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Communications in Theoretical Physics

Application of Eigenkets of Bosonic Creation Operator in Deriving Some New Formulas of Associated Laguerre Polynomials

Fan Hong-Yi, Wang Tong-Tong

Published Aug 1, 2008• 3 citations
Publisher

Abstract

Using the resolution of unity composed of bosonic creation operator's eigenkets and annihilation operator's un-normalized eigenket, which is a new quantum mechanical representation in contour integration form, we derive new contour integration expression of associated Laguerre polynomials L ρ m (| z | 2 ) and its generalized generating function formula. A series of recursive relations regarding to L ρ m (| z | 2 ) are also deduced in the context of the Fock representation by algebraic method.

DOI
10.1088/0253-6102/50/2/06

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Using the resolution of unity composed of bosonic creation operator's eigenkets and annihilation operator's un-normalized eigenket, which is a new quantum mechanical representation in contour integration form, we derive new contour integration expression of associated Laguerre polynomials L ρ m (| z | 2 ) and its generalized generating function formula. A series of recursive relations regarding to L ρ m (| z | 2 ) are also deduced in the context of the Fock representation by algebraic method.
In this paper, we derive two new generating functions of Laguerre-polynomials, which look like the negative binomial theorem for the Laguerre function Lnx, by adopting the bi-partite entangled state representation and the operator-Hermite-polynomial HnX method, where X is the coordinate operator. Their application in investigating the density operator of photon loss of chaotic light is presented, and a new partition of Fock space in terms of the density operator of the negative binomial field is obtained. We also point out that the power-series definition of the Laguerre-polynomial originates from the Hermite polynomial. As a by-product, we derive a new Laguerre-polynomials-operator generating formula by virtue of the method of integration within an ordered product of operators.
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.
Using the completeness relation composed of the coherent state and of the eigenket of bosonic creation operator, we establish a one-to-one correspondence between the z -transform and the quantum-mechanical transform from the representation by number states to the representation by coherent states (Bargmann representation). In this way, the quantum-mechanical version of the various properties of z -transform are obtained and the operators for embodying these properties in the Fock space are derived, which may find applications in quantum states engineering.
In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (a†a^\daggera†) and annihilation (aaa) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator (a†x+ay)n(a^\dagger x+ay)^n(a†x+ay)n with nnn as the exponent and x, yx,\,yx,y are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.
We give a general expression for the normally ordered form of a function F[ŵ(a,a†)] where ŵ is a function of boson creation and annihilation operators satisfying [a,a†]=1. The expectation value of this expression in a coherent state becomes an exact generating function of Feynman-type graphs associated with the zero-dimensional quantum field theory defined by F(ŵ). This enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences. We give several examples of the use of this technique, including the applications to Kerr-type and superfluidity-type Hamiltonians.