Weak quantization

Abstract

Quantization is called weak when a motion apparently allowed by the equation ∫pdq=nh, has less than the normal a-priori weight. It is believed that the deficiency in a-priori weight is taken over, either by neighboring classically allowed motions, or by neighboring strongly quantized motions when such are present in the region of the phase-space considered. Weak quantization is to be expected when uncertainties arise as to the period that should be used in determining the limits of the phase integral ∫pdq. Several cases are considered; (a) when the period is so long that there is considerable chance of interruption by a quantum transition; (b) when a system has two apparent periods, a long true period T and a short quasi-period θ; (c) when the periodicity is disturbed frequently in a fortuitous manner as by molecular collisions. In case (b), the tendency towards quantization with respect to T may be gradually replaced by quantization with respect to θ as T is lengthened, and then the probability of quantum transitions which correspond to quantization with respect to T is weakened while that of transitions related to θ is strengthened. This suggests the possibility that the strengthening of the probability of transitions related to a period θ may be accompanied by a strengthening of quantization with respect to that period.