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Results in Physics

The Perturbation Theory of Matrix Mechanics Based on its Canonical Transformations

Yong-Yi Huang

Published Jan 1, 2022• 1 citation
Publisher

Abstract

We elucidate that the canonical transformations of matrix mechanics are just the transformations of different representations of the standard quantum mechanics, reproduce the non-degenerate, degenerate and time-dependent perturbation theory in three men’s paper, we solve the mystery of time-dependent perturbation theory of matrix mechanics. The Kramers’ dispersion formula from time-dependent perturbation theory of matrix mechanics is also derived from that of wave mechanics.

DOI
10.1016/j.rinp.2022.106047

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We elucidate that the canonical transformations of matrix mechanics are just the transformations of different representations of the standard quantum mechanics, reproduce the non-degenerate, degenerate and time-dependent perturbation theory in three men’s paper, we solve the mystery of time-dependent perturbation theory of matrix mechanics. The Kramers’ dispersion formula from time-dependent perturbation theory of matrix mechanics is also derived from that of wave mechanics.
Abstract The time-dependent quantum perturbation theory developed by Born, Heisenberg and Jordan in 1926 is revisited. We show that it not only reproduces the standard theory formulated in the interaction picture, but also allows one to construct more accurate approximations if time averaging techniques are employed. The theory can be rendered unitary even if the expansion is truncated by using a transformation previously suggested by Heisenberg. We illustrate the main features of the procedure on a simple example which clearly shows its advantages in comparison with the standard perturbation theory.
The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
The Breakdown of Classical Mechanics Review of Classical Mechanics Elementary Systems One-Dimensional Problems More One-Dimensional Problems Mathematical Foundations Physical Interpretation Distributions and Fourier Transforms Algebraic Methods Central Force Problems Transformation Theory Non-Degenerate Perturbation Theory Degenerate Perturbation Theory Further Approximation Methods Time-Dependent Perturbation Theory Particle in a Uniform Magnetic Field Applications Scattering Theory-Time Dependent Scattering Theory-Time Independent Systems of Identical Particles Quantum Statistical Mechanics.