General Questions Related to Quantum Measurement

[Efuhvex]

I've recently been wondering about the notion of measurement in quantum mechanics. While I'm aware that understanding and precisely defining the concept of a quantum mechanical measurement is at the heart of interpreting the subject, I would like some feedback on some aspects of this topic I find confusing, ambiguous, or both.
My questions are:

Can the particular variable being measured in an experiment at times be ambiguous? For example, suppose I am able to make some measurement of the energy of a particle by having it collide with a small sensor of some kind. Could I also view the particle's collision with the small sensor as a measurement of its position? If yes, does this imply the hamiltonian of the particular experimental setup we're considering commutes with the position operator?

My next question pertains to imprecise measurements and wavefunction collapse. Suppose I measure the position of a particle to be within some small error. Does the wavefunction collapse to some eigenvector of position within that same error, collapse to an eigenvector of another operator, or not collapse at all in this case?

My last question is a bit strange but I would like to hear your opinions nonetheless. I was wondering if one's conscious perception of the world could be viewed as one illustrious slew of quantum measurements. Are the particles making up all the matter and fields in my immediate vicinity constantly being forced into eigenstates as a result of my conscious perception, or is a more formal and precise definition of measurement required for collapse?

Thanks

 

[jfizzix]

To start, measurement is a physical interaction like any other.

Instead of imagining a measurement as causing the wavefunction of a particle to collapse, one could go "one turtle further", and imagine the measurement as an entangling of the joint wavefunction of the particle-plus-measurement device. The outcome states of the measurement device are correlated to "collapsed states" of the particle. One could them imagine the observing of the measurement device to collapse the joint wavefunction to one particular measurement outcome and corresponding quantum state of the particle.

That said, the particle's collision with the small sensor could be considered as a measurement of position, since the state of the particle would be localized in position as a result of the detection. This does not imply that the Hamiltonian of the experiment commutes with position. In fact, a Hamiltonian that is dependent on both position and momentum cannot commute with either since position does not commute with momentum.

If you make an imprecise measurement, the wavefunction does not collapse to a single eigenstate. The physical interaction can be imagined like a filtering out all the components of the wavefunction, except those components within the narrow band of outcomes corresponding to a particular measurement with a particular precision. The better the precision, the closer you get to filtering out everything except a particular single eigenstate. In position and momentum, it is not actually possible to measure an eigenstate of position or of momentum, but one can use these projections or filters to describe what happens.

Your last question goes to the metaphysical heart of interpreting quantum mechanics.
There are no easy answers to this question.
A simple way out is to treat quantum mechanics as the tool that it is in predicting the statistics of any conceivable experiment. Eventually you might even develop intuition for which experiments produce what statistics (an instrumentalist viewpoint).
It is also possible to imagine our subjective experience as being the result of the information physically recorded in our brains as a product of interaction with the quantum world around us (a many-worlds viewpoint).
Not knowing enough about the mystery of consciousness to comment on it with authority, it is still reasonable to assume that particles are not being forced into eigenstates as a result of your (or my) conscious perception. Rather, it may be that as a result of interaction, the particles' states are correlated to corresponding quantum states of information-recording elements in the brain.

 

[Demystifier]

If Hamiltonian commutes with position, you can measure them both (if this is what your first question was).

If the measurement of position is not precise, it can still be described as a "collapse" (in the sense of information update), but this collapse is not a collapse into a position eigenstate. It is a collapse into a certain mixed state which represents a mixture of different positions.

The answer to the last question depends very much on the interpretation, so I will refrain from answering it.

 

[Denis]

What is named "measurement" in many QT introductions, and described by self-adjoint operators, or projector-valued measures (PVM), is in fact only a very special type of measurement, which can be used also for state preparation - the state after the measurement is defined by the measurement result.

There is another notion of measurement, which does not care at all about what happens after this, thus, cannot be used for state preparation, only gets results. The object which is measured may be even destroyed. Such measurements are described by a positive-operator valued measure (POVM). Mathematically, it appears that there is nonetheless not much new, because a POVM appears to be a PVM on some larger space.

The most interesting example is imprecise common measurement of position and momentum. It appears to reduce to the scheme that you use some additional test particle, which with some approximation is at rest near the point $q_t=0$, and measure the operators $p+p_t, q-q_t$, which commute, as approximations for p and q. This has rather beautiful mathematics if you use the ground state of a harmonic oscillator for the test particle, because this gives holomorph representations of the commutation relations.

But this does not give any information about the resulting state. So, if you want such information, you have no other choice than to model the imprecise measurement by some interaction and to solve the Schroedinger equation for the whole process. The result will, then, depend on the initial wave function too (different from the standard measurement, where the initial wave function defines only the probabilities of the possible outcomes, but not the resulting state for a given value of the outcome). And will probably depend on a lot of details about the interaction and the measurement device.

 

[AlexCaledin]

Feynman's extremely remarkable words about "Nature's Knowledge" as the Process consciousness participates in (to some extent) :

"You must never add amplitudes for different and distinct final states. Once the photon is accepted by one of the photon counters, we can always determine which alternative occurred if we want, without any further disturbance to the system. Each alternative has a probability completely independent of the other. To repeat, do not add amplitudes for different final conditions, where by “final” we mean at that moment the probability is desired—that is, when the experiment is “finished.” You do add the amplitudes for the different indistinguishable alternatives inside the experiment, before the complete process is finished. At the end of the process you may say that you “don’t want to look at the photon.” That’s your business, but you still do not add the amplitudes. Nature does not know what you are looking at, and she behaves the way she is going to behave whether you bother to take down the data or not.
. . .
...You may argue, “I don’t care which atom is up.” Perhaps you don’t, but nature knows..."

http://www.feynmanlectures.caltech.edu/III_03.html

 

[bhobba]

I've recently been wondering about the notion of measurement in quantum mechanics. While I'm aware that understanding and precisely defining the concept of a quantum mechanical measurement is at the heart of interpreting the subject, I would like some feedback on some aspects of this topic I find confusing, ambiguous, or both.

The formalism alone these days can be used to determine when a measurement has occurred - its just after decoherence.

It doesn't resolve any interpretational issues - just clarifies the idea of what a measurement is.

For simplicity you can say the mixed state that occurs after decoherence is a proper mixed state, without detailing exactly what it means or how it becomes one as opposed to an improper mixed state. There is no way to observationally tell the difference which is why you can neither disprove or prove it. If you do that - its called the ignorance ensemble interpretation - then you have an interpretation and most QM issues are resolved. But that is an interpretive assumption.

BTW its exactly the same as the ensemble interpretation you can look up except its applied to just after decoherence.

Thanks
Bill