Abstract
After developing the mathematical means for the correspondence of classical phase-space function to quantum-mechanical operators with symmetrical ordering of the basic canonical operators in the sense of Weyl the approach is applied to an infinite series of classical monomial functions of the canonical variables. These include as well as pure powers of the amplitude as also basic periodic functions of the phase φ with their quantum-mechanical correspondence. In the representation by number states, all the considered operators involve the Jacobi polynomials as the essential formative element. Whereas the quantity in normal ordering due to its indeterminacy leads to the introduction of the notions of sub- and super-Poissonian statistics the analogous quantity in (Weyl) symmetrical orderingis positive definite and satisfies an inequality. The notions of sub- and super-Poissonian statistics are problematic when they are used for the definition of nonclassicality of states since the mentioned measure in normal ordering does not determine the Poisson statistics in their middle in unique way but determines only a large set of statistics which may be very far in the sense of the Hilbert-Schmidt distance from a Poisson statistics that is discussed.
Highlights
After developing the mathematical means for the correspondence of classical phase-space function to quantum-mechanical operators with symmetrical ordering of the basic canonical operators in the sense of Weyl the approach is applied to an infinite series of classical monomial functions of the canonical variables
There are some technical difficulties to implement the explicit calculation of the symmetrically ordered operators corresponding to given classical phase-space functions in general cases, in particular, in the Fock or number state representation
In present article we have investigated in some detail the classical to quantum correspondence in the sense of Weyl for an important class of classical phase-space functions and quantum-mechanical operators and derived different representations, mainly representations by number states
Summary
IV, §14) proposed a general rule for the translation of arbitrary functions A(q, p) of the canonical phase-space variables in a unique way into quantum-mechanical operator-ordered functions A(Q, P) of the operators (Q, P). His way was via the Fourier transform of the classical function A(q, p) (denoted there in the way = f ( p, q). There are some technical difficulties to implement the explicit calculation of the symmetrically ordered operators corresponding to given classical phase-space functions in general cases, in particular, in the Fock or number state representation. { } { }2 quantity a†2a2 − a†a which is indefinite and leads in dependence of its negativity or positivity to the definition of sub- and super-Poissonian quantum statistics which are problematic when they are used for the definition of non-classicality of states
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