Measurements in quantum mechanics are made like for any system, in equilibrium with a bath of fluctuations. What matters is that the presence of the bath implies that what one can measure are probability distributions of any quantity of interest.

What quantum mechanics does differently from the case of any other bath of fluctuations is how the probability distributions are calculated.

But that's nothing surprising-for each bath the rules are different. That's nothing new.

Once the probability distributions have been calculated, how they are used isn't different than for any other probability distribution.

It's often forgotten that solving the Schrödinger equation is a means to an end, not an end in itself. The end is the calculation of the probability distributions of the quantities of interest.

So the statements about measuring a particle's energy or momentum, or position and so on are wrong: What one is really measuring is the probability distribution of these quantities.

Thank you Stam Nicolis , but I still have the same question. What procedure must be followed in order to produce a measurement that also prepares the system state to be a definite state of the measured observable? If there is no such thing, then why do textbooks tell us that a measurement or observation forces a system state to collapse into an eigenstate of the measured observable?

To further clarify my question, I know of three kinds of postulates in QM. One set leads to the deterministic evolution (the Schrödinger equation) between measurements. Another set leads to the calculations of the probabilities of various outcomes when measurements are performed. The third postulate is the collapse of a state immediately after a measurement or observation. My question is about the third postulate. What kind of measurement procedure cause a state collapse in real applications?

If a system is in an eigenstate of some observable, a measurement of that observable produces as result the eigenvalue and the system remains in that state.

If a system isn't in an eigenstate of some observable, a measurement of that observable produces the eigenvalue corresponding to the eigenstate selected

by the measurement with probability that of the system being in the particular eigenstate defined the measurement process-assuming the measuring process selects a particular eigenstate.

If the measurement process itself isn’t in an eigenstate, but a superposition of eigenstates, the result is a ``mixture" weighted according to the relative probabilities.

I understand this. Especially your second paragraph. But how is a measurement performed? I'm not getting an answer to this.

You have a state, |ψ>, the statement that a measurement occurs means the action on it by an operator, M, resulting in the state M|ψ>. This is the state of the system after the measurement. Now expand this state in the basis of the eigenstates of the observable O: The probability of the measurement yielding as result the eigenvalue λ_i, of O, corresponding to the state being the eigenstate |φ_i> of O, is ||^2/||^2.

What matters is that the state before the measurement was |ψ> and after the measurement is M|ψ>. How the measurement is carried out is defined by the operator M-which is an operator that acts on the space of states of the system.

It doesn't matter how you do the measurement-what you do with the result only matters. The result of the measurement is the average value of some quantity and the system is found in one of the possible states, consistent with having measured that quantity. There's nothing mysterious going on. The reason it doesn't make sense to talk about ``the collapse of the wavefunction'' is because the wavefunction is NOT a measurable quantity, so neither is its collapse.

What does make sense to talk about is the transition probability from one state to another. That probability can be measured.

Much of the confusion is the identification of what's apparently happening in real space with what's actually happening in phase space-the space of states of the system.

Everything that you said I already understood very well but doesn't answer my question. The reason that it matters how the measurement is done is simply that I want to know. So I asked the question of how the measurement is done. I asked this because measurement procedures that I have seen done in practice do not collapse a state into an eigenstate of the observable being measured. Not even close. Nothing like it. So I ask, what kind of measurement will collapse a state into an eigenstate of the measured observable? Textbooks say that any ideal measurement will do that, so I ask how do we at least come close to this kind of measurement in practice? I am sure that people familiar with experimental methods can answer that, but so far none of these people have responded to this thread. I ask that anyone responding to this thread read all of the posts given so far so they will have a better understanding of the question.

To make the question easier to answer we can be more specific about the observable. Say it is energy. I know that photon emission provides information but suppose there isn't any. Suppose the particle is a free particle. How do we measure its energy in such a way so that the particle energy becomes equal to the measured value after the measurement is made? Let us focus on that specific question if the answer to a more general question is too complicated.

The statement about ``collapse" doesn't have anything to do with quantum mechanics and everything to do with ordinary properties of probability.

If you take a coin and you flip it, it has a probability of being ``heads" and a probability of being ``tails"; when you actually do a measurement, it will be in a definite state. That's what ``collapse" is in this case. In quantum mechanics it's exactly the same, only the space of states is much larger, since all linear combinations are possible states in this case-though, once more, what is, also, important is the states the detector can be in. That's all there is to ``collapse".

So, by definition, any measurement ``collapses" the state of a system to something definite, related to the properties of the detector. Just like a thermometer provides a way of measuring the (average!) kinetic energy of the molecules in a volume. There's nothing different in principle than that for a quantum gas. The difference in practice is that, for a gas of molecules the fluctuations are thermal, while for a quantum gas, they're quantum. Two different baths. In both cases one computes probabilities, the only difference is the space of states is different. How the probabilities are manipulated, once they've been computed and that any measurement puts the system in a definite state is the same, whatever the bath-whose degrees of freedom aren't resolved in any case.

