Aharonov–Bohm effect

The Aharonov - Bohm effect ( according to David Bohm and Yakir Aharonov ) is a quantum mechanical phenomenon in which a magnetic field affects the interference of electron beams, although they are not in the classically expected sphere of influence. Main cause of the effect is that the influence is caused by the magnetic vector potential, and not by the magnetic field itself

The Aharonov - Bohm effect has been selected as one of the seven wonders in the quantum world by the magazine New Scientist.

Aharonov and Bohm published their work in 1959. However Werner Ehrenberg and Raymond E. Siday could predict the effect in 1949

Theory

In quantum mechanics one describes ( for the use of cgs units ) the behavior of a charged particle in a magnetic field by the following Hamiltonian

The variables here are the particle charge q, the canonical momentum operator (and the kinetic momentum operator ), vector potential, scalar potential, time t, location and mass of the particle m.

Classically, however, carried the influence of the Lorentz force of the magnetic field, according to the equation of motion

With or velocity or acceleration of the particle and the vector product means. Classic therefore an effect can be expected only in areas where the magnetic field is of non-zero (except for the electric field, this is insignificant, as the electric potential ).

Vector potential and magnetic field are related to the mathematical concept of rotation:

The vector potential is therefore generally determined up to the gradient of a scalar function arbitrary, since the rotation of a gradient for two times continuously differentiable scalar fields vanishes (see gauge transformation ). The symbol (*) to number the equation above for future reference.

Experiment

In the experiment, charged particles (electrons) run on opposite sides of a cylinder past, in which there is a magnetic field. The cylinder is surrounded by a wall that can not be penetrated by the particles; outside the magnetic field is zero. The output of the experiment, while it determines whether the magnetic field is switched on or off, for the vector potential in the first case and outside of the cylinder available. Imagine this in front of a radially extending vector potential. Meanwhile, rotation, and thus the magnetic field is zero outside the cylinder, yet is nowhere zero.

The superposition of the wave functions behind the cylinder results in an interference pattern which is affected by the vector potential, since the wave functions of receive paths on the right and left of the cylinder have a different phase shift. Mathematics arises from this, that the rotation of the vector potential - ie the magnetic field - is indeed equal to zero, but because of not simply connected space ( the cylinder interior is the hole in space) in the calculation occurring path integrals do not vanish on closed curves, and thus a net effect remains. Experimentally, this effect was demonstrated in the early 1960s, inter alia, of Mollenstedt and Robert G. Chambers.

Interpretation

Sometimes is drawn from the effect of the conclusion that the vector potential in quantum mechanics have a more fundamental than the corresponding force field. This is not true essentials: Lastly, the magnetic flux is crucial. This can be expressed by a line integral, where the path of integration may be located outside of the area with. It is certainly the case

Which must be closed, which is indicated by the circle in the integration symbol. This is a gauge invariant quantity, ie it does not depend on the function mentioned above. The closed curve is the edge of the surface F.

By the theorem of Stokes

This is the line integral of the closed curve for equation (* ) are identical to the flow of the magnetic field strength by the enclosed area ( the normal vector to the surface, and the two-dimensional surface element). The gauge freedom is related to the fact that in a closed curve different surfaces F can be clamped, all of which are bounded by.

It is important that the integration surface is not simply connected, since the cylinder interior does not belong to the spatial domain of particles. Because otherwise would be above zero integrals, because the rotation of the vector field vanishes by assumption ( no magnetic field ).

One can also interpret the effect that the geometry is changed to a non-commutative geometry by the magnetic flux. The closed curve corresponds to the Burgers loop in the theory of dislocations and the magnetic field of the dislocation density.

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