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College Physics

Derivation of wave functions of particle number state by using Hermite polynomial operator

Fan Hong-Yi

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

By introducing the Hermite polynomial operator and deriving its normally ordered expansion,we derive the wave functions of particle number state in the coordinate representation and momentum representation,respectively,which can directly lead to the Fock representation of coordinate and momentum eigenstate.Our way is brand-new and concise.

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By introducing the Hermite polynomial operator and deriving its normally ordered expansion,we derive the wave functions of particle number state in the coordinate representation and momentum representation,respectively,which can directly lead to the Fock representation of coordinate and momentum eigenstate.Our way is brand-new and concise.
By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials.We also derive the new reciprocal relation of Laguerre polynomials n=0(1) n l n L n(x) = x l l!,and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated.Some new expansion identities about the operator Laguerre polynomial are also derived.This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.
The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.
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.