂Ψ/∂t

where H is the Hamiltonian operator, Ψ is the wave function, ħ is the reduced Planck's constant, and t is time. The Hamiltonian operator H includes the kinetic energy and potential energy of the system, and in the presence of an electromagnetic potential, it takes the form:

H = (p - qA)²/2m + qφ

where p is the momentum operator, q is the charge of the particle, A is the vector potential, m is the mass of the particle, and φ is the scalar potential. The term qA represents the interaction of the charged particle with the vector potential, and this is the term that gives rise to the Aharonov-Bohm effect.

Experimental Observations

The Aharonov-Bohm effect has been observed in various experimental setups, most notably in electron interference experiments. In these experiments, a beam of electrons is split into two paths by a thin film of material with two slits. The paths are recombined on a detector screen, creating an interference pattern. When a solenoid (a coil of wire that produces a magnetic field when an electric current is passed through it) is placed between the slits, the interference pattern shifts, even though the magnetic field B is zero in the region of the electron paths. This shift is the Aharonov-Bohm effect 4(https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.56.792).

Significance and Applications

The Aharonov-Bohm effect has profound implications for our understanding of quantum mechanics and the nature of physical reality. It shows that the electromagnetic potentials, which are considered to be mere mathematical tools in classical physics, have real physical effects in quantum mechanics. This has led to a deeper understanding of the role of potentials in quantum mechanics, and has influenced the development of quantum field theory, the framework for the modern description of particle physics.

In addition to its theoretical significance, the Aharonov-Bohm effect also has practical applications. It is used in the design of quantum interference devices, such as the Aharonov-Bohm oscillators, which are used in quantum computing and quantum information processing 5(https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.79.2730).

References

1. "Observation of Aharonov-Bohm Effect in a Quantum Ring", Nature Physics. 2. "Significance of Electromagnetic Potentials in the Quantum Theory", Physical Review. 3. "The Refractive Index in Electron Optics and the Principles of Dynamics", Proceedings of the Royal Society A. 4. "Observation of Aharonov-Bohm Electron Interference in Magnetic Field Free Region", Physical Review Letters. 5. "Aharonov-Bohm Oscillations in the Presence of Strong Spin-Orbit Interaction", Physical Review Letters.