Abstract: Because the coordinate and the momentum operator do not communte，the re-ordering of the coordinate and the momentum operator functions is a fundamental problem in quantum mechanics. Using the completeness of coordinate and momentum representation and the integration within ordered product of operators，the re-ordering formulas of the coordinate operator and momentum operator function can be easily derived. In particular，a bivariate Hermite polynomial can be used to express the re-ordering of the power of coordinate and momentum exponential operator neatly. Furthermore，the structure of the coordinate-momentum intermediary representation can be derived by the integration within ordered product of operators.

Key words: representation, completeness, integration within ordered product, Hermite polynomial