Orderings of Operators in Quantum Physics

For well-known three orderings (normal ordering, anti-normal ordering, and Weyl ordering) in Quantum Physics, we give a simple proof for the intertwining formulas among them. Moreover, for the creation operator a† and the annihilation operator a of a harmonic oscillator, we show Weyl ordering expression of (a†a)n as different types of polynomials with respect to the number operator N ≡ a†a. Nontrivial relations including the Stirling number of the first kind are also obtained.