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Journal of Physics A Mathematical and General

Representation of quantum mechanical wavefunctions by transformation generators. II. One-dimensional time-dependent case

C Jung, H. Krüger

Published May 11, 1983• 1 citation
Publisher

Abstract

For pt.I see ibid., vol.16, no.11, p.3509-23 (1982). Sufficient conditions are given for constructing quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given Hamiltonian onto the new position variable. The transformation kernel for each individual step of this sequence is given by the semiclassical limit expression; it is a function of the generator of this transformation step only. The transformation kernel for the total transformation and the time-dependent wavefunction are obtained as a multiple integral over the product of the transformation kernels of the various intermediate steps. The practicability of this procedure is demonstrated by several examples. The authors consider explicitly time-dependent systems with one degree of freedom.

DOI
10.1088/0305-4470/16/7/016

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For pt.I see ibid., vol.16, no.11, p.3509-23 (1982). Sufficient conditions are given for constructing quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given Hamiltonian onto the new position variable. The transformation kernel for each individual step of this sequence is given by the semiclassical limit expression; it is a function of the generator of this transformation step only. The transformation kernel for the total transformation and the time-dependent wavefunction are obtained as a multiple integral over the product of the transformation kernels of the various intermediate steps. The practicability of this procedure is demonstrated by several examples. The authors consider explicitly time-dependent systems with one degree of freedom.
Sufficient conditions are given for the possibility to construct quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given Hamiltonian onto the new position variable. The transformation matrix element for each individual step of this sequence is given by the semiclassical limit expression of these matrix elements; it is a function of the generator of this transformation step only. The wavefunction, i.e. the transformation matrix element for the total transformation, is obtained as a multiple integral over the transformation matrix elements of the various intermediate steps. The practicability of this procedure is demonstrated by several examples. The authors consider time-independent systems with one degree of freedom.
For pt.II see ibid., vol.16, no.7, p.1397-408 (1983). It is shown how quantum mechanical wavefunctions can be obtained from a sequence of simple canonical transformations, which map the given system onto a simple reference system. The resulting wavefunctions are at least uniformly valid up to order h(cross). Under some more restrictive conditions for the individual transformation steps the authors even find the exact wavefunctions. The essential point of the paper is to enlarge the conventional coordinate-momentum phase space by taking time and energy as an additional conjugate pair. In this extended space they exploit the possibility of using transformations which intermix energy and time with position coordinates and momenta. Compared with transformations in the conventional position-momentum phase space, they gain the advantage that scattering states and bound states can be treated in a unified way. Therefore this method is appropriate for systems with mixed spectra. In addition it allows for more flexibility in choosing the individual transformation steps. The practicability of the method is demonstrated by several examples.
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.
Continuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n-dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space Lrn−12(0,∞) is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut-Girardello, one-dimensional Bargmann and Hankel transforms. The transform kernels provide a series of representations of a subsemigroup of SL(2,C) and allow the construction of coherent states for the harmonic oscillator with an extra centrifugal force. We present a hyperdifferential operator realization of these transforms which yields new Baker-Campbell-Hausdorff and special function relations.