The energy of particles in particle physics experiments is done in one of these ways: https://cds.cern.ch/record/1712874/files/393-403_Riegler.pdf

Momentum is measured in these ways: https://scholarcommons.sc.edu/cgi/viewcontent.cgi?article=1054&context=senior_theses

That these methods do what they're supposed to be doing, however, relies on theoretical understanding.

The reason ``collapse" doesn't get mentioned is because, when discussing measurements, we're doing it, so to speak, from the point of view of the detector, so ``collapse" to a definite state has, already, occurred.

L.D. Edmonds, if I have understood your question correctly, you ask how a quantum system is prepared for a quantum measurement and how a measurement is made. I'm not an experimentalist, but I would like to add the following to the already posted answers. Your question is related to what is known as the measurement problem in quantum mechanics, which has also to do with the interpretation of the theory (quantum mechanics). Apart from the Copenhagen interpretation of quantum mechanics, according to which the wave function of the examined system "collapses" when a measurement is performed on the system, there also exists the many-worlds interpretation, according to which the wave function does not collapse and all possible measurement outcomes are realized, each one in a different "world" along with a "copy" of the observer.

Regarding your question, the following links, and the links therein, may be useful:

https://physics.stackexchange.com/questions/29702/what-counts-as-a-measurement (the 1st answer contains points relevant to your question).

https://physics.stackexchange.com/questions/27/what-is-the-difference-between-a-measurement-and-any-other-interaction-in-quantu?rq=1 (the link to Sidney Coleman's lecture seems to be broken, but you can find the lecture here https://www.youtube.com/watch?v=EtyNMlXN-sw).

It's amazing how the confusion of the past gets perpetuated, instead of being replaced by understanding.

In the 1920s and 1930s people were obsessed, on the one hand, with ``interpretations'' and, on the other hand, with a reluctance to use probabilities as a means for describing natural phenomena.

Quantum mechanics doesn't change the rules, how probabilities are used; it changes the rules of how they are computed. It describes physical systems in equilibrium with a bath. In that sense, performing a measurement of a quantum system doesn't differ, in principle, with performing a measurement of a ``classical'' system in equilibrium at some temperature; the only difference is in the space of states. Indeed the correspondence between the description of a quantum system and a classical system at finite temperature can be made mathematically precise.

Whatever the interpretation of any formalism may be, the only thing that matters is the result, in the case of quantum mechanics, of the calculation of the probability distribution of the quantities of interest-not how it is calculated. If the result provided by the different interpretations-assuming they are self-consistent-is the same, which one uses is a matter of taste, that can't be resolved. If the result differs, then the question is, whether the difference is due to an error-conceptual or technical-of the calculation, or does it represent a new theory, in which case an experiment can be set up to see which is the description that's appropriate for our world.

The ``many-worlds'' interpretation doesn't differ from the ``Copenhagen'' interpretation so much with regards to physics, as with regards to mathematics:The way it uses probabilities isn't consistent with how probabilities are defined, mathematically. That's its problem.

The statement about a wavefunction ``collapsing'' is, simply, meaningless; as mentioned, it's not a measurable quantity, so its ``collapse'' doesn't make sense; and, as, also, mentioned, the ``collapse'', from the many possible states a system can, potentially, be, in to the state that the system is found in, upon performing a measurement, doesn't have anything to do with quantum mechanics. It has to do with probability theory, which the people that worked in quantum mechanics, thought represented a particular, ``undesirable'', property of quantum phenomena-whereas it is a fundamental property of all natural phenomena. And they got confused by trying not to use it, so they made mistakes. The bath of quantum fluctuations is a new bath, from the point of view of physics and its new properties reflect themselves in the space of states, not how probabilities are manipulated.

Thank you Stam Nicolis. Your two posts submitted 3 hours before my post here are exactly what I was looking for. You answered my question.

Glad it was useful! It's very easy to get sidetracked...

When it comes to a physical theory, the interpretation of the underlying mathematical theory is always an essential, and often complicated, issue. Quantum mechanics is not an exception to that. Issues related to the interpretation of quantum mechanics that have to do with making measurements and interpreting the results are still open, as one can easily find out by searching and asking. The next-to-last post of the previous, full-time poster seems to be a typical instance of scientific dogmatism, regrettably.

