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ON THE PROBLEM OF DISCRETE QUANTUM NUMBERS FOR THREE PARTICLE STATES

Andrzej Szymacha, J. Werle

Published Jan 1, 1963• 0 citations
OpenAlex

Abstract

A procedure for finding the discrete quantum numbers that, as previously noted, can replace continuous quantum numbers in a representation for systems of three particles of arbitrary mass and spin values is outlined. (D.C.W.)

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Consequences of the topology of the configuration space of a Hamiltonian dynamical system are considered for a coherent system of trajectories. It is shown that when the space is multiply-connected and therefore the action integral is multivalued, the allowed states of motion (labeled by the initial data) are constrained to a discrete set by the requirement that the action be single-valued. One thus obtains a quantum-like discretization of allowed states of motion even in classical mechanics. Such discrete “allowed” and “forbidden” states have indeed been observed in the classical mechanical system of charged particles in a magnetic field. The relationship of this formalism with a Schrödinger-like formalism for the latter problem given earlier is discussed.
We report here experimental results which point to a rather unexpected, astonishing behavior of charged particle motion in a magnetic field in the presence of a retarding electric potential. An electron beam, when energy scanned with respect to its ‘parallel’ energy using a retarding potential, the response of the transmitted current as a function of the retarding potential is found to be oscillatory as against the monotonic response, expected according to the standard initial value paradigm of classical mechanics. The unexpected ‘peaks’ and ‘dips’ in the transmitted current plots are interpreted as “allowed” and “forbidden” energy states of the system. The peaks (or dips) are found to fit very well with a theoretical relationship which motivated the search for such a behavior; also “quantum numbers” for the peaks have been identified. A very astonishing aspect of the relationship is the dependence of the energies of the allowed states on the length of the box.