I get that the probability of event $A$ is $\langle A|A\rangle$.

Given that $A$ has happened what is the probability of B happening at $t$ seconds in the future? My first guess is:

$$ P(A|B) \stackrel{?}{=} \frac{\langle A\vert U(t)\vert B\rangle}{\langle B|B\rangle}$$

where $U(t)$ is the time evolution operator. But isn't this a complex number?

And how is this related to the transition Green's functions $G(A,B)$ i.e. the amplitude going from state $A$ to state $B$. ?


You should think of quantities like $\langle B \vert A \rangle $ as describing probability amplitudes, where the probability is recovered by doing the "norm-squared" action $P(B\vert A) = \vert \langle B \vert A \rangle \vert^2 $. So $\vert \langle A \vert A \rangle \vert^2 = 1$ and resembles $P(A\vert A)$, not your best-guess equivalent $P(A)$.

Language like "happening" is kinda imprecise, but more importantly it dodges the big issue of uncertainty in QM. We might say that given an initial state $A$, the probability of measuring some condition/state $B$ after time $t$ is: $$ P(B\vert A) = \vert \langle B \vert U(t) \vert A \rangle \vert^2 $$

but we didn't know what "happened" without the notion of some measurement (or distinguishable effect) of the time-evolved state (this is addressed in Kyle's link). Use of the time-evolution operator like this implies that $B$ is some observable feature that $A$ is at least partly composed of; i.e. $A$ and $B$ are not simply conditionally linked events.

  • $\begingroup$ But what if the sates are not normalised so that $\langle A|A\rangle \neq 1$. Then do we divide by this to get the answer? Suppose there are two possible starting states $|A_1\rangle$ and $|A_2\rangle$ ? $\endgroup$ – zooby Sep 5 '17 at 17:35
  • $\begingroup$ The way we interpret $\langle A | A \rangle$ is as a probability (amplitude) that a particle is in the state that it is in; so its a definition that $ |\langle A | A \rangle |^2 =1$, from which we develop other probabilities. If there's two possible starting states, we either need to characterize the probability of an initial state $| \phi \rangle$ being one or the other: $| \phi \rangle = c_1 | A_1 \rangle + c_2 | A_2\rangle $. Or determine the evolution of each of the possible cases separately $\endgroup$ – forky40 Sep 9 '17 at 2:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.