# Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations

## Abstract

Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac's symbols (ket versus bra, e.g., |q><q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |><| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac's symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are presented.

- Authors:

- CCAST (World Laboratory), P.O. Box 8730, Beijing 100080 (China)
- (China)
- Department of Physics, Shanghai Jiao Tong University, Shanghai 200030 (China). E-mail: luhailiang@sjtu.edu.cn
- Intel Corporation 2200 Mission College Blvd., Santa Clara, CA 95052-8119 (United States)

- Publication Date:

- OSTI Identifier:
- 20766990

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 321; Journal Issue: 2; Other Information: DOI: 10.1016/j.aop.2005.09.011; PII: S0003-4916(05)00189-2; Copyright (c) 2005 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; CLASSICAL MECHANICS; FUNCTIONS; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; QUANTUM OPERATORS; TRANSFORMATIONS

### Citation Formats

```
Fan Hongyi, Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, Lu Hailiang, and Fan Yue.
```*Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations*. United States: N. p., 2006.
Web. doi:10.1016/j.aop.2005.09.011.

```
Fan Hongyi, Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, Lu Hailiang, & Fan Yue.
```*Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations*. United States. doi:10.1016/j.aop.2005.09.011.

```
Fan Hongyi, Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, Lu Hailiang, and Fan Yue. Wed .
"Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations". United States.
doi:10.1016/j.aop.2005.09.011.
```

```
@article{osti_20766990,
```

title = {Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations},

author = {Fan Hongyi and Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026 and Department of Physics, Shanghai Jiao Tong University, Shanghai 200030 and Lu Hailiang and Fan Yue},

abstractNote = {Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac's symbols (ket versus bra, e.g., |q><q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |><| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac's symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are presented.},

doi = {10.1016/j.aop.2005.09.011},

journal = {Annals of Physics (New York)},

number = 2,

volume = 321,

place = {United States},

year = {Wed Feb 15 00:00:00 EST 2006},

month = {Wed Feb 15 00:00:00 EST 2006}

}