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Modern Physics Letters A

SOLVING OPERATOR DIFFERENTIAL EQUATIONS IN TERMS OF THE WEYL-ORDERED POLYNOMIALS

Zeng‐Bing Chen, Huai-Xin Lu, Jun Li

Published Sep 28, 2002• 2 citations
Publisher

Abstract

A systematic approach to integrate the Heisenberg equations of motion is proposed by using the Weyl-ordered polynomials. The solutions of the Heisenberg equations of motion, i.e. P(t) and Q(t), are expanded as a sum over the Weyl-ordered polynomials Tm,n(P(t),Q(t)) at time t = 0. The coefficients of the expansions satisfy two sets of first-order ordinary differential equations resulting from the Heisenberg equations of motion for time-independent systems. This general approach for time-independent systems is also tractable in obtaining the adiabatic invariants of the time-dependent systems. In this paper, interest is mainly focused on the formal aspect of the approach.

DOI
10.1142/s0217732302008459

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A systematic approach to integrate the Heisenberg equations of motion is proposed by using the Weyl-ordered polynomials. The solutions of the Heisenberg equations of motion, i.e. P(t) and Q(t), are expanded as a sum over the Weyl-ordered polynomials Tm,n(P(t),Q(t)) at time t = 0. The coefficients of the expansions satisfy two sets of first-order ordinary differential equations resulting from the Heisenberg equations of motion for time-independent systems. This general approach for time-independent systems is also tractable in obtaining the adiabatic invariants of the time-dependent systems. In this paper, interest is mainly focused on the formal aspect of the approach.
A direct integration of the Heisenberg equations of motion yields expansions of the position and momentum operators x(t) and p(t), each as an infinite series in terms of the initial Weyl-ordered basis set {Sm,{\mathit{S}}_{\mathit{m},}Sm,​n} formed from x(0) and p(0), with c-number time-dependent coefficients. This method is applied to the problem of tunneling in a symmetric and an asymmetric quartic potential. The expectation values of the position and momentum operators with minimum uncertainty wave packet 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 can be calculated accurately for a maximum time that is short compared to the period of oscillation of the wave packet. From the result of this calculation, with the help of Prony's method one can determine the level spacings for the low-lying states. In addition, in this formulation the wave packet retains its shape; therefore, one can study the trajectory 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 as the quantal analogue of the motion of the system in phase space.
The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-ordered basis set {Sm,n{\mathit{S}}_{\mathit{m},\mathit{n}}Sm,n​(t)} formed from r(t) and pr{\mathit{p}}_{\mathit{r}}pr​(t), one can express the time evolution of any member of the set as an infinite sum involving the operators {Sm,n{\mathit{S}}_{\mathit{m},\mathit{n}}Sm,n​(0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.
A method of integration of the Heisenberg equation of motion is proposed in which the time evolution of the basis set {Tm,{\mathit{T}}_{\mathit{m},}Tm,​n(t)} of the Weyl-ordered operators introduced by Bender and Dunne [Phys. Rev. D 40, 3504 (1989)] is obtained from the Taylor expansion and is expressible in terms of the initial {Tm,{\mathit{T}}_{\mathit{m},}Tm,​n(0)}'s. In the absence of damping forces, the constant values of the energy and the position-momentum commutation relation are used to check the accuracy of the integration. This method is applied to obtain the mean position and velocity of the particle as a function of time as well as the dwell time of the particle inside the barrier. In the example that is considered here, the potential is assumed to be the sum of a harmonic and a cubic term, and the calculation is done with and without dissipative coupling.
We investigate the properties of the Weyl-ordered polynomials Tm,n by the technique of integration within a Weyl-ordered product (IWWOP) and the Weyl quantization rule. The introduction of the Weyl-ordering symbol ⋮⋮ and the IWWOP provide us with a convenient approach for deriving the Weyl-ordered form of the product of Tm,n and other operators. The Weyl-ordered Dirac's δ-function form of the Wigner operator can also directly lead us to the Weyl quantization formalism for the functions of radial vaiables.
The solution of the quantum dynamical equation $\frac{i\ensuremath{\hbar}\mathrm{dT}}{\mathrm{dt}}=HT$, for the time-displacement operator TTT, is given, when the Hamiltonian HHH is a polynomial of the second degree in canonically conjugate variables, with arbitrary time-dependent coefficients. Heisenberg's equations of motion are then solved, and the general integral of Schr\"odinger's equation in coordinate space is expressed by the Green's function corresponding to TTT. An example is given.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.