Which basis does the wavefunction collapse to?w X7SsXA fCc Cc5 Y
When we measure position for example, how does the system "know" that we're measuring position in order to collapse to a position eigenvector? Does the wave function always evolve from the state that it collapsed to? For example, if we measure the position (whatever that means) does the wave evolve from a delta function?
3 Answers
The system doesn't "know" anything.
The only uncontroversial statement one can make about the (strong) measurement of a quantum system is that you will make the correct predictions if you assume that the state after the measurement was the eigenstate corresponding to the measured value of the observable (so, for position, indeed a $\\delta$-function, if we ignore issues with that not being a real function, which would be a distraction here). But what we mean by "state" in the first place - i.e. what ontology, if any, corresponds to the statement "the system is in the quantum state $\\lvert \\psi\\rangle$" - is ambiguous to begin with:
Whether the original state "collapsed" to this new state, whether the "state" is just an imperfect representation of our knowledge and the "collapse" is just updating our information (cf. "$\\psi$-ontic" vs "$\\psi$-epistemic", see e.g. this answer by Emilio Pisanty) instead of an actual physical process, or something else entirely, is a matter of quantum interpretation. In some interpretations, there is collapse, in others there isn't, but in any case, the formalism of quantum mechanics itself does not provide a single "correct" interpretation.
That is, your question is essentially unanswerable unless you specify the interpretation within which it is to be answered. But none of the predictions of quantum mechanics depends on it anyway - you do not need to have a concept of "how" collapse works to compute the outcome of measurements.
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$\\begingroup$ Interesting I've never heard it explained that way. So the precision of your observation must play a part then, right? If you don't carefully pin something down to exactly one location how could the wave localize to that point? However, I thought that the wave needed to collapse to a basis vector, so which basis does it use? I assume I'm misunderstanding what constitutes a basis vector. Is it more like the wave "constricts" to match the updated possibilities for the state? $\\endgroup$ – Jeff Bass 49 mins ago
The collapse happens in all bases. What I mean by that is that the wavefunction can be expressed in any basis you want to. It's just that the easiest basis to look at right after measurement is the one corresponding to what you measured, since the state is the eigenstate corresponding to your measurement.
Always remember the wavefunction isn't physical. It's an abstract thing that we can only describe and "look at" as shadows from their projections. We can choose any projection we want to, but that choice doesn't change the wavefunction
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$\\begingroup$ This does not answer the question. The question is not about representing a wavefunction in different bases. $\\endgroup$ – eigenvalue 4 hours ago
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$\\begingroup$ @eigenvalue I answer the question at the beginning and then qualify and explain my answer. The point is that there isn't a basis that the wavefunction collapses to. You can choose any basis your want. It's just how you represent it. It doesn't go into some basis. $\\endgroup$ – Aaron Stevens 2 hours ago
I'm just posting a quick answer, mainly to say this question is about what is called the "preferred basis problem" and it is a well-studied aspect of quantum measurement theory. The main thing to say is that it can happen that for one basis an off-diagonal density matrix element such as $\\langle \\phi_i |\\psi\\rangle \\langle \\psi | \\phi_j \\rangle$ (where $\\phi_i$ are states of the basis) will either evolve very quickly or can be sensitive to very small disturbances, whereas for another basis this may not be so. In this case the off-diagonal elements of the density matrix average to zero over any practical timescale, so we have decoherence between states of such a basis. It is called a pointer basis.
