San José State University

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Thayer Watkins
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 The Aharonov-Bohm Effect

The equipment for demonstrating the Aharonov-Bohm effect consists of a source of uniform energy electrons, a screen with two slits in it, a screen to capture the interference pattern and a solenoid. The solenoid is an electromagnet encased in an iron tube. The iron tube captures all of the magnetic field created by the electromagnet. Outside of the tube the magnetic field is zero.

Without the electromagnet on, the equipment generates the standard interference pattern for the two-slit experiment. A schematic diagram for the experiment is shown below.

When the current in the solenoid is increased there is a shift in the interference pattern on the screen. This is quite surprising because the iron shielding of electromagnet confines the magnetic field entirely to the solenoid itself. The magnet field in the paths of the electrons is zero. In any real experiment the magnetic field would be zero except for background levels.

Paul A.M. Dirac proved in 1931 that in such arrangement there would be a phase shift for the wave function of the electron based not upon the level of the magnet field for the region through which it passes but upon the level of the vector potential function.

If B(X) is the magnetic field function then the vector potential function is such that

#### ∇×A(X) = B(X)

For any A(X) equal to the gradient of a scalar function ∇G the curl is zero,

#### ∇×(∇G) = 0

Thus the vector potential function in a region can be nonzero even though the magnetic field is zero.

The wave function ψ(X) is a complex-valued function of the point in space X. The magnitude squared of the wave function |ψ(X)|2 is the probability density for the electron at point X. A wave function can be multiplied by a function of the form exp(-iφ) without affecting the magnitude and thus without affecting the probabilities of the electron being found in any region of space. The quantity φ is called the phase angle.

It was generally thought before the Aharonov-Bohm experiment that changes in phase angle do not affect the behavior of electrons. The experiment showed that the phase angle of electrons could be modified even though the magnetic field through which they pass is zero, and that the modification can be detected.

What Dirac showed is that the change in phase angle of an electron passing through a path S is

#### φ = eJ/h

where e is the charge of the electron, h is Planck's constant divided by 2π and J is the line integral

#### J = ∫SA·ds

where A is the vector potential for the magnetic field. The phase difference between electrons traveling on path 1 compared to path 2 is thus based upon the difference in the line integrals. This is equivalent to computing the line integral forward on path 1 and then backward on path 2. This in turn is equivalent to computing the line integral around the path created by path 2 to path 1 in a reverse direction. This creates a closed path. By Stoke's Theorem the line integral around a closed path is equal to the integral of the curl of the vector quantity over the surface enclosed by the path. In this case the curl of the vector potential is the magnet field and this is nonzero for the cross-section of the solenoid.

What the Aharonov-Bohm experiment established is that it is not only the electric and magnetic field that can have observable effects. The vector potential can also produce observable effects. Originally the vector potential function was only a mathematical artifact, a convenience. What the Aharonov-Bohm effect shows is that the vector potential function has a primacy, an existence in its own right.