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Journal of Physics A Mathematical and General

Propagator for quantum systems involving spin–orbit coupling

M. Moshińsky, Emerson Sadurní

Published May 16, 2006• 2 citations
Publisher

Abstract

In the present paper we compute the propagator of a quantum mechanical system whose Hamiltonian consists of two commuting terms, the spin–orbit coupling being one of them. We assume that the propagator corresponding to the first part can be cast into a closed form. A detailed treatment is given when such term is set as the simple harmonic oscillator. Some applications are also included.

DOI
10.1088/0305-4470/39/22/016

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In the present paper we compute the propagator of a quantum mechanical system whose Hamiltonian consists of two commuting terms, the spin–orbit coupling being one of them. We assume that the propagator corresponding to the first part can be cast into a closed form. A detailed treatment is given when such term is set as the simple harmonic oscillator. Some applications are also included.
The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.
The ring model of the coupled oscillator has enormously studied from the perspective of quantum mechanics. The research efforts on this system contribute to fully grasp the concepts of energy transport, dissipation, among others, in mesoscopic and condensed matter systems. In this research, the dynamics of the quantum propagator for the ring of oscillators was analyzed anew. White noise analysis was applied to derive the quantum mechanical propagator for a ring of four harmonically coupled oscillators. The process was done after performing four successive coordinate transformations obtaining four separated Lagrangian of a one-dimensional harmonic oscillator. Then, the individual propagator was evaluated via white noise path integration where the full propagator is expressed as the product of the individual propagators. In particular, the frequencies of the first two propagators correspond to degenerate normal mode frequencies, while the other two correspond to non-degenerate normal mode frequencies. The full propagator was expressed in its symmetric form to extract the energy spectrum and the wave function.
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix.The quantization of a Hamiltonian describing such a system has been done in the canonical case.In this paper, we take a more general approach and look at the system as a Wigner quantum system.Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations.Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n).We determine the spectrum of the considered Hamiltonian in specific representations of these Lie superalgebras and discuss the results in detail.We also make the connection with the well-known canonical case.
The evolution operator of the spin–orbit coupled system in an external magnetic field is factorized according to the operator equation:[Formula: see text]where HSO is the spin–orbit coupled Hamiltonian and HZ is the Zeeman Hamiltonian. The preceding equation is expressed in matrix form using the theory of the O(4) group representations. For the special case of [Formula: see text], an explicit form of the matrix C(t) is found. The operator corresponding to the matrix C(t) is found in a closed form using the projection operators constructed in the Appendix. Two special cases, corresponding to a weak and a strong external field, are considered, and possible applications of the results obtained are indicated.
White noise path integration is applied successfully to derive the quantum propagator of a three body particle system consisting of three harmonic oscillators coupled to each other, or tri-coupled harmonic oscillators. It is shown that the full quantum propagator is a product of three simple harmonic oscillator propagators. These propagators were obtained after two successive transformations of the coordinates for the first two harmonic oscillators, and the coordinates for the coupling between the third and one of the first two harmonic oscillators in the system. Furthermore, when the third coordinate and coupling constants are set to zero the resulting propagator reduces to that of the propagator of the coupled harmonic oscillators.
We present a purely analytical method to calculate the propagator for the quantum harmonic oscillator using Feynman’s path integral. Though the details of the calculation are involved, the general approach uses only matrix diagonalization and well-known integrals, techniques which an advanced undergraduate should understand. The full propagator, including both the prefactor and the classical action, is obtained from a single calculation which involves the exact diagonalization of the discretized action for the system.