Conceptual question on the expansion postulate

[Void123]

Homework Statement

This isn't a homework question, but here goes:

Suppose I have a system in an initial state |$\Psi_{initial}$> within the basis ${|\Phi_{n}>}$ and later on I have a final state |$\Psi_{final}$> within the same basis, and I want to know the probability of ending up in the latter state.

Now, since $<\Phi_{n}|\Psi> = c_{n}$, would the probability just be $|<\Psi_{final}|\Psi_{initial}>|^{2}$?

 

[Matterwave]

Er...isn't the final state defined such that your system is in the "final" state at the end? ...

What you have there is the probability that the final state "is still the initial state".

 

[Void123]

I see what your'e saying, but the wording confuses me. For the question is "what is the probability that you end up in the final state." If we had been given a different problem with an initial wavefunction (for the infinite square well, for example) and instead we were asked "what is the probability that the particle will be found in the ground state", then we would naturally say $<\phi_{1}|\Psi(x)>$. So, to the laymen, wouldn't it seem logical to ask what is $\Psi_{final}$ acting on the ket of $\Psi_{initial}$?