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I want to calculate the transition amplitude for a particle to start at position $q_1$ at time $t_1$ to position $q_2$ at time $t_2$ in the Heisenberg picture.

As we are in the Heisenberg picture, the time dependence is in the states, so the position operator is given by

$$Q(t)=e^{iHt}Q(0)e^{-iHt}$$

We have instantaneous eigenstates $|q_1;t_1\rangle$ such that

$$Q(t_1)|q_1;t_1\rangle=q_1|q_1;t_1\rangle. $$

Now because this is the Heisenberg picture the states don't evolve with time. They are fixed entities that tell us the entire spacetime history of a particle (if we could "solve" them, in a sense.)

The transition amplitude is given by $\langle q_2;t_2|q_1;t_1\rangle$. Now, as I said, the instantaneous eigenstates describe the entire spacetime history of the particle. The only condition on $|q_1;t_1\rangle$ is that

$$Q(t_1)|q_1;t_1\rangle=q_1|q_1;t_1\rangle$$

We have no idea how $Q(t)$ acts on $|q_1;t_1\rangle$ for times $t\neq t_1$.

So, is it then possible to view the transition amplitude as the probability amplitude that if I act on $|q_1;t_1\rangle$ with $Q(t_2)$ I will find $q_2$? i.e the probability amplitude for

$Q(t_2)|q_1;t_1\rangle=q_2|q_1;t_1\rangle$ being true?

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No. There are multiple problem with your idea that $\langle q_2;t_2\vert q_1;t_1\rangle$ represents "the probability amplitude for $$ Q(t_2)\lvert q_1;t_1\rangle = q_2\lvert q_1;t_1\rangle$$ being true":

  1. You have no reason at all to believe that $\lvert q_1;t_1\rangle$ will be an eigenstate of $Q(t_2)$, so...this will probably never hold.

  2. There are no "probability amplitudes" for certain equations being true or not. There are only probability amplitudes for measurement results, given by the overlap $\langle f \vert \psi \rangle$ of the measurement result state $\lvert f \rangle$ with the original state $\lvert \psi \rangle$ (and, as a slight extension, there are probabilities associated to certain ranges of measurement results, but that's it. You don't get probabilities for equations being true or not).

The amplitude $\langle q_2;t_2\vert q_1;t_1\rangle$ very simply gives the probability amplitude for a particle known to be located at $q_1$ at time $t_1$ to be detected by a position measurement at $q_2$ at time $t_2$. Nothing else.

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