A review of canonical transformations in quantum mechanics is outlined with a view toward their use in the solution of specific problems. Several typical problems are worked out by the method of canonical transformations to emphasize the utility of this approach. In this way one may obtain an insight into the nature and limitation of the connection that exists between the canonical transformations of classical mechanics and the unitary transformations of quantum mechanics. The purpose of this paper is to achieve certain expository and pedagogical objectives, and no original contribution to the theory is intended.A review of canonical transformations in quantum mechanics is outlined with a view toward their use in the solution of specific problems. Several typical problems are worked out by the method of canonical transformations to emphasize the utility of this approach. In this way one may obtain an insight into the nature and limitation of the connection that exists between the canonical transformations of classical mechanics and the unitary transformations of quantum mechanics. The purpose of this paper is to achieve certain expository and pedagogical objectives, and no original contribution to the theory is intended.

American Journal of PhysicsSep 1, 19651 citation

Classical Mechanics

Newton C. A. da Costa, Francisco Antônio Dória

Synthese Library/Synthese libraryAug 9, 20210 citations

Nonlinear canonical transformations in classical and quantum mechanics

Alastair Brodlie

p-mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how nonlinear canonical transformations affect p-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of p-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.p-mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how nonlinear canonical transformations affect p-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of p-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.

arXiv:quant-ph/0401160Journal of Mathematical PhysicsJul 9, 20044 citations

Matrix Mechanics

University of California Press eBooksNov 9, 20230 citations

Non-Linear Canonical Transformations in Classical and Quantum Mechanics

Alastair Brodlie

$p$ -Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect $p$ -mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of $p$ -mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.$p$-Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect $p$-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of $p$-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.

Jan 26, 20046 citations

p

-Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect

p

-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of

p

-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.