Theory Of Operator Hermite Polynomials And Its Application In Quantum Optics

The development of modern physics theory no doubt needs advanced mathematics,theoretical physicists should invent advanced mathematics correspondingly by them-selves.,and display their independent thinking.Hermite polynomials,as a kind of spe-cial functions,has wide applications in quantum mechanics and quantum optics.For instance,wave functions of the eigenstates of quantum oscillator are described by Her-mite polynomials.Special functions are so-called due to their specia recurssive rules,and are easily remembered.The set of Hermite polynomials Hm(x)constitute an or-thonormal and complete function space,so it is important in mathematical physics.In this paper we propose to replace the argument x of Hm(x)by the coordinate opera-tor X,and we name Hm(X)the operator Hermite polynomial.Because operators in general are not commutable,the ordering problem of operator special functions is a complete new one.I employed the integration method within ordered operators to sys-tematically study this problem,and found new ordering features of Hm(X),based on which I derived some new operator identities that are useful in constructing quantum optical states.Furthermore,when I transit the ordering rule of Hm(X)to classical case,the new generalized binomial theorem involving Hm(x)is established and some new generating functions are obtained.The theory of operator Hermite polynomials includes:1.With the aid of new formula regarding to operator special function we can find new relationship connecting various special functions,e.g.,deriving the Laguerre polynomial from the Hermite polynomial.2.By comparison of various ordering forms of operator special functions and using quantum mechanical representations' completeness we can derive quite a few new useful integration formulas(without directly perform these integrations).3.By virtue of operator special functions' identities we can develop quantum me-chanical representation and transform theory.4.Develop classical-quantum mechanical correspondence and phase space theory.5.Find new series expansion of some special functions and its reciprocal relation.6.Conveniently evaluating various physical quantuties,such as moment function,cumulant functions.In the paper I also study the two-variable Hermite polynomials and develop it to the operator Hermite polynomials,then study its properties and applications.As peoples' physical understanding relies on advanced mathematics,this paper focuses on constructing operator Hermite polynomial theory on the basis of physical conception.The paper is arranged as follows:In Chapter 1 I introduced the method of integration within ordered product of op-erators which was invented by chinese scientist Fan Hong-yi.In Chapters 2-3 I pro-pose the theory of operator Hermite polynomial(single variable),its kernel is the nor-mally ordered and antinormally ordered expansion,which is remarkable.In Chapter 4 I propose how to employ operator Hermite polynomial method to introduce the La-guerre polynomial,this deffers from the usual way that the Laguerre polynomial was defined independently.In Chapter 5 I present physical application of the operator iden-tity Hn(X)= 2n:Xn:.Chapter 6 is devoted to derive the new binomial theorem which involves Hermite polynomial Hn(x).In Chapter 7 I employ operator Hermite polyno-mial method to deduce the generating function of even-and odd-Hermite polynomial,which are new.Chapter 8-11 I present the way of introducing the two-variable Hermite polynomials and derive its various generating functions.In Chapter 12,in view of the fact that Fock space is usually expanded in terms of the pure states(number states),I in-stead explore the possibility if the Fock space can be expanded in terms of some mixed states,and find that it can be Fock space can be expanded in terms of binomial state and negative-binomial state.In summary,this paper creates the operator Hermite polynomial theory,and the newly found classification of Fock space enriches the fundamental theory of quantum mechanics.