ℏ-independent universality of the quantum and classical canonical transformations

Abstract

A theory of non-unitary-invertible and also unitary canonical transformations is formulated in the context of Weyl's phase space representations. It is shown in the phase space that all quantum canonical transformations without an explicit ℏ dependence are also classical mechanical and vice versa. Contrary to some earlier results, it is also shown that the quantum generators and their classical counterparts are identical in this ℏ-independent universal class. © 2006 Elsevier B.V. All rights reserved

Constraints on the choice of position dependent effective mass and external potential for the existence of Lewis-Riesenfeld invariance and quantum canonical transformation

“…First let us recall the action of a time-dependent quantum canonical transformation (QCT) on quantum system [38][39][40][41][42][43][44][45][46]. One can note that the time dependent Schrödinger…”

“…In particular, only a mass profile which varies with position (x) as ∼ 1 x+a under either Coulomb potential or potential well/barrier are allowed for the existence of a close form LR-invariant operator (LRIO). Interestingly, time dependent unitary quantum canonical transformation (TDQCT) [38][39][40][41][42][43][44][45][46] exists for the the concerned PDEM can be transformed into an equivalent time independent Hamiltonian under TDQCT.…”

Quantum Canonical Transformations in Weyl–wigner–groenewold–moyal Formalism

“…is used for the QCT (1), the corresponding transformation in the c-number phase-space can be written as: [14] F…”

“…In these general terms, for the sake of generality, we assume that there always exists a one to one correspondence between arbitrary F (q, p) and F (q, p). If this general correspondence which can be summarized as F (q, p) ↔ F (q, p), F (q, p) ⋆ G(q, p) ↔ F (q, p) Ĝ(q, p) (15) is used for the QCT (1), the corresponding transformation in the c-number phase-space can be written as: [14] F (q, p) ⋆ q ⋆ F −1 (q, p) = Q(q, p), F (q, p) ⋆ p ⋆ F −1 (q, p) = P (q, p)…”