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arXiv:1407.3826

Quantum Mechanics on a Ring: Continuity versus Gauge Invariance

A. Davidson

Published Jul 14, 2014• 0 citations
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Abstract

Remarkably we find that for a ring with linear boundary conditions such that the eigenvector and its derivative are continuous, there does not seem to be a way for the well-known de Broglie relation to be gauge invariant. Certain nonlinear boundary conditions assure gauge invariance, and lead to eigenfunctions with a discontinuous but differentiable phase and a continuous spectrum. A discrete subset of this spectrum forms a Hilbert space, while another subset is excluded by the nonlinear boundaries. We conclude that discontinuous momentum eigenfunctions are tenable, and that it is possible that quantum mechanics can have nonlinear boundary conditions in some circumstances.

DOI
10.48550/arxiv.1407.3826

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Remarkably we find that for a ring with linear boundary conditions such that the eigenvector and its derivative are continuous, there does not seem to be a way for the well-known de Broglie relation to be gauge invariant. Certain nonlinear boundary conditions assure gauge invariance, and lead to eigenfunctions with a discontinuous but differentiable phase and a continuous spectrum. A discrete subset of this spectrum forms a Hilbert space, while another subset is excluded by the nonlinear boundaries. We conclude that discontinuous momentum eigenfunctions are tenable, and that it is possible that quantum mechanics can have nonlinear boundary conditions in some circumstances.
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A consensus exists for small tunnel junctions that a discrete spectrum follows from a continuous wave on a circle. There is another view about an electron on a 1D metal ring that a discrete spectrum follows from a singular Bloch wave. We find using an often ignored form of gauge function that continuity on a ring violates gauge invariance of the de Broglie wavelength! This same gauge function shows that Lagrangians for the electron on a ring and the charge on a junction are mutual gauge transforms. Thus persistent current on a metal ring and the Coulomb blockade on a tunnel junction seem to be the same dynamical theory based on discontinuous Bloch waves on the compact perimeter of a circle.
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We argue that continuity and discontinuity in special relativity play an important role in the probabilistic wave properties of quantum mechanics. Special relativity is involved with transformations from a rest frame to a frame moving with a constant -v, for example . The discontinuity appears in the form of a change from v=0 to v, while the continuity arises from Lorentz invariants which indicate one has the same physical state whether it is viewed from a frame with one velocity or another. This invariance applied to E, p (momentum) as a modulus forces a minus sign between the two, in order to allow for linkage with E=mo (c=1) and p=0, i.e. -EE + pp = -momo (c=1). Another invariant, however, is A= -Et’ + px’, and in this case the minus sign and the discontinuity lead to a new x’,t’ (in comparison with the rest frame for a particle with rest mass mo), but also to t’ -> t’+ b/E and x->x’ +b/p leaving A unchanged (b=constant). The discontinuity in v, which leads to a jump in x’,t’ (as seen in the moving frame) also manifests itself in a time-length interval which ultimately leads to exp(-iEt+ px), we argue. In other words, classically one considers constant velocity as smooth continuous motion, but the introduction of two frames, v=0 and -v, leads to a discontinuity which leads to “chunkiness” in the form of a frequency and wavelength. If these cannot be perceived classically (as is often the case), then one considers only smoothness. Relative differences in time or length or in frequency or wavelength, however, can appear in nature for length scales of the order of the frequency/length scale. In the case of n-slit interference-diffraction, a relative difference leads to a “jump” (so to speak) which manifests the wavelength. In particular, a photon moves along the y direction with a phase px (considering a steady state picture). It reaches a two-slit system along the x-axis and there is probability for it to interact with both slits. If one considers two parallel rays from each slit moving towards a point on a screen very far away (as is done traditionally), there is again smooth motion with a rotated p vector (same magnitude), but for a point on the screen to the left of the center, there is a jump in the length associated with the right slit. In other words, there is a jump similar to the v=0, -v jump. This leads to a relative difference in length associated with both rays (linked to an OR probability situation using exp(ip dot r). This relative difference in length displays the phase difference initiated by special relativity. In the case of a bound state, different impulse hits by a potential create different momenta which have different phases exp(ipx) because of special relativity and one has interference. The same phase A= -Et + p dot r, which is linked with a chunkiness due to a discontinuity, is also associated with continuity because A is a Lorentz invariant. Thus A=constant may be used to solve for refraction/reflection problems with or without moving mirrors/interfaces with different indices of refraction. Continuity appears further in a 1-dimensional reflection/refraction problem when applied to the exp(ipx), i.e. Aexp(ipx) + Bexp(-ipx) = C exp(ip2x) where E=pc = p2 c2, p2=n2p and c2=c/n2 with n1=1, together with a continuity equation for the first derivative. Thus, both continuity and discontinuity (or jumps), which follow from special relativity, seem to play an important role in quantum mechanics.