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Physical Review

Quasi-Classical Transformation Theory

Ralph Schiller

Published Feb 1, 1962• 41 citations
Publisher

Abstract

The quasi-classical theory is the theory of the WKB approximation as defined by Van Vleck. This approximate wave theory is invariant under canonical transformations. We derive the transformation laws for wave functions and operators in this theory. The connection problem of the ordinary configuration space is discussed from the point of view of Van Vleck. The usual method of WKB quantization is then contrasted with another method of quantization in a classical configuration space of creation and destruction operators. In this new configuration space we find that, for some problems, the quasi-classical solutions are exact solutions of the Schr\"odinger equation. Finally, we show that the canonical transformations of the quasi-classical theory may be put into one-to-one correspondence with a group of approximate unitary transformations.

DOI
10.1103/physrev.125.1109

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The quasi-classical theory is the theory of the WKB approximation as defined by Van Vleck. This approximate wave theory is invariant under canonical transformations. We derive the transformation laws for wave functions and operators in this theory. The connection problem of the ordinary configuration space is discussed from the point of view of Van Vleck. The usual method of WKB quantization is then contrasted with another method of quantization in a classical configuration space of creation and destruction operators. In this new configuration space we find that, for some problems, the quasi-classical solutions are exact solutions of the Schr"odinger equation. Finally, we show that the canonical transformations of the quasi-classical theory may be put into one-to-one correspondence with a group of approximate unitary transformations.
We develop a modified Hamiltonian-Jacobi theory of classical mechanics following the early work of Van Vleck. This modified Hamiltonian-Jacobi theory, or quasi-classical theory, permits us to exhibit in classical mechanics many features that in the past have been exclusively associated with quantum mechanics. We deal with classical wave functions, classical operators, classical "eigenvalue" equations, a classical "sum over paths" formulation of classical mechanics, and with classical creation and destruction operators. Following Van Vleck, one can derive the WKB approximate solutions to the Schr\"odinger equation from the solutions of the classical Hamilton-Jacobi equation. If we apply the methods of Keller to the nonrelativistic and relativistic Kepler problem, we derive eigenvalues from the requirement of single-valuedness imposed on the WKB solutions. It turns out that the energy eigenvalues are those given by the Schr\"odinger equation and the Klein-Gordon equation, respectively. In the particular case of the harmonic oscillator there exists a canonical transformation which transforms the quasi-classical equation into an exact equation of quantum mechanics. We conjecture that if the WKB approximation and the Schr\"odinger equation predict the same eigenvalues, then there always exists a canonical transformation which transforms the quasi-classical equation into the corresponding Schr\"odinger equation. Finally we derive the quasi-classical equations in momentum space.
The quasi-classical spin theory is a modified Hamilton-Jacobi theory for a classical relativistic dipole whose space-time trajectory is coupled to its spin motion. The equations of this quasi-classical theory are almost identical with the second-order (squared) Dirac equation, and, in fact, form a new WKB approximation to the second-order Dirac equation. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single valued, and that it satisfy an eigenvalue equation. We find for a particle in a uniform magnetic field, and in a Coulomb field, that this quasi-classical theory predicts the same energy and angular momentum eigenvalues as the Dirac theory. The quasi-classical theory is invariant under Lorentz transformation, spin rotations, and charge conjugation. The spinor wave functions are form invariant with respect to arbitrary canonical transformations.
New exact quasi-classical asymptotic of solutions to the n -dimensional Schrödinger equation beyond WKB-theory and beyond Maslov's canonical operator theory is presented. Quantum jump nature is considered successfully. We pointed out that an explanation of quantum jumps can be found to result from Colombeau solutions of the Schrödinger equation alone without additional postulates. Such jumps, does not depend on any single measurement of particle position q ( t ) at instant t and completely obtained only from Schrodinger equation, without reference to any phenomenological master-equation of Lindblad form.
Abstract We explore the relation of Van Vleck-Primas perturbation theory of quantum mechanics with the Lie-series-based perturbation theory of Hamiltonian systems in classical mechanics. In contrast to previous works on the relation of quantum and classical perturbation theories, our approach is not based on the conceptual similarities between the two methods. Instead, we show that for quantum systems with a finite-dimensional Hilbert space, the Van Vleck-Primas procedure can be recast exactly into a classical perturbation problem. As a non-obvious consequence, this approach gives a new way of calculating the geometric phase of quantum systems using tools from the theory of classical canonical transformations.
In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one-to-one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one-dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x̄, p̄) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group 𝒞κ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group 𝒞κ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single component function, in the new Bargmann Hilbert space we must take a κ component one. In this way we can map in a one-to-one fashion the states and operators in the old and new Bargmann Hilbert spaces. When translating these results to ordinary Hilbert space, we get in an unambiguous way the quantization of the observables appearing in the equations that determine the representation of the classical canonical transformation relating oscillators of frequencies κ−1 and k−1. Furthermore, we also get the solutions of these equations, and the resulting representation is unitary. While our discussion is restricted to the problem mentioned above, in the concluding section we indicate our surmise for deriving systematically the unitary representation in quantum mechanics of arbitrary canonical transformations.