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Title: 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:
 [1];  [2];  [2];  [3];  [4]
  1. CCAST (World Laboratory), P.O. Box 8730, Beijing 100080 (China)
  2. (China)
  3. Department of Physics, Shanghai Jiao Tong University, Shanghai 200030 (China). E-mail: luhailiang@sjtu.edu.cn
  4. 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}
}
  • We show that the technique of integration within normal ordering of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480-494] applied to tackling Newton-Leibniz integration over ket-bra projection operators, can be generalized to the technique of integration within Weyl ordered product (IWWOP) of operators. The Weyl ordering symbol is introduced to find the Wigner operator's Weyl ordering form {delta}(p,q) = {delta}(p - P){delta}(q - Q) , and to find operators' Weyl ordered expansion formula. A remarkable property is that Weyl ordering of operators is covariant under similarity transformation, so it has many applications in quantum statistics andmore » signal analysis. Thus the invention of the IWWOP technique promotes the progress of Dirac's symbolic method.« less
  • We show that Newton-Leibniz integration over Dirac's ket-bra projection operators with continuum variables, which can be performed by the technique of integration within ordered product (IWOP) of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480], can directly recast density operators and generalized Wigner operators into normally ordered bivariate-normal-distribution form, which has resemblance in statistics. In this way the phase space formalism of quantum mechanics can be developed. The Husimi operator, entangled Husimi operator and entangled Wigner operator for entangled particles with different masses are naturally introduced by virtue of the IWOP technique, and their physical meaningsmore » are explained.« less
  • The Newton-Leibniz integration over Dirac's ket-bra operators introduced in Ref. [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480-494] is generalized to Newton-Leibniz-Berezin integration over fermionic ket-bra projection operators, the corresponding technique of integration within an ordered product (IWOP) of Fermi operators is proposed which is then used to develop fermionic quantum statistics. The generalized partition function formula of multi-mode quadratic interacting fermion is derived via the fermionic coherent state representation and the IWOP technique. The two-mode fermionic squeezing operators and their group property studied by their fermionic coherent state representation as well as fermionic permutation operator aremore » also deduced in this way. Thus Dirac's symbolic method for Fermi system can also be developed, which exhibits Bose-Fermi supersymmetry in this aspect.« less
  • Via the route of applying Newton-Leibniz integration rule to Dirac's symbolic operators, we show that the density operator e{sup -{beta}}{sup H}, where H is multi-mode quadratic interacting boson operators, is a mapping of symplectic transformation in the coherent state representation appearing in the form of non-symmetric ket-bra operator integration. By virtue of the technique of integration within an ordered product (IWOP) of operators, we deduce its normally ordered form which directly leads to the generalized partition function formula and the Wigner function. Some new representations, such as displacement-squeezing correlated squeezed coherent states, constructed by the IWOP technique, also bring conveniencemore » in deriving partition functions.« less