A New Symmetric Expression of Weyl Ordering

Abstract

For the creation operator $\adag $ and the annihilation operator $a$ of a harmonic oscillator, we consider Weyl ordering expression of $(\adag a)^n$ and obtain a new symmetric expression of Weyl ordering w.r.t. $\adag a \equiv N$ and $a\adag =N+1$ where $N$ is the number operator. Moreover, we interpret intertwining formulas of various orderings in view of the difference theory. Then we find that the noncommutative parameter corresponds to the increment of the difference operator w.r.t. variable $N$. Therefore, quantum (noncommutative) calculations of harmonic oscillators are done by classical (commutative) ones of the number operator by using the difference theory. As a by-product, nontrivial relations including the Stirling number of the first kind are also obtained.

15 pages, Latex2e, the title before replacement is "Orderings of Operators in Quantum Physics", new proofs by using a difference operator added, some references added, to appear in Modern Physics Letters A

References in corpus (1)

• Equivalence of Weyl Vacuum and Normal Ordered Vacuum in the Moyal Quantization

Cited by in corpus (7)

• Wick's theorem for q-deformed boson operators
• Commutation relations of operator monomials
• Emergence of the Born rule in quantum optics
• Quantum eigenstates from classical Gibbs distributions
• Self-rotating wave approximation via symmetric ordering of ladder operators
• Quantum Diagonalization Method in the Tavis-Cummings Model
• Intensity correlations in the Wigner representation