Dual meaning of "measurement" in QM

[zonde]

Reading about collapse and Born rule it gives strong feeling that there is semantic mess in QM with the word "measurement".
Wikipedia describes Born rule in following way:
In its simplest form it states that the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point.
So the born rule speaks about finding the particle at a given point. Operationally this is obviously detection in particle detector with amplification to classical signal.

But when measurement is defined using operator it leads to different operational definition of measurement. Again from wikipedia Measurement in quantum mechanics:
The state of a system after measurement is assumed to "collapse" into an eigenstate of the operator corresponding to the measurement. Repeating the same measurement without any evolution of the quantum state will lead to the same result.
This type of projective measurement operationally is defined as filter or splitter of particle beam.

Both measurements are essential in QM and are clearly different operationally and mathematically. But in ordinary language they go by the same name.

Is it possible that big part of "measurement problem" is semantic in nature?

 

[DrClaude]

For one, it is Wikipedia.

For two, there is no significant difference between the two. The Born rule is described in the context of a wave function, meaning specifically the measurement related to the position operator, which is of course continuous. Forgetting for a moment the peculiarities of measurements of continuous variables (where you can't, in practice, actually measure the system in a single eigenstate), that corresponds to what is described in the second Wikipedia article.

 

[zonde]

For two, there is no significant difference between the two. The Born rule is described in the context of a wave function, meaning specifically the measurement related to the position operator, which is of course continuous. Forgetting for a moment the peculiarities of measurements of continuous variables (where you can't, in practice, actually measure the system in a single eigenstate), that corresponds to what is described in the second Wikipedia article.

If you take position observable the two meanings seem similar (well, except you can measure position with aperture). But take another observable than position. What then?

 

[DrClaude]

If you take position observable the two meanings seem similar (well, except you can measure position with aperture). But take another observable than position. What then?

The Born rule was stated a long time ago, so it is based on concepts of the time. It was instrumental in understanding exactly what the function solving Schrödinger's wave equation represented. The explanation for the Born rule stated in the Wikipedia article reflects this, since the wave function is considered only as a function of position. But the same could be said by using a momentum representation for the wave function, where the Born rule tells you that $|\psi(p)|^2$ is the probability that the particle will have momentum $p$.

To generalize, given a quantum system in state $| \psi \rangle$ and an observable $\hat{A}$, the probability that a measurement of $\hat{A}$ will result in a value $a$, where $\hat{A} | a \rangle = a | a \rangle$ (i.e., the measured value must correspond to an eigenvalue of the observable $\hat{A}$), is given by the Born rule,
$$
P(a) = | \langle a | \psi \rangle |^2
$$
and, after measurement, the system will be found to now be in state $| \psi\rangle = |a\rangle$. That takes care of the two statements from Wikipedia.

 

[zonde]

There are no momentum detectors in laboratories. Instead what experimentalists do is use some sort of momentum splitter or filter and then detect particles with position detectors appearing at different places. So this mathematical operation has no counterpart in reality.

 

[DrClaude]

There are no momentum detectors in laboratories. Instead what experimentalists do is use some sort of momentum splitter or filter and then detect particles with position detectors appearing at different places.

How is that different than what is used to measure spin in the Stern-Gerlach experiment?

So this mathematical operation has no counterpart in reality.

I'm sorry, but I don't see how the fact that its a "mathematical operation has no counterpart in reality" is related to your original question.

 

[vanhees71]

To generalize, given a quantum system in state $| \psi \rangle$ and an observable $\hat{A}$, the probability that a measurement of $\hat{A}$ will result in a value $a$, where $\hat{A} | a \rangle = a | a \rangle$ (i.e., the measured value must correspond to an eigenvalue of the observable $\hat{A}$), is given by the Born rule,
$$
P(a) = | \langle a | \psi \rangle |^2
$$
and, after measurement, the system will be found to now be in state $| \psi\rangle = |a\rangle$. That takes care of the two statements from Wikipedia.

