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Chinese Physics B

Ordered product expansions of operators (AB)±m with arbitrary positive integer*

Shi-Min Xu, Yushan Li, Xing-Lei Xu, Lei Wang, Ji‐Suo Wang

Published Jun 5, 2020• 3 citations
Publisher

Abstract

We arrange quantum mechanical operators (a † a) m in their normally ordered product forms by using Touchard polynomials. Moreover, we derive the anti-normally ordered forms of (a † a)± m by using special functions as well as Stirling-like numbers together with the general mutual transformation rule between normal and anti-normal orderings of operators. Further, the ℚ- and ℙ-ordered forms of (QP)±m are also obtained by using an analogy method.

DOI
10.1088/1674-1056/ab99aa

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We arrange quantum mechanical operators (a † a) m in their normally ordered product forms by using Touchard polynomials. Moreover, we derive the anti-normally ordered forms of (a † a)± m by using special functions as well as Stirling-like numbers together with the general mutual transformation rule between normal and anti-normal orderings of operators. Further, the ℚ- and ℙ-ordered forms of (QP)±m are also obtained by using an analogy method.
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In this paper, we provide a new kind of operator formula for anti-normally and normally ordering bosonic-operator functions in quantum optics, which can help us arrange a bosonic-operator function f ( λ + ν ) in its anti-normal and normal ordering conveniently. Furthermore, mutual transformation formulas between anti-normal ordering and normal ordering, which have good universality, are derived too. Based on these operator formulas, some new differential relations and some useful mathematical integral formulas are easily derived without really performing these integrations.
The problem of normal ordering of the following boson operator functions (a+a + ) m , (a r +a + ) m , (a+N) m , (a 2 +N) m , where N=a + a, is solved explicitly. Simple algebraic methods are used. As an intermediate result the ordering formulae for the symmetrised products of boson operators are presented. There is agreement with earlier results of other authors and generalisation of these results. A new representation of the generalised Stirling numbers used for the ordering of the binomial (a 2 +N) m is given and several principal properties of these numbers are derived.