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Note: If it is not visible, click the at the top of the pane, choose Object Type and locate the query under the Queries heading.

lead to a discontinuity, the wave function vanishes anyway. This is not relevant to free electrons, but may be relevant, for example, for an electron trapped by an impurity in a metal. (b) The electron is "extended," with a wave function that is phase-coherent around a closed path about the hole, but the flux is quantized in integer multiples of the flux quantum

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(11.73)

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In this case, (11.72) becomes a periodic function of () and represents a legitimate gauge transformation. Thus, the vector potential can be transformed away, and the electron does not "know" there is flux through the hole. This is the basis of the Aharanov-Bohm effect,* and the flux quantization in superconductivitY,t which have been experimentally verified.

The relevance of flux quantization to the Landau levels lies in the fact that the degeneracy (11.69) is just the total magnetic flux measured in units of the flux quantum:

(11.74)

*Y. Aharanov and D. Bohm, Phys. Rev. U5, 465 (1959). tSee P. G. De Gennes, Superconductivity of Metal and Alloys (Benjamin, New Yark, 1966), p.149.

Fig. 11.9 Different bases for electron states in a Landau level, which is highly degenerate: (a) Member wave functions are peaked at different elevations (y direction), and are eigenstates of momentum in the x direction. (b) Member wave functions are eigenstates of orbital angular momentum. They are peaked at concentric circles, with equal areas between successive circles. Each ring between circles supports one magnetic flux quantum. (c) Member wave functions are "vortices" of flux quanta, forming a lattice.

We have been working in a gauge in which the wave functions have definite linear momentum in the x direction. Thus the probability densities are independent of x, and peaked about parallel ridges at y = Yo' The spacing between successive values of Yo is hc/eHL, and hence the area of the strip in the xy plane between two neighboring ridges is hc/eH. Thus, exactly one flux quantum hc/e goes through the strip. Since the energy levels are highly degenerate, we can make linear transformations on the wave functions belonging to the same Landau level to obtain equivalent sets. Such transformations are equivalent to gauge transformations. We can make them eigenfunctions of angular momentum about the z axis, in which case the probability distributions will be peaked about circles in the xy plane about the z axis, with exactly one quantum of flux going through the annular ring between two successive circles. We can also make them into individual orbitals, whose centers form a regular lattice in the xy plane. In this case each orbital will link exactly one flux quantum (a "vortex.") These different bases are illustrated in Fig. 11.9. The qualitative fact relevant to our immediate purpose is that the flux quantum sets a finite minimum size of an orbit, and thereby provides the escape from Van Leeuwen's theorem.

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