2005/11/19 | [Aharonov-Bohm Effect]
类别(Ω〖物理〗) | 评论(0) | 阅读(98) | 发表于 12:56
An effect that occurs in small disordered metallic conductors and causes conductance fluctuations in small (nonsuperconducting) rings and wires. It arises from the presence of a vector potential produced by an applied magnetic field.

To understand the effect, consider the arrangement illustrated above. In this experiment, a wall with two narrow splits intercepts electrons from the source, and a detector on the other side registers the rate at which electrons arrive at a small region at a distance x above the axis of symmetry. The rate is proportional to the probability that an individual electron will reach the region, which can be understood as the interference of the wavefunctions and passing through each slit. The phase difference produces an interference pattern and, as shown above, the phase difference is given by
(1)

where a is the difference in the path lengths and is the wavelength of the space variation of the probability amplitude.
Now, in the presence of a magnetic field, the interference of the phases of arrival of the two waves from the source will determine the location of the maxima in probability. In this case, the phase of trajectory 1 is given by the line integral
(2)

where q is the charge on the particles emitted from the source, is h-bar, and A is the magnetic vector potential. Similarly, the phase of trajectory 2 is
(3)

The interference of the waves at the detector depends on the phase difference, so
(4)

where the difference of the two integrals has been written as a single line integral over a closed path that goes forward along and back along . Therefore, the phase difference is changed by the magnetic field.
The question raised by Aharonov and Bohm (1959) was that, given two different vector potentials A and representing the same magnetic field (since vector potential is uniquely determined only up to a gauge transformation), which of the possible potentials is correct? In other words, by changing A to in (4), the integral over A becomes
(5)

Now, the integral of is around the closed path but the integral of the tangential component of a gradient on a closed path is always zero by Stokes' theorem.Therefore, A and give the same phase differences and the same quantum mechanical interference effects. Both classically and quantum mechanically, it is only the curl of A that matters, so any choice of the function of A which has the correct curl gives the current physical result. Note that equation (5) can be written in terms of the magnetic field B,because the line integral of A around a closed path is the flux of B through the path. It appears that if we can write the equation in terms of B as well as in terms of A, the B can stand on its own as a "real" field, while A can still be thought of as an artificial construction. But, according to equation (5), if we arrange a situation where electrons are to be found only outside of a long solenoid carrying an electric current, there still can be an influence on the motion. This is classically impossible, because there is only A in that region. The force should depend only on B, and in order to know that the solenoid is carrying current, the particle must go through it.
Observing this effect proved to be a very difficult experiment. Aharonov-Bohm oscillations were observed by Webb et al. (1985) in ordinary (i.e., nonsuperconducting) metallic rings, showing that electrons can maintain quantum mechanical phase coherence in ordinary materials (Schwarzschild 1986).
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