Now again this comes up :-(. It's not what the Born rule says. It says what $P(a)$ is, not more and not less. When measuring an observable on a system, it obviously depends on the apparatus you use to measure the observable what happens to the system. Maybe it's destroyed or maybe it's even constructed such that really you have this filter measurement, but the collapse hypothesis is at best a FAPP ("for all practical purposes", Bell) description of such filter measurements. It's incompatible with the very foundations of relativistic QFT which describes all of the known matter on the fundamental level in terms of local interactions, and is obeying the linked-cluster principle. So, if you just stick to the formalism without adding unjustified and unnecessary ideas like collapse, there is no problem.

Of course, now all the collapse proponents will come up and violently disagree with my point of view, and we end up with a nonsense discussion, we've already have had zillions of times without any conclusion. Just stick to the physics, which is what physicists around the world do in the lab when interpreting there measurements with QT, and no nonsense discussions and big confusions result!

 

[zonde]

How is that different than what is used to measure spin in the Stern-Gerlach experiment?

There is no difference.

I'm sorry, but I don't see how the fact that its a "mathematical operation has no counterpart in reality" is related to your original question.

There is measurement equipment in laboratory that is represented by projection in theory. You are saying we can replace projection with Born rule. I say in doesn't work. That is my point and of course it's related to my original question because the two meanings of measurement are "projection" and Born rule with corresponding equipment in laboratory. I'm saying they are different things.

 

[zonde]

collapse hypothesis is at best a FAPP ("for all practical purposes", Bell) description of such filter measurements.

Please, don't start collapse topic. Collapse was just an intro for the question. The question is about projection vs Born rule and corresponding equipment.

 

[vanhees71]

If I'm not supposed to start the collapse topic, why did you? SCNR.

 

[atyy]

There are no momentum detectors in laboratories. Instead what experimentalists do is use some sort of momentum splitter or filter and then detect particles with position detectors appearing at different places. So this mathematical operation has no counterpart in reality.

The "position" measurement is a measurement of momentum, so the measurement of momentum does exist.

 

[vanhees71]

It's an example for entanglement between position and momentum. Most measurements involve entanglement. E.g., in the Stern-Gerlach experiment by using the magnetic field you entangle the spin component in direction of this field with the position of the particles and then you can easily filter out the particles with the wanted spin component by blocking all the unwanted partial beams.

 

[Fra]

The question is about projection vs Born rule and corresponding equipment.

I do not see the born rule much coupled to the measurement problems, at least not unless you start digging and attempt a reconstruction of things and understand the origin of the born rule, and wikipedia leaves no clues on that.

Born rule just explains how to get actual probabilities from "quantum state", regardless of details of measuement. The probability distribution clearly does not follow from a single measurement, we need either ensemble statistics OR an information representation, which then also encodes an interaction history.

But as another step, I agree that it is interesting to ask yourself, from the perspective of original actual measumrents, HOW to construct the information state from available measurements - rather than postulating it?

This is a question that i have been working on for some time, and one possible "perspective" or "interpretation" is to consider how an observer = agent, with limited information capactity and processing power, processes a stream of data which is its own interaction history. In this perspective, it is not hard to imagine that the environment will prefer certain optimal representations in order to not destabilished the agent - the agent needs to align or face destabilisation. And the agent can choose to store x or p data, or both. And there is an internal transformation here which keeps the total information constant, BUT it is lossy, and the CHOICE is in which perspective you are better of "truncating". A kind of change is basis where the relevant information loss is minimal.

Out of this, i expect natural explanations of the burn rule. I do not see the born rule as a "big problem", i see much bigger headaches. I know where are various explanations of Born rule, but i am not sure if there is a paper which the idea i describe here is proven. I think this will solve itself.

This also relates to your issue of what "computations" that has a correspondence in nature? Like, how does "nature" perform a Fourier transform? or does it? One approach is to just say that the math is just a way to compute things. And the computations match experiment and we should ask for no more. Anything beyond that is not science. This is a respectable position, but also one which seems without further ambitions.To just understand QM as it stands, we are certainly not allowed ask these things.

/Fredrik