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.
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.
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.
Abstract The phase gradient of the wave describing the Aharonov–Bohm effect (AB) is proportional to the local canonical momentum. This vector field contains vortices (phase singularities), whose strengths cannot be detected in quantum mechanics because they increase (discontinuously) with the magnetic flux, violating gauge invariance. The analogous quantity which is gauge-invariant is the kinetic momentum field, proportional to the local electron velocity. Investigation of the streamlines (integral curves) of this velocity field reveals that as the flux increases from 0 to 1/2 (in quantum units), a vortex V is generated at the flux line, accompanied by a stagnation point (saddle) S that emerges from V and then collapses back into V. The VS pair is always small: the maximum distance between V and S is approximately 0.0209 de Broglie wavelengths. The VS phenomenon survives generalization to a superposition of AB waves. If the flux is confined within an impenetrable tube of radius R , S persists if R < 0.004 de Broglie wavelengths, and is swallowed by the tube for larger R . An experiment is envisaged.
A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A family of two-dimensional walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these discrete-time quantum walks (DTQWs) exhibit an exact discrete local U(1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called $\mathbf{E}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.
A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A new family of 2D walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local U(1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called E×B drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.
One of the fundamental questions resolved in the early development of quantum mechanics was whether the electron motion was wavelike or particle-like. The answer was provided by De Broglie who showed that the particle linearmomentum, p, was related to an electromagnetic wavelength, l, as follows: l ¼ h p= , where h¼Planck’s constant ¼ 6:65 1034 J sec. This relationship was important, because it says that the motion of a particle can be described in terms of wave propagation. Thus, for example, a particle enclosed in a microwave cavity can be described in terms of an electromagnetic standing mode or as a particle in a box. De Broglie’s equation proved to be a great triumph in the early development of quantum physics, especially when applied to atomic physics. Let’s apply the De Broglie relationship to a ring resonator. The ring resonator consists of a simple wire loop and is coupled to a microwave source. Further, let’s imagine a hypothetical situation in which the loop is so small that very few, perhaps one single electron, is flowing around the wire loop. Thus, the motion is constrained to be circular. Remarkably, it radiates maximum energy at resonance, as determined by nl ¼ 2pr, where n ¼ 1, 2, 3, . . . , l is the electrical wavelength of the radiated energy or the wavelength of the particle within the wire loop, and r is the radius of the loop. If we assume that De Broglie’s equation is applicable, l is related to the linear momentum, p, of the few particles ‘‘entrapped’’ in the resonator. Since the motion is circular, the angular momentum, G, is simply G ¼ pr. This implies that (applying the De Broglie relationship) 2pr ¼ h G rn or G ¼ nh: G is in units of h and discrete. This implies that the magnetic moment of the ring resonator must also be quantized, since G is discrete.
This talk is assumed to exhibit an overview of the quantum theory for mesoscopic electric circuits and some of its further developments. In the theory the importance of the discreteness of electronic charge in mesoscopic electric circuit is addressed. The mesoscopic LC-design is quantized in accord with the charge discreteness. The uncertainty relation for electric charge and current is given. Because the stationary Schrödinger equation is turned to be Mathieu equation in p-representation, the wave function and energy spectrum is formally solved. As further applications, the persistent current is obtained by considering the mesoscopic ring as a pure L-design. The Coulomb blockade phenomenon occurs when applying the theory to the pure C-design. Concerning the time evolution of the states for mesoscopic electric circuit, we are able to study it by the method of characteristics. In order to study the dissipative effect in the circuit, we use density-matrix formulation. In this formulation, several type of ``off diagonal'' dissipations are expected to be discussed.