Tuesday, October 13, 2009

Aharonov-Bohm Effect

Thomson Reuters speculated that the 2009 Nobel Prize for Physics might go in part to Yakir Aharonov of Chapman University, in Orange County, California. Fifty years ago, Yakir Aharonov and his thesis adviser David Bohm devised an astonishing experiment that showed that electrons can sense a magnetic field without passing through it (which had, unknown to them, been described a decade earlier by Werner Ehrenberg and Raymond Siday).

The experiment--since demonstrated experimentally--requires a long coil or solenoid. The results depend on the magnetic field threading along the axis of the coil, not on any fields leaking out the sides or ends.

When electrons are shined at the coil, they reveal their wave nature by passing on both sides simultaneously. On the far side, the chances of an electron appearing at a particular position depends on the relative "phase," which is the difference in the number of wavelike oscillations for electrons going on either side. If the peaks from one side match the troughs from the other, few electrons will be seen, even if the wave from each side alone is strong. This interference effect is well known for other waves, like light.

What quantum mechanics predicted, and experiments confirmed in great detail, is that the relative phase is directly proportional to the magnetic field passing through the solenoid. The surprising thing is that this is true even though on neither side do the electrons pass through any magnetic field. They respond to the field between the paths, at a place where they never go.

Now a mathematical digression: It is customary to describe this result using the "vector potential." As a vector field, this quantity has both a magnitude and a direction at each point in space. The vector potential is related to the magnetic field, while its "scalar" counterpart is related to the electric field. But the exact values of the potentials are somewhat arbitrary, so in classical physics they are regarded as poor cousins of the fields. That's less true in quantum mechanics.

The arbitrariness is easiest to describe for the scalar, or electrostatic, potential, which is closely related to voltage. The electric field is the negative of the gradient, or slope, of this potential. The field is large where the potential varies rapidly with position and points in the direction where the potential decreases fastest. But there are an infinite number of potentials that give the same field, because shifting the potential by the same constant everywhere doesn't change how rapidly it varies in space.

The arbitrariness of the vector potential that determines the magnetic field is more subtle, because their relationship is more subtle. The magnetic field is the "curl" of the vector potential, which is how much it swirls around in a circle(the field points in the direction perpendicular to the swirling). This means that you can add to the vector potential any vector field that has no curl, in what is called a gauge transformation, and the magnetic field will be the same.

Quantum mechanics stipulates that the momentum of the electron (and thus its inverse wavelength) should be corrected by the addition of the vector potential (times a constant). A convenient form (gauge) for the vector potential outside a solenoid is one that everywhere points tangentially along concentric circles. For electrons on one side, this adds to the phase, and on the other side it subtracts, causing the experimentally measured phase shift of the Aharonov-Bohm effect. The size of the relative phase shift depends on the vector potential.

It is common to conclude that in quantum mechanics, in contrast to classical physics, the vector potential is "more real" than the magnetic field. I regard this conclusion as misguided. The phase shift can only be observed by interference between complete paths that pass on opposite sides of the solenoid, which reflect the total phase shift around a loop (one that encloses the solenoid). Because the magnetic field represents the swirliness of the vector potential, this change around a loop is just equal to the total magnetic field passing through the loop, or in this case the solenoid.

Another clue that the vector potential is not really important is that its value changes with different choices of gauge, but the results do not.

So what is a better way to think about the Aharonov-Bohm effect? I like to think about that wonderful print by M.C. Escher called Ascending and Descending. As one passes around the path, there is no clue that anything unusual is happening. But on completing a circuit, it is apparent that something is different. Similarly, a magnetic field changes something subtle (the quantum-mechanical phase) of any electron that passes around it. But that hardly makes the effect less mysterious.

[Note: the September 2009 Physics Today has a story on Aharonov-Bohm effects.]



  1. Hi Don - The "realness of the vector potential" argument is usually made in a particular limit of the thought experiment. For the case of a very long solenoid and far-apart slits, you can have an interference experiment that shows AB effects, even though the probability of ever finding the charged particle in a region containing magnetic field can be arbitrarily low. In other (classical) words, the particle is always in a B-field-free region, but nonetheless experiences a phase shift. The "local" picture for this is that the particle does experience A even though it never sees B, and therefore A has to be the origin of the phase shift.

  2. Hey Doug,

    I would say that the particle doesn't experience A; it experiences the line integral of A around the loop, which is always equal to the integrated B-field, and which, unlike A itself, is independent of gauge.

    It is true that if you insist on the effect being local, then you have no choice to attribute it to A, even though A--and therefore the phase--is only meaningful up to a gauge transformation (which is itself nonlocal). I prefer to regard it as a nonlocal effect that depends on B, not on an incompletely defined quantity.