In reference to the Weyl ordering , where X and P are coordinate and momentum operator, respectively, this paper examines operators' s -parameterized ordering and its classical correspondence, finds the fundamental function-operator correspondence and its complementary relation where H m,n is the two-variable Hermite polynomial, a , a † are bosonic annihilation and creation operators respectively, s ′ is a complex parameter. The s -ordered operator power-series expansion of s -ordered operator in terms of the two-variable Hermite polynomial is also derived. Application of operators' s -ordering formula in studying displaced-squeezed chaotic field is discussed.
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.
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.
The s-ordered form of any product of single-mode boson creation and annihilation operators, containing only a single annihilator, is computed explicitly. The s-ordering concept originated in quantum optics, but subsumes normal, symmetric (Weyl), and anti-normal ordering for any two operators satisfying a canonical commutation relation. Because the s-ordering map can be viewed as producing a function of a complex variable, its inverse is a quantization map that takes such “classical” functions to quantum operators. The explicit s-ordered expressions are derived with the aid of a parametric family of Sheffer polynomial sequences (or equivalently a parametric exponential Riordan array of polynomial coefficients), called the Hsu–Shiue family. To yield orderings interpolating between normal and anti-normal, this family must be extended.