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arXiv:2403.06287 · Physica Scripta

Quantization of resistivity as consequence of symmetry invariance

Jorge A. Lizarraga

Published Jul 3, 2024• 1 citation
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Abstract

Abstract The Schrödinger equation for a spinless electron under the influence of a constant electromagnetic field is analyzed based on the conserved operators of the system when the magnetic field is described by Landau’s gauge. For this specific situation, the Lorentz force can be recovered if two conserved generalized momentum operators are considered: one along the x-axis and the second along the y-axis. From the general solution obtained, a time-dependent ground state is constructed characterized by quantized resistivity proportional to integer multiples of von Klitzing’s constant when an invariance condition under a unitary transform is satisfied.

DOI
10.1088/1402-4896/ad5ed8

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Abstract The Schrödinger equation for a spinless electron under the influence of a constant electromagnetic field is analyzed based on the conserved operators of the system when the magnetic field is described by Landau’s gauge. For this specific situation, the Lorentz force can be recovered if two conserved generalized momentum operators are considered: one along the x-axis and the second along the y-axis. From the general solution obtained, a time-dependent ground state is constructed characterized by quantized resistivity proportional to integer multiples of von Klitzing’s constant when an invariance condition under a unitary transform is satisfied.
Abstract Two systems for a charged particle are studied using an adaptation of Lewis and Riesenfeld invariant method. The first system involves a particle under the effect of a constant electric field, and the second system adds a parallel magnetic field. In both cases, time-dependent conserved operators are identified, which can be used to derive time-dependent wave functions for the Schrödinger equation, where the time variable is not separable from the space coordinates. These conserved operators are also used to construct unitary operators, which define the symmetries of the systems. Finally, it is shown that the invariance of the wave function under the action of these unitary operators leads to the quantization of resistance equal to an integer number times the Klitzing’s constant.
The ordinary Landau problem consists of describing a charged particle in time-independent magnetic field. In the present case the problem is generalized onto time-dependent uniform electric fields with time-dependent mass and harmonic frequency [1].The spectrum of a Hamiltonian describing this system is obtained. The configuration space wave functions of the model is expressed in terms of the generalised Laguerre polynomials. To diagonalize the time-dependent Hamiltonian we employ the Lewis-Riesenfeld method of invariants. To this end, we introduce an unitary transformation in the framework of the algebraic formalism to construct the invariant operator of the system and then to obtain the exact solution of the Hamiltonian. We recover the solutions of the ordinary Landau problem in the absence of the electric and harmonic fields, for a constant particle mass. The quantization of this system exhibits many symmetries such as U(1),SU(2),SU(1,1)U(1), SU(2), SU(1,1)U(1),SU(2),SU(1,1). We therefore construct the coresponding coherent states and the associated photon added and nonlinear coherent states.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
The motion of a charged particle moving on a flat surface is studied through the constants of motion associated with the system, given the magnetic gauge. The usual Landau’ solution and the non-separable solution for the Landau’s gauge are found, and a new non-separable solution for the symmetric gauge is given. As a consequence of this, the quantization of the magnetic flux results from the invariance of the solutions under the unitary transformations which arise from the operator’s constants of motion.
This paper examines how Maxwell electrodynamics can be recovered as the field-theoretic structure required to consistently couple quantum matter to electromagnetism, while distinguishing what follows from quantum mechanics from what must be supplied independently. Local phase invariance introduces electromagnetic potentials through minimal coupling and yields the homogeneous Maxwell equations as geometric identities, while quantum probability conservation provides a conserved electric current and Ehrenfest dynamics recover the Lorentz-force form. The inhomogeneous Maxwell equations require an additional Lorentz- and gauge-invariant electromagnetic field action, whose vacuum constants satisfy μ0ε0=1/c2\mu_0\varepsilon_0=1/c^2, linking the construction to the SQC/MSC electric–magnetic prefactor relation. Hydrogen is used as an explicit microscopic source: the 1s1s state generates a quantum charge distribution, 2p2p states provide orbital currents and magnetic moments, and 1s1s–2p2p superpositions produce radiative dipole transitions. The treatment is then extended from individual atoms to a hydrogen gas, where polarization and magnetization generate the macroscopic fields DD and HH. The resulting framework shows how quantum sources and material response fit consistently within Maxwell electrodynamics without claiming that quantum mechanics alone uniquely derives the complete electromagnetic field dynamics.
In this chapter, we consider a quantum particle wave function with a bound spectrum of velocity c, and obtain the relativistic momentum based on the group velocity of this wave function. With a space-time isometry condition, the Lorentz transformation and the relativistic dynamics were obtained. Considering a field interacting with a quantum particle as a four-vector conjugated to the space-time vector in the time-dependent phase, we obtain the Lorenz force and the Maxwell equations. It is interesting that only the Ampère-Maxwell equation of a magnetic circuit is specific to the electromagnetic field, while the other equations are general for a field interacting with a charged quantum particle. Considering the time-dependent phase of a quantum particle interacting with an electromagnetic field with a space-time homogeneity condition, we obtain Lorentz transformations for this field. For a quantum particle at a non-relativistic velocity, we obtain a wave function with a very rapidly-varying factor, of a frequency proportional to the rest energy of this particle. From the Schrödinger equation of a particle with a relativistic Hamiltonian, we obtain a split of the wave function into four components, describing a proper rotation of this particle with an angular momentum called spin (Dirac’s relativistic electron theory). Moreover, we also calculate electron potential in the magnetic field, and two-electron interaction potential.