So I came across a question on conditional probability in quantum mechanics: There's an interesting comment which tells why this does not work for "the non-commutative case".

I was wondering, however, since there are more than operators in quantum mechanics one could ask about their relation. For example, there is time which is a parameter. It seems straightforward to compute the conditional probability of an outcome given the time was say $t$ by (for example):

$$ P( A|T_1) = |\langle x_A, t_1 | \psi, t_1 \rangle|^2 $$

where $A$ denotes the event of say measuring the position at a $x = x_a$, $T_1$ represents the time being say $t_1$ and let pre-measurement state be $\psi$. But what if one swaps things as:

$$P(T_1|A) = \frac{| \langle x_a, t_1| \psi, t_1\rangle |^2}{\int_{t_{A-}}^{t_{A+}} | \langle x_a, t| \psi, t\rangle |^2 dt}$$

This something I figured out here: Physics SE: Conditional probability between parameter and operator in quantum mechanics


How does this change if we update the question to multiple measurements. For example what is the probability of $P(A|T_1)$ and $P(B|T_2)$ (with $T_2 > T_1$ as the time evolution of the wave-function to see $B$ at $T_2$ depends on the outcome A) - here $A$ and $B$ can be different observable's outcomes? How is this related to probability of the times being $T_1$ and $T_2$ when $A$ and $B$ are observed?


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