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Technical Physics

Canonical averaging of the Schrödinger equation

A. G. Chirkov

Published Feb 1, 2002• 0 citations
Publisher

Abstract

The representation of the Schrödinger equation in the form of a classical Hamiltonian system makes it possible to construct a unified perturbation theory that is based on the theory of canonical transformations and covers both classical and quantum mechanics. Also, the closeness of the exact and approximate solutions of the Schrödinger equation can be approximately estimated with such a representation.

DOI
10.1134/1.1451981

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The representation of the Schrödinger equation in the form of a classical Hamiltonian system makes it possible to construct a unified perturbation theory that is based on the theory of canonical transformations and covers both classical and quantum mechanics. Also, the closeness of the exact and approximate solutions of the Schrödinger equation can be approximately estimated with such a representation.
Many physical systems cannot be solved exactly. In this case, we look for approximate methods to gain insight into their properties. Perturbation theory is one such method. When the Hamiltonian of the system doesn't change over time, one approximate method is the Time Independent Perturbation Theory, which is the subject of Chapter 10 . The infinite square well, the harmonic oscillator, and the hydrogen atom are among the few physical systems for which the Schrodinger equation can be solved exactly. For this reason, they are described in detail in most of the introductory quantum mechanics textbooks. However, most physical systems are not exactly solvable, i.e., they are described by the Schrodinger equation, whose solutions cannot be expressed in terms of commonly used functions. In order to study such systems, we try to decompose the corresponding Hamiltonian into a solvable Hamiltonian plus a small perturbation. For such cases, we say that the Hamiltonian is “close enough” to an exactly solvable Hamiltonian to warrant using perturbative methods to approximate its eigenenergies and eigenfunctions.
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.