Help with intuition concering braket sandwich

[Hymne]

Hi there!
I have a hard time getting any sense out of what the sandwich:

<q'', t'' | Q(t1) | q', t'>
where t' < t1 < t'', actually means..

If we skip Q(t1) we get the usual transition amplitude. But how should i interpret this when we use an operator that is timedependet but act on a statevector in the "wrong time"?

Thanks!

 

[Avodyne]

IMO, it doesn't have a nice interpretation. In the Schrodinger picture, you are starting in a position eigenstate at time t', evolving for a time t1-t', acting with the position operator, the evolving for a time t''-t1, then taking the inner product with another position eigenstate. But "acting with the position operator" does not have a nice physical interpretation; in particular, it's not the same as measuring the position.