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Journal of Experimental and Theoretical Physics

Feynman disentangling of noncommuting operators in quantum mechanics

Vladimir S Popov

Published Nov 1, 2005• 3 citations
Publisher

Abstract

Feynman’s disentangling theorem is applied to noncommuting operators in the problem of quantum parametric oscillator, which is mathematically equivalent to the problem of SU (1, 1) pseudospin rotation. The number states of the oscillator correspond to unitary irreducible representations of the SU (1, 1) group. Feynman disentangling is combined with group-theoretic arguments to obtain simple analytical formulas for the matrix elements and transition probabilities between the initial and final states of the oscillator. Feynman disentangling of time evolution operators is also discussed for an atom or ion interacting with a laser field and for a model Hamiltonian possessing the “ hidden” symmetry of the hydrogen atom.

DOI
10.1134/1.2149062

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Feynman’s disentangling theorem is applied to noncommuting operators in the problem of quantum parametric oscillator, which is mathematically equivalent to the problem of SU (1, 1) pseudospin rotation. The number states of the oscillator correspond to unitary irreducible representations of the SU (1, 1) group. Feynman disentangling is combined with group-theoretic arguments to obtain simple analytical formulas for the matrix elements and transition probabilities between the initial and final states of the oscillator. Feynman disentangling of time evolution operators is also discussed for an atom or ion interacting with a laser field and for a model Hamiltonian possessing the “ hidden” symmetry of the hydrogen atom.
Feynman's method for disentangling noncommuting operators is discussed and applied to nonstationary problems in quantum mechanics, including the excitation of a harmonic oscillator by an external force and/or by time-varying frequency; spin rotation in a time-varying magnetic field; the disentangling of an atom (ion) Hamiltonian in a laser field; a model with the hidden symmetry group of the hydrogen atom; and the theory of coherent states. The Feynman operator calculus combined with simple group-theoretical considerations allows disentangling the Hamiltonian and obtaining exact transition probabilities between the initial and final states of a quantum oscillator in analytic form without cumbersome calculations. The case of a D-dimensional oscillator is briefly discussed, in particular, in application to the problem of vacuum pair creation in an intense electric field.
Disentangling is the essential operation of Feynman's operational calculus for noncommuting operators. Thus formulas which simplify this operation are central to the subject. In a recent paper the procedure for 'extracting a linear factor' has been established in the setting of Feynman's operational calculus for time independent operators A1,...,AnA_1, ... , A_nA1​,...,An​ and associated probability measures μ1,...,μn{\mu}_1,..., {\mu}_nμ1​,...,μn​ . While the setting just described is natural in many circumstances, it is not natural for evolution problems. There the measures should not be restricted to probability measures and it is worthwhile to allow the operators to depend on time. The main purpose for this paper is to extend the procedure for extracting a linear factor to this latter setting. We should mention that Feynman's primary motivation for developing an operational calculus for noncommuting operators came from a desire to describe the evolution of certain quantum systems.m systems.
Time evolution operator for a {h(1)⊕h(1)}⊎u(2)\left\{ h(1) \oplus h(1) \right\} \uplus u(2){h(1)⊕h(1)}⊎u(2) time-dependent quantum Hamiltonian; a self-consistent resolution method based on Feynman's disentangling rules
In this article the time evolution operator of two interacting quantum oscillators, whose Hamiltonian is an element of the complex {h(1)⊕h(1)}⊎u(2)\left\{ h(1) \oplus h(1) \right\} \uplus u(2){h(1)⊕h(1)}⊎u(2) algebra, is analyzed using the Feynman time ordering operator techniques. This method is consistently used to generate the conditions and to formally find explicit disentangled expressions for such operator. In this way, it is shown that all the problem reduces to solve a complex Riccati-type differential equation. Some closed solutions to this differential equation are found and then concrete disentangling expressions for the time-ordered evolution operator are given. Finally, the time evolution of the coherent states linked to the isotropic 2D quantum oscillator are analyzed under alternative time-independent an time-dependent Hamiltonian systems.
The formalism of non-Abelian quantum kinematics is applied to the Newtonian symmetry group of the harmonic oscillator. Within the regular ray representation of the group, the Schrödinger operator, as well as two other (new) invariant operators, are obtained as Casimir operators of the extended kinematic algebra. Superselection rules are then introduced, which permit the identification (and the explicit calculation) of the physical states of the system. Next, a complementary ray representation, attached to the space-time realization of the group, casts the Schrödinger operator into the familiar time-dependent space-time differential operator of the harmonic oscillator and thus, by means of the superselection rules, one obtains the time-dependent Schrödinger equation of the sytem. Finally, the evaluation of a Hurwitz invariant integral, over the group manifold, affords the well known Feynman space-time propagator 〈t′,x′‖t,x〉 of the simple harmonic oscillator. Everything comes out from the assumed symmetries of the system. The whole approach is group theoretic and ‘‘relativistic.’’ No classical analog is used in this ‘‘quantization’’ scheme.