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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Effective generating functions for quantum canonical transformations

G. I. Ghandour

Published Feb 15, 1987• 37 citations

Abstract

An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent. Using ${e}^{\mathrm{iF}}$ as a kernel, an integral transform relates the wave functions of the corresponding quantum systems. The function F reduces in the classical limit (\ensuremath{\Elzxh}\ensuremath{\rightarrow}0) to the generating function of the classical transformation. Conversely, starting with the classical form, F can be calculated in a recurrent fashion, order by order in powers of \ensuremath{\Elzxh}. For the canonical transformation that relates a particle moving in an exponential (Liouville) potential to a free particle, the effective quantum generating function is identical to its classical counterpart. The generalization to quantum field theory is possible using the Schr\"odinger wave-functional formalism.

DOI: 10.1103/physrevd.35.1289

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An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent. Using

{e}^{\mathrm{iF}}

as a kernel, an integral transform relates the wave functions of the corresponding quantum systems. The function F reduces in the classical limit (\ensuremath{\Elzxh}\ensuremath{\rightarrow}0) to the generating function of the classical transformation. Conversely, starting with the classical form, F can be calculated in a recurrent fashion, order by order in powers of \ensuremath{\Elzxh}. For the canonical transformation that relates a particle moving in an exponential (Liouville) potential to a free particle, the effective quantum generating function is identical to its classical counterpart. The generalization to quantum field theory is possible using the Schr"odinger wave-functional formalism.

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.

The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.

This paper is a generalization of previous work on the use of classical canonical transformations to evaluate Hamiltonian path integrals for quantum-mechanical systems. Relevant aspects of the Hamiltonian path integral and its measure are discussed, and used to show that the quantum-mechanical version of the classical transformation does not leave the measure of the path-integral invariant, instead inducing an anomaly. The relation to operator techniques and ordering problems is discussed, and special attention is paid to incorporation of the initial and final states of the transition element into the boundary conditions of the problem. Classical canonical transformations are developed to render an arbitrary power potential cyclic. The resulting Hamiltonian is analyzed as a quantum system to show its relation to known quantum-mechanical results. A perturbative argument is used to suppress ordering-related terms in the transformed Hamiltonian in the event that the classical canonical transformation leads to a nonquadratic cyclic Hamiltonian. The associated anomalies are analyzed to yield general methods to evaluate the path integral's prefactor for such systems. The methods are applied to several systems, including linear and quadratic potentials, the velocity-dependent potential, and the time-dependent harmonic oscillator.

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

{\mathrm{p}}_{\mathrm{k}}

are equal to those of the other Q and

{\mathrm{P}}_{\mathrm{k}}

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.