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Orderings of Operators in Quantum Physics

Kazuyuki Fujii, Tatsuo Suzuki

Published Apr 13, 2003• 0 citations
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Abstract

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. 1

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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. 1
For the creation operator a†and the annihilation operator a of a harmonic oscillator, we consider Weyl ordering expression of (a†a)nand obtain a new symmetric expression of Weyl ordering w.r.t. a†a≡N and aa†=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.
We give the normal and anti-normal order expressions of the number operator to the power kkk by using the commutation relation between the annihilation and creation operators. We use those expressions to give general formulae for functions of the number operator in normal and anti-normal order.
Normal-ordered equivalent of the Weyl ordering of q^jp^k\hat{q}^j \hat{p}^kq^​jp^​k
The problem of quantizing a bivariate dynamical system can be reduced to evaluating the ordering of q^jp^k\hat{q}^j \hat{p}^kq^​jp^​k. Here, we consider the Weyl ordering of q^jp^k\hat{q}^j \hat{p}^kq^​jp^​k that is then expressed in term of the annihilation a^\hat{a}a^ and creation a^†\hat{a}^\daggera^† operator. The explicit formula for the normal-ordered equivalent (all a^†\hat{a}^\daggera^†'s preceeding all a^\hat{a}a^'s) of the resulting expression is then given, and some relations are discussed.
We address a systematic combinatorial approach to the anti-normal ordering problem. In this way, we use the Stirling numbers and their generating function, the so-called Bell polynomials, together with the operational methods to anti-normal the operator eλa†a{e^{λ{a^†}a}}eλa†a. In fact, we exploit the new theorem given by Shähandeh et al. in a special case of anti-normal ordering.
Two calculation formulas of expansion each other for normal ordering and antinormal ordering of creation operator and annihilation operator are given in this paper as () n( +) m=∑pk=0m!n!(m-k)!(n-k)!k!( +) m-k () n-k  and ( +) n() m=∑pk=0(-1) km!n!(m-k)!(n-k)!k!() m-k ( +) n-k , where p=min(n,m). It is right that we prove by using mathematical inductive method. The two formulas of this paper are used in a wide class of study of quantum statistical properties for various excited optical fields.