I'm watching MIT online lectures Quantum Physics I (roughly from one hour mark in the video). The lecturer explains wave functions that describe "Stationary States" that consist of a single energy eigenfunction then he points out that no such thing exists in real life. At some point a student asks a question: "Wait a minute, you said that if we measured the energy of the system, the wave function would collapse into one of the eigenfunctions. So wouldn't measuring the energy of the system would cause it to collapse into a single eigenfunction, thus creating a Stationary State that you just said didn't exist?" The lecturer answers that you can't measure energy with arbitrary accuracy and leaves it at that.
My question is this, though: so what does happen if you measure the energy with some inaccuracy? The wave function does collapse, right? But what does it collapse into? Sounds like it depends on how accurate your measurement was: if you did a crappy job, then it collapses a 'little' (some eigenfunctions are eliminated from the superposition), if you did a good job, it collapses 'a lot'. Am I on the right track or is this completely wrong reasoning? (Either case, it's weird: it's as if the system knew about the accuracy with which a measurement was performed.)