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David griffiths quantum mechanics6/1/2023 ![]() ![]() We will find there is a finite probability that accumulates as $x$ goes from infinitesimally less than $t_0$, to infinitesimally more. ![]() Then $\pi(t_0)$ is infinite, so we need to regularize the probability $\pi(x) dx$ near $x=t_0$ to find out what is happening. $$ \text(x)$ is a normal, not-singular PDF. Technically, we’re talking about infinitesimal intervals.) Thus ![]() If the baby boom lasted six hours, we’ll take intervals of a second or less, to be on the safe side. ( Unless, I suppose, there was some extraordinary baby boom 16 years ago, on exactly that day - in which case we have simply chosen an interval too long for the rule to apply. For example, the chance that her age is between 16 and 16 plus two days is presumably twice the probability that it is between 16 and 16 plus one day. If the interval is sufficiently short, this probability is proportional to the length of the interval. The only sensible thing to speak about is the probability that her age lies in some interval - say, between 16 and 17. If I select a random person off the street, the probability that her age is precisely 16 years, 4 hours, 27 minutes, and 3.333. But it is simple enough to generalize to continuous distributions. So far, I have assumed that we are dealing with a discrete variable - that is, one that can take on only certain isolated values (in the example, $j$ had to be an integer, since I gave ages only in years). From Introduction to Quantum Mechanics by Griffiths: ![]()
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