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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Exact solutions to operator differential equations

Carl M. Bender, Gerald V. Dunne

Published Oct 15, 1989• 55 citations
Publisher

Abstract

This paper considers the Heisenberg equations of motion $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}=\ensuremath{-}i[q, H]$, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{p}=\ensuremath{-}i[p, H]$, for the quantum-mechanical Hamiltonian $H(p, q)$ having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle-variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of the anharmonic oscillator whose Hamiltonian is $H=\frac{{p}^{2}}{2}+\frac{{q}^{4}}{4}$.

DOI
10.1103/physrevd.40.2739

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This paper considers the Heisenberg equations of motion q\ifmmode˙\else˙\fi=\ensuremath−i[q,H]\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}=\ensuremath{-}i[q, H]q\ifmmode˙\else˙\fi​=\ensuremath−i[q,H], p\ifmmode˙\else˙\fi=\ensuremath−i[p,H]\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{p}=\ensuremath{-}i[p, H]p\ifmmode˙\else˙\fi​=\ensuremath−i[p,H], for the quantum-mechanical Hamiltonian H(p,q)H(p, q)H(p,q) having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle-variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of the anharmonic oscillator whose Hamiltonian is H=p22+q44H=\frac{{p}^{2}}{2}+\frac{{q}^{4}}{4}H=2p2​+4q4​.
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.
For a class of problems it is shown that a solution to Newton's equation of motion may be interpreted as the generating function for the matrix elements of the coordinate operators in the Heisenberg representation. For three one-dimensional problems, the harmonic oscillator, the particle in a box, and the x4{x}^{4}x4 anharmonic oscillator, solutions to the quantum-mechanical problem are given in terms of the classical solution. A brief examination is made of a two-dimensional problem in which the angular momentum is conserved.
Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.
Exact Heisenberg operator solutions for independent “sinusoidal coordinates” as many as the degree of freedom are derived for typical exactly solvable multiparticle quantum mechanical systems, the Calogero systems [J. Math. Phys.12, 419 (1971)] based on any root system. These Heisenberg operator solutions also present the explicit forms of the annihilation-creation operators for various quanta in the interacting multiparticle systems. At the same time they can be interpreted as multivariable generalization of the three term recurrence relations for multivariable orthogonal polynomials constituting the eigenfunctions.
Quantum dynamical systems typically deal with huge num-bers of degrees of freedom, particularly functional integrals, which makes them very difficult to simulate using deter-ministic methods. Modeling such systems with stochas-tic differential equations (SDEs) is an attractive alterna-tive to Markov chain Monte-Carlo, but there remain stabil-ity problems. An example of a multi-dimensional complex SDE arises from an anharmonic oscillator with a one-mode Bose-Einstein condensate (BEC) Hamiltonian given in [1], for which the numerical solution becomes unreliable after approximatively t = 0.3 for all tested numerical methods. Splitting of drift methods as introduced in [3] have been tried on this problem and seemed to improve the stability of the results, but not to the desired extent (see [1]). To improve the results, new numerical splitting methods have recently been developed. However, it seems that the choice of the numerical scheme alone might not guarantee the reliability of the results. The use of carefully chosen stochastic gauges as in [1] improves the accuracy of the simulation greatly by taking into account some aspects of the dynamics of the system. Non-vanishing boundary terms from partial integration can be problematic and have to be taken care of, for example considering them as constraints and using projected SDEs techniques [2]. 1. Simulation of Bose-Einstein condensates Consider the case of an anharmonic oscillator with one-mode BEC Hamiltonian treated in [1]. From the master equation ∂ρ̂