一种新的Glauber公式的证明方法  

A NEW PROOF METHOD OF GLAUBER FOMULA

作　　者：陶俊琦 王蒙 程剑剑 郑华[1] TAO Junqi;WANG Meng;CHENG Jianjian;ZHENG Hua(School of Physics&Information Technology,Shaanxi Normal University,Xi'an,Shaanxi 710119)

机构地区：[1]陕西师范大学物理学与信息技术学院,陕西西安710119

出　　处：《物理与工程》2021年第1期25-27,30,共4页Physics and Engineering

基　　金：中央高校基本科研业务费专项资金项目(GK201903022)。

摘　　要：Glauber公式在量子力学中有着重要的应用,教科书中证明它的方法可以分为两种。一种是构造一个含有参变量的算符指数的函数,然后对参变量微分并利用Baker-Hausdorff公式,最后得到微分方程并积分求解得证。此方法存在一点瑕疵,因为在积分的过程中需要将算符放在分母,然而算符所对应的矩阵是没有除法的。另一种是先证明考虑算符对易性质的两个算符相加的二项式定理与不考虑算符对易性质的两个算符相加的二项式定理之间的关系,然后直接将Glauber公式中两个算符和的指数做展开并利用上述关系直接证明。此方法的证明过程略显复杂。本文通过构造、利用Baker-Hausdorff公式和算符的指数展开公式,给出了一种新的Glauber公式的证明方法。Glauber formula has an important application in quantum mechanics.There are two ways to prove it shown in textbooks.One is to construct an operator function consisting of the product of two exponentials of a variable and an operator,then make the derivative referring to the variable and apply the Baker-Hausdorff formula to solve the differential equation and set the variable to 1 to prove Glauber formula.There is a flaw in this method because the operator function has been divided when the differential equation is solved,however there is not division law for matrices corresponding to the operators in the operator function.The other needs to prove the relationship between the binomial theorem for two operators and the binomial theorem for two numbers first.Then one can prove the Glauber formula by doing Taylor expansion of the exponential of the summation of two operators and using the relationship proved above.The whole proof procedure of this method is a little bit complex.In this paper,we propose a new proof method of Glauber formula by utilizing the Baker-Hausdorff formula and the Taylor expansion of exponential of operator.

关 键 词：Glauber公式 算符 Baker-Hausdorff公式 

分 类 号：O413.1[理学—理论物理]