The Aharonov-Bohm Effects: Variations on a Subtle Theme

© 2009 American Institute of Physics, S-0031-9228-0909-020-6 Quantum phenomena that were once thought to limit measurement capabilities are now being harnessed to enhance them and to improve the sensitivity of nanometer-size devices. Persistent questioning and probing of quantum phenomena has yielded many such advances. However, the Aharonov–Bohm effect, a modern cornerstone of quantum mechanics, is not yet a good example of that kind of progress. The AB effect was already implicit in the 1926 Schrödinger equation, but it would be another three decades before theorists Yakir Aharonov and David Bohm pointed it out.1 And to this day, the investigation and exploitation of the AB effect remain far from finished. Our discussion of the AB effect begins with a seemingly innocent question: What happens to an electron as it passes by an infinitely long ideal solenoid? One might expect that the electron is unaffected. Outside the solenoid, the magnetic and electric fields B and E, and thus the Lorentz force, are all zero. But in quantum mechanics one describes an interaction not by the forces involved but rather by the Hamiltonian. The canonical momentum in the electromagnetic Hamiltonian has a term proportional to the magnetic vector potential A, which is defined (with some gauge freedom) by B = ∇ × A. Before the AB paper, the vector potential was generally thought of as a useful mathematical artifice without independent physical reality. Aharonov and Bohm changed all that. The Aharonov–Bohm effects: Variations on a subtle theme