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You can do two things: You can express the wave function (the old ground state in terms of the new angular frequence) in terms of the new angular frequency and then integrate that with the new ground state to get the probability amplitude.

Or you can express the new ground state in terms of the old angular frequence and integrate up to get the probability amplitude. Either way you get the correct answer.

What my problem was, is that I can't explain that the two procedures are equivalent. It should be pretty basic, after all it is just a matter of which angular frequence you choose to work with. But I still find it a bit miracolous that in the end you get the same... Please explain :)