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I think it'll be clearest if I start with the case that I understand. Suppose I start in some initial state |ψ>, and I let it evolve over time t to some state e

^{iHt/ħ}|ψ>. Now, if I want to know the probability that I measure some particular value of, say, momentum, then I project it onto the momentum eigenbasis and the probability amplitude of measuring a momentum of p is

<p|e

^{iHt/ħ}|ψ>

where p is, and I stress this because it is key to the confusion that follows, a momentum eigenstate. If I want the probability that I measure a value a for some quantity other than momentum, I do the same thing but I use <a|, which is an eigenstate with well-defined a.

Now my confusion is that I often see discussions where people talk about "the probability amplitude for going from state |ψ> to state |φ>" where |φ> is quite often NOT an eigenstate of any kind. And they'll just write this as

<φ|e

^{iHt/ħ}|ψ>

But I don't really see any sensible way to interpret the above statement if |φ> is not an eigenstate of some sort. For example, I have seen the above expression used where |φ> is a wave packet. And I just don't understand why, mathematically, the above gives you the probability that state |ψ> evolves to state |φ>. Even when |φ> is an eigenstate, I wouldn't call that "the probability amplitude that |ψ> evolves to |φ>", because the final state could have all sorts of momentum components if it is, for example, a wave packet. I would just get the probability amplitude of measuring a momentum of p. But the above expressions seems to be used FAR more generally than my narrow interpretation, and I'm at a loss as to why.

Thanks.