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Physical Review

On the Momentum Operator in Wave Mechanics

Aurel Wintner

Published Apr 15, 1947• 5 citations
Publisher

Abstract

A simple example shows that a standard tacit hypothesis concerning quadratic integrability in momentum space is not in general legitimate, if only quadratic integrability in configuration space is assumed. Inasmuch as only the latter assumption is justified, there arises the need for criteria which prevent the occurrence of this difficulty. Such a criterion is developed in terms of the qualitative behavior of the potential. The conditions on the potential are of such a nature as to be satisfied in all cases which seem to be reasonable.

DOI
10.1103/physrev.71.547

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A simple example shows that a standard tacit hypothesis concerning quadratic integrability in momentum space is not in general legitimate, if only quadratic integrability in configuration space is assumed. Inasmuch as only the latter assumption is justified, there arises the need for criteria which prevent the occurrence of this difficulty. Such a criterion is developed in terms of the qualitative behavior of the potential. The conditions on the potential are of such a nature as to be satisfied in all cases which seem to be reasonable.
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An effort is continued to define quantum probabilities for continuous histories. The sum-over-paths approach used in Prog. Theor. Phys. 87, 77 (1992) is critically reviewed, improved, and applied again. Consistency between wave nature and particle nature is the criterion used to judge the definability of the probabilities and is formulated as the path classifiability condition (C1) and the no-interference condition (C2). A set of classes of histories satisfying these two conditions is considered as a space-time analog of observables. In particular, Feynman's paths for a nonrelativistic particle are considered as histories and the definability of probabilities for classes of paths is investigated, where classes are defined by classifying paths according to their behavior with respect to a rectangular space-time region \ensuremath{\Omega}\ensuremath{\equiv}\ensuremath{\Delta}X\ifmmode\times\else\texttimes\fi{}\ensuremath{\Delta}T. Confining ourselves to cases where C1 is satisfied, we examine C2 for some sets of classes of paths. Although C2 does not hold in general, some examples are found where C2 holds. In all the examples, the initial wave function is restricted. In some examples, the location of \ensuremath{\Omega} and/or the potential in the region also affect the success of C2. Due to these restrictions, the resultant probabilities for classes of paths take reasonable values. A puzzling example in which probabilities cannot be defined for histories contrary to intuition is resolved by considering an appropriate coarse graining of classes of paths. Considering a rectangular potential barrier, we show that the reflection and the transmission probabilities are special cases of probabilities for histories, and also that probability densities of transmission and reflection times cannot be defined. This study may be taken to be a study of the consistent-histories approach not with discrete histories defined by products of projection operators but with continuous histories defined by Feynman's paths in configuration space. \textcopyright{} 1996 American Institute of Physics.