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Chinese Physics B

Recast combination functions of coordinate and momentum operators into their ordered product forms*

Lei Wang, Xiang‐Guo Meng, Ji‐Suo Wang

Published Mar 9, 2020• 3 citations
Publisher

Abstract

By using the parameter differential method of operators, we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings, which is more ecumenical, simpler, and neater than the existing ways. These products are very useful in obtaining some new differential relations and useful mathematical integral formulas. Further, we derive the normally ordered form of the operator (fQ + gP)−n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate–momentum representation. In addition, general mutual transformation rules of the normal and anti-normal orderings, which have good universality, are derived and hence the anti-normal ordering of (fQ + gP)−n is also obtained. Finally, the application of some new identities is given.

DOI
10.1088/1674-1056/ab7dbd

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By using the parameter differential method of operators, we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings, which is more ecumenical, simpler, and neater than the existing ways. These products are very useful in obtaining some new differential relations and useful mathematical integral formulas. Further, we derive the normally ordered form of the operator (fQ + gP)−n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate–momentum representation. In addition, general mutual transformation rules of the normal and anti-normal orderings, which have good universality, are derived and hence the anti-normal ordering of (fQ + gP)−n is also obtained. Finally, the application of some new identities is given.
In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator ( fQ + gP ) n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.
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.
Since the foundation of quantum mechanics, operator-ordering identities for mutual transformation of power of coordinate-momentum operators have been a fundamental and tough topic. To the best of our knowledge, this topic has not been tackled smoothly because there is no elegant and direct way to investigate it. In this paper we report a very concise and novel method to handle this topic, i.e., we employ the generating function of two-variable Hermite polynomial and the characteristics of ordered operators to derive a series of operator-ordering identities for mutual transformation of power of coordinate-momentum operators: they surly possess potential applications. The essence of our method lies in the fact that coordinate-momentum operators can be permutable within ordered product of operators, just as the scenarios in P-Q ordering, Q-P ordering and Weyl ordering. We also derive the integration transformation formula about two-variable Hermite polynomial in phase space. The correspondence relation between operator ordering and quantization recipe is established. The beauty of theoretical physics is embodied extensively in the paper.