Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical Physics

An ħ-Expansion of a Unitary Transformation and Quantum Corrections to a Canonical Transformation

Shinji Iida

Published Jul 1, 1986• 2 citations
Publisher

Abstract

A relation between classical canonical transformations and quantal unitary transformations is investigated with the use of the phase space method. A unitary transformed operator expressed in the phase space is expanded into a power series of ħ, which shows that a canonical transformation can be regarded as a classical approximation (the limit ħ→0) to the unitary transformation. The expansion provides us with a procedure to successively evaluate quantum corrections starting with the classical limit. As an illustrative example, the lowest-order quantum correction to the Birkhoff-Gustavson transformation for a one-dimensional anharmonic oscillator is calculated in the case of Weyl correspondence rule.

DOI
10.1143/ptp.76.115

Related papers

Canonical Transformations in Quantum Mechanics

Arlen Anderson

Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schr\\"odinger equation is discussed. Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can be realized quantum mechanically as a product of these transformations. Each transformation corresponds to a familiar tool used in solving differential equations, and the procedure of solving a differential equation is systematized by the use of the canonical transformations. Several examples are done to illustrate the use of the canonical transformations. [This is an extensively revised version of hep-th-9205080: the first third of the paper is new material; the notation has been simplified, and further discussion has been added to the remainder.]

•May 13, 1993•100 citations

Mapping of classical canonical transformations to quantum unitary operators

Fan Hong‐Yi, Jack Vanderlinde

Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.

Physical Review A•Mar 1, 1989•61 citations

Wigner distribution functions and the representation of canonical transformations in quantum mechanics

Gastón García‐Calderón, M. Moshińsky

Shows how for classical canonical transformations the authors can pass, with the help of Wigner distribution functions, from their representation U in the configurational Hilbert space to a kernel K in phase space. The latter is a much more transparent way of looking at representations of canonical transformations, as the classical limit is reached when h(cross) to 0 and the successive quantum corrections are related with the power of h(cross) 2n , n=1,2, et seq.

Journal of Physics A Mathematical and General•Jun 1, 1980•49 citations

Quantum harmonic analysis on phase space

Reinhard F. Werner

Relative to an irreducible representation of the canonical commutation relations, convolutions between quantum mechanical operators and between functions and operators are defined, for which the usual Weyl transform acts as a Fourier transform. Basic properties of these operations are developed in close analogy to harmonic analysis on R2n. Using the quantum version of Wiener’s approximation theorem, a natural one-to-one correspondence between the closed, phase-space translation invariant subspaces of classical and quantum observables is established.

Journal of Mathematical Physics•May 1, 1984•171 citations

Quantum mechanics as classical mechanics plus quantum corrections: the cubic anharmonic oscillator

Gabriel Álvarez

The Rayleigh-Schrodinger perturbation series for the cubic anharmonic oscillator can be formally rearranged as the classical Birkhoff series plus quantum corrections proportional to successive powers of h(cross) 2 . Each of the individual quantum corrections is a convergent power series expansion (with the same radius of convergence as the classical series) of the corresponding term in the Jeffreys-Wentzel-Kramers-Brillouin series.

Journal of Physics A Mathematical and General•Mar 21, 1989•12 citations

Canonical transformations and Hamiltonian path integrals

Прохоров Лев Васильевич

The behavior of Hamiltonian path integrals under canonical transformations defined by a generator is investigated. The exact form of the kernel of the unitary operator which implements the quantum transformation is indicated. Equivalence rules are found (for the Hamiltonian formalism in the one-dimensional case) allowing one to get rid of the nonstandard terms in the action. It is shown that under canonical transformations the Hamiltonian path integral changes its form: in the transformed expression one cannot get along only with the corresponding classical Hamiltonian function.

•Feb 1, 1982•0 citations
A relation between classical canonical transformations and quantal unitary transformations is investigated with the use of the phase space method. A unitary transformed operator expressed in the phase space is expanded into a power series of ħ, which shows that a canonical transformation can be regarded as a classical approximation (the limit ħ→0) to the unitary transformation. The expansion provides us with a procedure to successively evaluate quantum corrections starting with the classical limit. As an illustrative example, the lowest-order quantum correction to the Birkhoff-Gustavson transformation for a one-dimensional anharmonic oscillator is calculated in the case of Weyl correspondence rule.
Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schr\\"odinger equation is discussed. Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can be realized quantum mechanically as a product of these transformations. Each transformation corresponds to a familiar tool used in solving differential equations, and the procedure of solving a differential equation is systematized by the use of the canonical transformations. Several examples are done to illustrate the use of the canonical transformations. [This is an extensively revised version of hep-th-9205080: the first third of the paper is new material; the notation has been simplified, and further discussion has been added to the remainder.]
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.
Relative to an irreducible representation of the canonical commutation relations, convolutions between quantum mechanical operators and between functions and operators are defined, for which the usual Weyl transform acts as a Fourier transform. Basic properties of these operations are developed in close analogy to harmonic analysis on R2n. Using the quantum version of Wiener’s approximation theorem, a natural one-to-one correspondence between the closed, phase-space translation invariant subspaces of classical and quantum observables is established.
The behavior of Hamiltonian path integrals under canonical transformations defined by a generator is investigated. The exact form of the kernel of the unitary operator which implements the quantum transformation is indicated. Equivalence rules are found (for the Hamiltonian formalism in the one-dimensional case) allowing one to get rid of the nonstandard terms in the action. It is shown that under canonical transformations the Hamiltonian path integral changes its form: in the transformed expression one cannot get along only with the corresponding classical Hamiltonian function.