摘要
在量子光学理论计算中,经常遇到算符的正规排序和反正规排序问题,我们从双变量厄米多项式Hm,n(x,y)的母函数出发,导出两个简洁的重要的基本算符恒等式并由此可以给出一些推论公式.
Abstract
Quantum optics theory needs an advanced method to tackle density operator'various physical quantities,such as expec-tation value,variance,cumulant,etc.To be specific,since photon creation and annihilation operators do not commute,we need to deal with the problems of how to convert normally ordered operators into anti-normally ordered operators,and how to convert anti-normally ordered operators into normally ordered operators.In short,the operator re-ordering problem is often encountered in quantum optics theory.In this paper we employ the generating function of two-variable Hermite polynomials to derive two basic operator identities.The first basic operator identity is ana+m=(-i)m+n:Hm,n(ia+,ia):,which converts anti-normally ordered operators into normally ordered operators.As an application of the basic operator identity we compute and get am|n〉=√n!/(n-m)!|n-m〉,meanwhile,we give the commutation relation of[am,a+n].The second basic operator identity is a+man =…Hn,m(a+,a)…,which converts nor-mally ordered operators into anti-normally ordered operators.When m=n,in virtue of laguerre's polynomials we get the equal-ity Hn,n(x,y)=(-1)nn!Ln(xy).We derive a formula for the transformation between normal product and the anti-normal product in the end.The two basic operator identities are easily remembered and useful in quantum optics.The application of two-variable Hermite polynomials,such as for studying quantum entangled state representation,is greatly developed by Fan Hong-yi in recent years.One can also apply the new basic operator identities to develop binomial and negative-binomial theory which involves two-variable Hermite polynomials.
基金项目
武夷学院引进人才科研启动经费项目(YJ201808)
NETL
NSTL
万方数据