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Physical Review A

Quantum correspondence for linear canonical transformations on general Hamiltonian systems

Kyu Hwang Yeon, Daniel F. Walls, Chung‐In Um, Thomas F. George, Lakshmi N. Pandey

Published Sep 1, 1998• 20 citations
Publisher

Abstract

General Hamiltonian systems related by linear canonical transformations, which are combinations of the scale and gauge transformations, are considered. Using path integrals, it is proven that each quantum Hamiltonian can be expressed by the classical Hamiltonian whose canonical variables are replaced by their corresponding quantum operators. The relation between the Schr\"odinger solution and the propagator for the transformed (new) and for the original (old) systems are evaluated by applying the unitary operator which describes the linear relationship between their quantum operators as the corresponding relations between their classical forms. It is shown that the uncertainty relations between the canonical position and momentum operator depend on the gauge function chosen, and satisfy Heisenberg's uncertainty principle. The uncertainty relations between the canonical position and kinetic momentum operator (distinguished from the momentum operator) do not depend on the gauge function chosen, and may not satisfy Heisenberg's uncertainty principle. By the gauge transformation, a single system has innumerable Schr\"odinger equations, but the quantum averages of the function of the position and kinetic momentum operators are invariant for all solutions, as for classical cases.

DOI
10.1103/physreva.58.1765

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General Hamiltonian systems related by linear canonical transformations, which are combinations of the scale and gauge transformations, are considered. Using path integrals, it is proven that each quantum Hamiltonian can be expressed by the classical Hamiltonian whose canonical variables are replaced by their corresponding quantum operators. The relation between the Schr"odinger solution and the propagator for the transformed (new) and for the original (old) systems are evaluated by applying the unitary operator which describes the linear relationship between their quantum operators as the corresponding relations between their classical forms. It is shown that the uncertainty relations between the canonical position and momentum operator depend on the gauge function chosen, and satisfy Heisenberg's uncertainty principle. The uncertainty relations between the canonical position and kinetic momentum operator (distinguished from the momentum operator) do not depend on the gauge function chosen, and may not satisfy Heisenberg's uncertainty principle. By the gauge transformation, a single system has innumerable Schr"odinger equations, but the quantum averages of the function of the position and kinetic momentum operators are invariant for all solutions, as for classical cases.
We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum pk{\mathrm{p}}_{\mathrm{k}}pk​ are equal to those of the other Q and Pk{\mathrm{P}}_{\mathrm{k}}Pk​ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.
The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.
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.