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Communications in Theoretical Physics

New Operator Ordering Formulas Related to Hermite Polynomials Derived by Virtue of IWOP Technique

Fan Hong-Yi

Published Sep 15, 2004• 13 citations
Publisher

Abstract

By virtue of the technique of integration within an ordered product of operators and the fundamental operator identity , where is the coordinate operator and is the -order Hermite polynomials, : : is the normal ordering symbol, we not only simplify the derivation of the main properties of Hermite polynomials, but also directly derive some new operator identities regarding to . Operation for transforming is also discussed.

DOI
10.1088/0253-6102/42/3/339

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By virtue of the technique of integration within an ordered product of operators and the fundamental operator identity , where is the coordinate operator and is the -order Hermite polynomials, : : is the normal ordering symbol, we not only simplify the derivation of the main properties of Hermite polynomials, but also directly derive some new operator identities regarding to . Operation for transforming is also discussed.
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For directly normalizing the photon non-Gaussian states (e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product (IWOP) of operators to derive some new bosonic operator-ordering identities. We also derive some new integration transformation formulas about one- and two-variable Hermite polynomials in complex function space. These operator identities and associative integration formulas provide much convenience for constructing non-Gaussian states in quantum engineering.
In this paper, by virtue of the entangled state representation we concisely derive some new operator identities regarding to two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.