Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom
Abstract
This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schroedinger equation. Using complex canonical variables, a formal proof of the quantization axiom p {sup {yields}} p = ih{nabla}, which is the kernel in constructing quantummechanical systems, becomes a oneline corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, AharonovBohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.
 Authors:
 Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan (China). Email: cdyang@mail.ncku.edu.tw
 Publication Date:
 OSTI Identifier:
 20845952
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Annals of Physics (New York); Journal Volume: 321; Journal Issue: 12; Other Information: DOI: 10.1016/j.aop.2006.07.008; PII: S00034916(06)001588; Copyright (c) 2006 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AHARONOVBOHM EFFECT; CARTESIAN COORDINATES; CLASSICAL MECHANICS; ELECTRONIC STRUCTURE; EQUATIONS OF MOTION; QUANTIZATION; QUANTUM MECHANICS; QUANTUM OPERATORS; SCHROEDINGER EQUATION; SPIN; TUNNEL EFFECT
Citation Formats
Yang, C.D.. Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom. United States: N. p., 2006.
Web. doi:10.1016/j.aop.2006.07.008.
Yang, C.D.. Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom. United States. doi:10.1016/j.aop.2006.07.008.
Yang, C.D.. Fri .
"Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom". United States.
doi:10.1016/j.aop.2006.07.008.
@article{osti_20845952,
title = {Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom},
author = {Yang, C.D.},
abstractNote = {This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schroedinger equation. Using complex canonical variables, a formal proof of the quantization axiom p {sup {yields}} p = ih{nabla}, which is the kernel in constructing quantummechanical systems, becomes a oneline corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, AharonovBohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.},
doi = {10.1016/j.aop.2006.07.008},
journal = {Annals of Physics (New York)},
number = 12,
volume = 321,
place = {United States},
year = {Fri Dec 15 00:00:00 EST 2006},
month = {Fri Dec 15 00:00:00 EST 2006}
}

EquationsofMotion Approach to Quantum Mechanics: Application to a Model Phase Transition
We present a generalized equationsofmotion method that efficiently calculates energy spectra and matrix elements for algebraic models. The method is applied to a fivedimensional quartic oscillator that exhibits a quantum phase transition between vibrational and rotational phases. For certain parameters, 10x10 matrices give better results than obtained by diagonalizing 1000x1000 matrices.