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American Journal of Physics

Classical Transformation Operators and the S Matrix

Philip M. Campbell

Published Nov 1, 1968• 0 citations
Publisher

Abstract

A novel formulation of classical dynamics in terms of differential operators is presented. The basis for this formulation is the direct application of the unfolding theorem for arbitrary times. The solution of a mechanical problem is given by the operator method as a Taylor series about the initial point. The classical operator formalism has similarities with quantum mechanics. It is shown that classical scattering can be described in terms of a classical S-matrix operator. Examples are given.

DOI
10.1119/1.1974357

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A novel formulation of classical dynamics in terms of differential operators is presented. The basis for this formulation is the direct application of the unfolding theorem for arbitrary times. The solution of a mechanical problem is given by the operator method as a Taylor series about the initial point. The classical operator formalism has similarities with quantum mechanics. It is shown that classical scattering can be described in terms of a classical S-matrix operator. Examples are given.
Basically, there are two approaches to the formulation of quantum mechanics, i.e., the operator and the Feynman path integral approach. The main object in quantum mechanics is referred to as the transition amplitude between those states, matrix elements of physical operators and, in particular, the S- matrix in the problem of quantum scattering. On the basis of quantum mechanics, the operator method is closed, well formulated mathematically, and so is an appropriate universal language describing quantum field theory, the quantum physics of condensed states and quantum relativistic statistical physics. The differential equation defining the path of the system in the configuration space is called the equation of motion of the system. The aim of classical as well as quantum mechanics is to predict the time evolution of the system, starting from knowledge of the state of the system at a given point in time.
We establish the general form of the evolution equation and study scattering and tunneling in real quantum mechanics. We prove an analogue of Stone's theorem for a real Kahler space and derive the corresponding general evolution equation, whose evolution operator is both orthogonal and symplectic. We formulate scattering theory in terms of real wave operators and the associated S-matrix. For the class of potentials considered, we show that the differential scattering cross section is equivalent to that obtained in standard complex quantum mechanics. We also show that the tunneling probability in real quantum mechanics is identical to that in complex quantum mechanics.
It is shown how differential cross sections can be obtained from the time dependence of phase space packets. This procedure is valid both for classical and quantum mechanics. Two methods are described. In one the trajectory of the packet is emphasized, while in the second the packet is appropriately spread to infinite size. Both methods are applicable to either mechanics. It is shown how the quantal results agree with those of the stationary state approach as formulated in terms of the density operator. The description is also used to elucidate the difference between the scattered flux and the generalized flux that arises naturally in the superoperator formulation.
This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.
An operator formulation of classical mechanics is given. Rather than being concerned with Lie algebraic or abstract Hilbert space properties of classical mechanics, the present formulation is directly concerned with wave packets in configuration space and is thus more similar to that of conventional quantum theory than other extant operator formulations of classical mechanics. Among other things, the following items are discussed for classical systems: Ehrenfest’s theorem, the uncertainty principle, dissipative systems, and field theory.
The information-theoretic maximal-entropy procedure for the analysis of collision processes is derived as a consequence of the dynamics, be they quantal or classical. The method centers attention on the minimal number of operators (the "dynamic constraints") whose expectation values are both necessary and sufficient to completely characterize the collision dynamics. For a given Hamiltonian and initial state, the constraints required to obtain an exact solution of the equations of motion are determined by a purely algebraic procedure. It is furthermore found possible to derive equations of motion for the conjugate Lagrange parameters. Immediate applications are noted, e.g., a family of similar reactions is shown to have a common set of dynamic constraints and simple illustrative applications are provided. The determination of the scattering matrix is discussed, with examples. The general formalism consists in solving the scattering problem in two stages. The first is purely algebraic. At the end of this stage one obtains the functional form of, say, the scattering matrix or of the density matrix after the collision expressed in terms of parameters whose number equals the number of dynamic constraints. The end result of this algebraic stage suffices to analyze the scattering pattern for any initial state. The second stage is the predictive procedure. Explicit coupled first-order nonlinear differential equations are obtained for the parameters.