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Point transformations in quantum mechanics

Kurt Bernardo Wolf

Published Jan 1, 1973• 3 citations
OpenAlex

Abstract

First a review is presented of the definition and properties of classical and quantum canonical transformations, from the point of view of a current program on the role of canonical transformations in quantum mechanics. The groups and the corresponding infinitesimal algebras are explored. The subgroup of point transformations (i.e. canonical transformations between pairs of conjugate observables, where one of a pair is function only of one of the other pair) is of special interest since it is a group of transformations for which classical and quantum mechanics follow each other. Only the group of inhomogeneous symplectic transformations has similar characteristics. Point transformations are treated in detail and a principal series of unitary representations is constructed. (auth)

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Journal of Mathematical Physics•Nov 1, 1969•35 citations
First a review is presented of the definition and properties of classical and quantum canonical transformations, from the point of view of a current program on the role of canonical transformations in quantum mechanics. The groups and the corresponding infinitesimal algebras are explored. The subgroup of point transformations (i.e. canonical transformations between pairs of conjugate observables, where one of a pair is function only of one of the other pair) is of special interest since it is a group of transformations for which classical and quantum mechanics follow each other. Only the group of inhomogeneous symplectic transformations has similar characteristics. Point transformations are treated in detail and a principal series of unitary representations is constructed. (auth)
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.]
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.
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 Schrödinger 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
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.
An isomorphism is shown to exist between the group of point transformations in classical mechanics and a certain subgroup of the group of all unitary transformations in quantum mechanics. This isomorphism is used to indicate that the quantum analogs of physically significant classical expressions can be constructed uniquely in any coordinate system. There is no ambiguity in the ordering of noncommuting quantum operators, and the method of constructing the quantum analogs is covariant under general coordinate transformations. The method is actually only applicable to systems having Lagrangians which are at most quadratic in the velocities, but this includes all systems which are presently of interest in physics. The method is applied to two intrinsically nonlinear examples, one of which is the gravitational field. The correct Hamiltonian operator for a quantized version of Einstein's gravitational theory is constructed.
The claim that unitary transformations in quantum mechanics correspond to the canonical transformations of classical mechanics is not correct. The spectra of operators produced by unitary transformation of Cartesian coordinate and momentum operators (q, k) are necessarily continuous over the entire real domain of their eigenvalues. Operators with spectra which are not everywhere continuous are generated from (q, k) by one-sided unitary transformations U for which U†U = 1 but for which UU† commutes with either q or k (but not both). If UU† commutes with k, the new coordinates and momenta (r, s) satisfy commutation relations [sm, rn] = 2πi1δm,n, [sm, sn] = 0, but [rm, rn] ≠ 0; (r, s) are canonical only for one-dimensional systems. The properties of one-sided unitary transformations are described; they are characterized by φ(K), the eigenvalue of UU†. The one-dimensional case for which the one-sided unitary transformation is canonical is discussed in detail. A prescription is given for obtaining the operator canonically conjugate to any one-dimensional observable. Generalization to higher dimensions is also discussed.