In quantum mechanics, what is the probability of $B$ given that $A$ has happened?

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.

• 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 $|A_2\rangle$ ? – zooby Sep 5 '17 at 17:35
• 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 – forky40 Sep 9 '17 at 2:06