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Journal of Experimental and Theoretical Physics

Feynman method for disentangling operators and a singular oscillator

Vladimir S Popov, M. A. Trusov

Published Feb 1, 2012• 1 citation
Publisher

Abstract

The problem of the evolution of a singular quantum oscillator with a frequency exhibiting an arbitrary time dependence has been solved. The probabilities w mn of transitions in the oscillator spectrum and generating functions have been calculated, and the sum rules for w mn have been derived. The calculations are based on the Feynman disentangling method and the theory of representations of the SU(1, 1) group.

DOI
10.1134/s1063776112010062

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The problem of the evolution of a singular quantum oscillator with a frequency exhibiting an arbitrary time dependence has been solved. The probabilities w mn of transitions in the oscillator spectrum and generating functions have been calculated, and the sum rules for w mn have been derived. The calculations are based on the Feynman disentangling method and the theory of representations of the SU(1, 1) group.
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.
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 problem of existence of a self-adjoint time operator conjugate to a Hamiltonian with SU(1,1) dynamical symmetry is investigated. In the space spanned by the eigenstates of the generator K3K_3K3​ of the SU(1,1) group, the time operator for the quantum singular harmonic potential of the form ω2x2+g/x2ω^2x2 + g/x2ω2x2+g/x2 is constructed explicitly, and shown that it is related to the time-of-arrival operator of Aharonov and Bohm. Our construction is fully algebraic, involving only the generators of the SU(1,1) group.