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Journal of Physics A Mathematical and General

Ordering of some boson operator functions

V V Mikhailov

Published Nov 11, 1983• 26 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/16/16/019

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The normal ordering formula for (N+a r ) n , where N=a + a, is derived for arbitrary r and n. This is a generalisation of the results recently obtained by Mikhailov (1983) for r=1, 2. Some other closely related results are also discussed.

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General expansion of natural power of linear combination of Bosonic operators in normal order

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arXiv:2305.18113•May 29, 2023•0 citations
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.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobihski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
The normal ordering formulae for powers of the boson number operator n are extended to deformed bosons. It is found that for the 'M-type' deformed bosons, which satisfy aa Dagger -qa Dagger a=1, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the 'P-type' deformed bosons, which satisfy aa Dagger -qa Dagger a=q -n are found to depend on the operator n. This distinction between the two types of deformed bosons is in harmony with earlier observations made in the context of a study of the extended Campbell-Baker-Hausdorff formula.
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.
In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (a†a^\daggera†) and annihilation (aaa) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator (a†x+ay)n(a^\dagger x+ay)^n(a†x+ay)n with nnn as the exponent and x, yx,\,yx,y are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.