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arXiv:2404.05115 · Physica Scripta

Time-dependent conserved operators for autonomous systems and quantization of resistance

Jorge A. Lizarraga

Published May 14, 2025• 0 citations
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Abstract

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.

DOI
10.1088/1402-4896/add8f4

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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.
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.
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.
We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.
The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.
The spectrum of a Hamiltonian describing the dynamics of a Landau particle with time-dependent mass and frequency undergoing the influence of a uniform time-dependent electric field is obtained. The configuration space wave function 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 a 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.
In this paper I have drawn out the steps to be followed in order to derive the exact Schr\"odinger wave function for a particle in a general one-dimensional time-dependent linear potential. To this end I have used the so-called Lewis and Riesenfeld invariant method, which is based on finding an exact quantum-mechanical invariant in whose eigenstates the exact quantum states are found. In particular, I have obtained the wave functions of a particle in the linear potential well, driven by a monochromatic electric field.