Mathematics – Probability
Scientific paper
Dec 1944
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1944phrv...66..343l&link_type=abstract
Physical Review, vol. 66, Issue 11-12, pp. 343-350
Mathematics
Probability
5
Scientific paper
Sec. I. This paper inverts certain quantum mechanical ideas by constructing a "probability of presence" from first principles. If xi=fi(s,a1,a2) represents a 2-parameter family of curves in 3-dimensional space, the density of the functions fi with respect to the parameters a1a2 is defined by means of an expression for the normal cross section of a sheaf of curves. This density plays the role of a probability. Its derivative along a curve of the family is independent of the parameters a1a2. This circumstance permits a special density to be constructed, called the "characteristic" or "smoothest" density, which is independent of the parameters and which can be found by solving a linear second-order partial differential equation. The extension to n-dimensional space and (n-1) parameters is given. Sec. II. The concept of orbital density is the dynamical equivalent of the density of Sec. I. Dividing the orbital density by the velocity of the representative point gives the quantity we call kinematical probability P, which satisfies the fundamental equation (2.8), and which has the same force as any other probability of presence. Equation (2.8) is used to derive from the Hamilton-Jacobi equation a "wave" equation for √P for a system of n planets moving around a sun under the condition that the space average of the mutual interactions of the planets is a minimum. This provides a new approach to planetary dynamics and yields an equation that is very much like Schroedinger's equation for the stationary states of n particles, and has the same basic interpretation. Sec. III. A non-conservative system of n particles is assumed to be in a state such that the action function can be written W=-Et+R(xi,t) in which R still contains the time explicitly. Starting with the Hamilton-Jacobi equation, using Eq. (2.8), and imposing the condition that the space average of ∂R∂t is a minimum, an equation [viz., (3.10)] is derived which differs from Schroedinger's equation for stationary states only in certain respects that cannot be regarded as fundamental. Despite this basic equivalence of Schroedinger's equation and our equation (3.10), conflicts in interpretation and formalism arise which are briefly discussed. Sec. IV. Complex U' s are introduced into Eq. (3.10) in a simple way which fails to bridge the gap between our formalism and that of quantum mechanics. The kinematical probability defined by Eq. (3.10) is shown to have the "smoothest" property. To account physically for the stationary states an electromagnetic wave system is proposed which has the property that a particle moving in it may be in "phase-mechanical" resonance with it. Macroscopic examples of this kind of resonance are given. This concept represents an inversion of physical ideas of quantum mechanics corresponding to the mathematical inversion with which we began.
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