In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (
a†) and annihilation (
a) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator
(a†x+ay)n with
n as the exponent and
x,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.
This study derives the normal ordering expansion of degenerate integral powers of the number operator, (a†a)_{n,λ}, using recurrence relations for the coefficients and an operator action on number states. Here a† and a are respectively the boson creation and annihilation operators. We also determine the inverse of this normal ordering expansion. By analyzing diagonal coherent state elements of the degenerate integral powers of the number operator, we establish a combinatorial identity which yields a Dobinski-like formula for the degenerate Bell numbers at a specific value, connecting degenerate quantum operator calculus with combinatorics.
The expansion of operators as ordered power series in the annihilation and creation operators
a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required
c-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values $s=+1,0,\ensuremath{-}1$, respectively, of an order parameter
s. In terms of this convention it is shown that for bounded operators the coefficients are finite when
s>0, and the series are convergent when
s>21. For each value of the order parameter
s, a correspondence between operators and
c-number functions is defined. Each correspondence is one-to-one and has the property that the function $f(\ensuremath{\alpha})$ associated with a given operator
F is the one which results when the operators
a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for
F are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the
P representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter
s.
We solve the boson normal ordering problem for ( q ( a † ) a + v ( a † )) n with arbitrary functions q ( x ) and v ( x ) and integer n , where a and a † are boson annihilation and creation operators, satisfying [ a , a † ] = 1. This consequently provides the solution for the exponential e λ( q ( a † ) a + v ( a † )) generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.
Normally ordered forms of functions of boson operators are important in many contexts mainly concerning quantum field theory and quantum optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento 10(13):565--567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure [see P. Blasiak, quant-ph/0507206]. In this paper, we apply the linear representation of noncrossing partitions to contractions of normally ordered forms. In this way, we define the notion of noncrossing normal ordering. This appear to be a natural mathematical concept and it results indeed to be linked with well-known combinatorial objects. We explicitely give the noncrossing normally ordered form of the functions (a^{r}(a^{dagger})^{s})^n) and (a^{r}+(a^{dagger})^{s})^n, plus various special cases. We establish bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths. Physical applications of the noncrossing normal ordering are desiderata.