# 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 quantum-mechanical systems, becomes a one-line 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, Aharonov-Bohm 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). E-mail: 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: S0003-4916(06)00158-8; 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; AHARONOV-BOHM 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 quantum-mechanical systems, becomes a one-line 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, Aharonov-Bohm 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}

}