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## Main Question or Discussion Point

A fundamental quantity that we calculate with QM is [tex]\langle \Phi|\Psi\rangle[/tex]-- the probability amplitude for observing a system to be in state [tex]|\Phi\rangle[/tex] given that it is in state [tex]|\Psi\rangle[/tex]. In the Schrodinger picture the states are time-dependent and we can ask, "What is the probability amplitude for observing the system to be in state [tex]|\Phi\rangle[/tex] at time t2 given that it is in state [tex]|\Psi\rangle[/tex] at time t1. And of course the answer is [tex]\langle \Phi(t_2)|\Psi(t_1)\rangle[/tex]

We can also refer them to a fixed time t=0, by writing [tex]|\Psi(t_1)\rangle=e^{-iHt_1}|\Psi_0\rangle[/tex]. Then the probability amplitude for observing the system to be in state [tex]|\Phi\rangle[/tex] at time t2 given that it is in state [tex]|\Psi\rangle[/tex] at time t1 may be written

[tex]\langle \Phi_0|e^{+iH\Delta t}|\Psi_0\rangle[/tex]

where [tex]\Delta t=t_2-t_1[/tex]

But [tex]|\Psi_0\rangle[/tex] and [tex]|\Phi_0\rangle[/tex] are the very same states we use to describe the system from the Heisenberg-picture approach. That said we finally come to my question:

We can also refer them to a fixed time t=0, by writing [tex]|\Psi(t_1)\rangle=e^{-iHt_1}|\Psi_0\rangle[/tex]. Then the probability amplitude for observing the system to be in state [tex]|\Phi\rangle[/tex] at time t2 given that it is in state [tex]|\Psi\rangle[/tex] at time t1 may be written

[tex]\langle \Phi_0|e^{+iH\Delta t}|\Psi_0\rangle[/tex]

where [tex]\Delta t=t_2-t_1[/tex]

But [tex]|\Psi_0\rangle[/tex] and [tex]|\Phi_0\rangle[/tex] are the very same states we use to describe the system from the Heisenberg-picture approach. That said we finally come to my question:

**What does the quantity [tex]\langle \Phi_0|e^{+iH\Delta t}|\Psi_0\rangle[/tex] mean in the Heisenberg picture?**The answer should avoid referring to "state at time t" because in the Heisenberg picture states are not time dependent; only operators are time dependent.