Weak Quantization

Quantization is called weak when a motion apparently allowed by the equation $\int \mathrm{pdq}=nh$, 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 $\int \mathrm{pdq}$. 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 $\theta$; (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 $\theta$ 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 $\theta$ is strengthened. This suggests the possibility that the strengthening of the probability of transitions related to a period $\theta$ may be accompanied by a strengthening of quantization with respect to that period.