New General Approach for Normally Ordering Coordinate-Momentum Operator Functions

Abstract

By virtue of integration technique within ordered product of operators and Dirac’s representation theory we find a new general formula for normally ordering coordinate-momentum operator functions, that is $f(g\hat {{Q}}+h\hat {P})= :\exp [\textstyle {g^{2}+h^{2} \over 4}\textstyle {{\partial ^{2}} \over {\partial (g\hat {{Q}}+h\hat {P})^{2}}}]f(g\hat {{Q}}+h\hat {P})$:, where $\hat {Q}$ and $\hat {P}$ are the coordinate operator and momentum operator respectively, the symbol :: denotes normal ordering. Using this formula we can derive a series of new relations about Hermite polynomial and Laguerre polynomial, as well as some new differential relations.