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Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & Astronomy

s-ordering operator expansions of quantum-mechanical fundamental representations and their applications

Fan Hong

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We show that the quantum-mechanical fundamental representations, say, the coordinate representation, the coherent state representation, the Fan-Klauder entangled state representation can be recast into s-ordering operator expansion, which is elegant in form and has many applications in deriving new operator identities. This demonstrates that Dirac’s symbolic method can be merged into Newton-Leibniz integration theory in a broad way.

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We show that the quantum-mechanical fundamental representations, say, the coordinate representation, the coherent state representation, the Fan-Klauder entangled state representation can be recast into s-ordering operator expansion, which is elegant in form and has many applications in deriving new operator identities. This demonstrates that Dirac’s symbolic method can be merged into Newton-Leibniz integration theory in a broad way.
We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.
Dirac's ket-bra formalism is the language of quantum mechanics and quantum field theory. In Refs.(Fan et al, Ann. Phys. 321 (2006) 480; 323 (2008) 500) we have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors. In this work by alternately using the technique of integration within normal, antinormal, and Weyl ordering of operators we not only derive some new operator ordering identities, but also deduce some useful integration formulas regarding to Laguerre and Hermite polynomials. This opens a new route of deriving mathematical integration formulas by virtue of the quantum mechanical operator ordering technique.
We develop quantum mechanical Dirac ket-bra operator's integration theory in Q-ordering or P-ordering to multimode case, where Q-ordering means all Qs are to the left of all Ps and P-ordering means all Ps are to the left of all Qs. As their applications,we derive Q-ordered and P-ordered expansion formulas of multimode exponential operator e iPlΛlkQk. Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation q1q2 → A B C D q1q2 is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.