Generalization of Quantum Mechanics

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Scientific Paper
Title Generalization of Quantum Mechanics
Author(s) Thomas E Phipps
Keywords {{{keywords}}}
Published 1960
Journal Physical Review
Volume 118
Number 6
Pages 1653-1658


The possibility of generalizing quantum mechanics in such a way as to retain its predictive results, while comprehending additional solutions, is examined. It is found that this can be done through a perfected formal correspondence with Hamilton-Jacobi mechanics, by which one is led to consider generalizations of the Heisenberg postulate of the form pk qj - qj pk = S (delta jk), where S is a quantum analog of Hamilton's principal function. The formalism is shown to be equivalent to a simple change in Hamiltonian, with transformed momentum operators satisfying conventional commutation relations, and with an additional relationship involving formal analogs of the classical "initial constants" adjoined. A particular choice of S (= h-bar/i) leads to a theory identical with wave mechanics apart from a constant (unobservable) phase factor on the wave function. The fact that S may possess other, nonconstant values, demonstrated by a specific example, suggests the ability of the mechanical equations to describe a broader class of physical states than has hitherto been investigated.