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OPERATOR METHOD FOR NONPERTURBATIVE DESCRIPTION OF QUANTUM SYSTEMS

И. Д. Феранчук, Alexey Ivanov

Published Nov 1, 2004• 1 citation
Publisher

Abstract

Nonperturbative method for description of quantum systems - the operator method (OM) \nand the conception of the uniformly suitable estimation (USE) are considered for the series of real physical systems. It is shown that the OM zeroth-order approximation permits one to find the analytical approximation for eigenfunctions and eigenvalues with high accuracy \nwithin the entire range of the Hamiltonian parameters and any quantum numbers. The \nOM subsequent approximations converge rapidly to the exact solutions of the Schrödinger equation. The generalization of OM for the quantum statistics is also developed.

DOI
10.1142/9789812702296_0011

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Nonperturbative method for description of quantum systems - the operator method (OM) \nand the conception of the uniformly suitable estimation (USE) are considered for the series of real physical systems. It is shown that the OM zeroth-order approximation permits one to find the analytical approximation for eigenfunctions and eigenvalues with high accuracy \nwithin the entire range of the Hamiltonian parameters and any quantum numbers. The \nOM subsequent approximations converge rapidly to the exact solutions of the Schrödinger equation. The generalization of OM for the quantum statistics is also developed.
We obtain a complete series solution of stationary Schrödinger's equation in the general quantum systems. It is exact in the sense that any approximation means is not used, or that the whole corrections or contributions from all order perturbations are involved if the perturbation concept is introduced. Furthermore, the useful forms of our exact solution and a new expression the complete Green operator are given out. As a universal and analytical solution, it is helpful for the theoretical derivations and practical calculations in quantum theory.
Accurate eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically. A special iteration procedure based on the operator method permits one to consider the solution within a wide range of the Hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues. Characteristic features of the model are considered, such as the level intersections, the population of the field states and the chaotization in the system through the doubling of the frequencies.
Approximation of eigen values and eigenvectors Many realistic problems that arise in theoretical physics lead to equations which cannot be given exact general solutions. Therefore, many attempts have been made to produce satisfactory methods to provide approximate solutions. The most common ones search for solutions in terms of power series and in terms of asymptotic expansions. The main idea behind approximation methods is to try to find approximate solutions of a given system by means of exact solutions of an approximate system (a ‘nearby’ system). We start with the eigenvalue equation associated with the Hamiltonian H and consider an approximating one, say H 0 , whose equations of motion can be explicitly solved. We introduce an interpolating family of Hamiltonians H ε = H 0 + ε H 1 such that, for a given value of ε, say ε, H ε ≡ H . Both eigenvalues and eigenvectors for H ε may be expanded as a series in ε and solved order by order. The simplest example to illustrate what we have in mind is provided by a second-order algebraic equation, e.g. x 2 − 3.03 x + 2.02 = 0. This equation may be written by means of an interpolating family x 2 − 3(1 + ε) x + 2(1 + ε) = 0, which yields the equation we want to solve when ε = 10 −2 . One can consider the approximate equation to be given by ε = 0, i.e. x 2 − 3 x + 2 = 0.
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a ccc-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.