It's interesting how many people continue to ascribe to quantum fluctuations some mystical quality. There's no conceptual issue in describing measurements of a quantum system, that's radically different from the case of any other physical system in equilibrium with a bath; the rules are clear and the results are unambiguous: They are probability distributions, whose properties are those of any probability distribution and that can be and have been compared to experiments. This, apparently, troubled many ``famous'' people, who didn't like probability distributions and liked even less the fact that a quantum system has a much larger-actually infinite-space of states than its classical limit, but these are just misconceptions, among the many misconceptions that arise in any subject. It's been known for many years how to deal with probability distributions and infinite-dimensional spaces of states and how the classical limit can be recovered. And that many questions-among them the so-called ``collapse of the wavefunction''-simply expressed the lack of understanding, that, usually, is followed by learning which questions make sense and which don't. Not all questions make sense in physics.

In the end, what matters are the rules for doing calculations and performing experiments. There are many ways of doing both and what matters is understanding that the different ways are the equivalent-so it doesn't matter which one is chosen-or why they might not be.

Stam Nicolis Said "the so-called ``collapse of the wavefunction''-simply expressed the lack of understanding, that, usually, is followed by learning which questions make sense and which don't. Not all questions make sense in physics." Now in steps condensed-matter laboratory-quantum-gravity. Gravity is very weak and measuring the gravitational binding energy released during “collapse of the wavefunction” is very hard to do in the laboratory. Cosmologically, new JWST data supports the concept of “zero active mass” in general relativity. This concept separates momentum from inertia using gravitational binding energy and defines inertia as a local vacuum interaction. We have tons of data that contributes the mass of the atomic nucleus to inertial mass generated by quarks and gluons moving through a gluon fluid inside the nucleus. This separation of inertia from momentum is just beginning to open our eyes to getting our grubby little hands onto the vacuum as the source of inertial mass that gravitates in a condensed matter physics experiment.

It may be useful for the readers to know what Gerard t Hooft has written about the interpretation of quantum mechanics. The following was taken from an answer written on Oct 1, 2021 in a discussion about General Relativity that can be found here https://www.researchgate.net/post/Why-is-GR-highly-non-linear-while-electrostatic-interactions-are-linear-even-in-the-comparatively-stronger-electric-force-limit.

"In general, a theory of physics could have defects of all sorts. I divide them in 3 categories:

A) There could be internal logical inconsistencies in its mathematical construction;

B) There could be conceptual difficulties in the physical interpretation of a theory, and

C) It may well be that the theory does not describe the physical world in the way it claims to.

...In quantum mechanics, for instance, A and C are perfect, but in B, many researchers have complaints; I nyself have serious objections to what is often said."

Clearly, "nyself" is a typo for "myself".

Physics isn't appeal to authority nor Scripture; there's no point in quoting scientific texts, i.e. texts that are meant to be understood that way-their meaning matters, not the exact words, nor who it is that says or writes them.

They're cited for attribution of priority of ideas and for their meaning, not their words.

And it's not useful to confuse opinion-which can't be subject to debate-with technical issues, that can. So, as long as the ``objections'' can't lead to anything impersonal-in the present case a calculation-there's nothing to discuss. Science isn't done by voting on opinions.

For historical reasons describing measurements in quantum mechanics is presented as something that's mystical and not fully understood. In fact it's nothing more or less than performing measurements when taking into account a bath of fluctuations and the rules for doing so are unambiguous. So the only technical issues pertain to inquiring what rules can be defined for describing the bath of fluctuations in this case. There's nothing conceptual about this, though, the new rules must be spelled out as must be the framework for carrying out the corresponding calculations.

The usual way of doing them leads to well-defined probability distributions for quantities that can be measured in real experiments. So any alternative framework must show what the result of using it leads to. If it leads to the same probability distributions, it's a matter of taste which framework is to be used. If it leads to different probability distributions, then an experiment can be set up to check which framework does, indeed, describe quantum phenomena in our universe-and what are the effects, that are responsible for the discrepancy with the usual description, that does describe known quantum phenomena in our universe, already. So any claim about an alternative formulation has to show how it happens that it does agree with the usual formulation of quantum mechanics, for the known experiments, while not agreeing with it in a new range of experiments, that can, then, be set up to check its validity.

Another typical example of how measurements are made in quantum mechanics-and how to go beyond it-is the so-called Lamb shift: If one considers the electromagnetic interaction between the proton and the electron as classical, but in the presence of quantum fluctuations for the particles alone, one can compute the spectrum of the hydrogen atom, find excellent agreement with the experiments made in the late 19th century-and predict, in particular, that certain energy levels, that are labeled by integers, are degenerate. Until 1947 experiments did, indeed, confirm this. While it was expected that quantum fluctuations of the electromagnetic interactions could lead to modifications, it wasn't known how to compute them and experiment didn't seem to be sensitive to them-until Lamb and Retherford did their experiment in 1947 and found two energy levels that weren't degenerate, but had a splitting they could measure to be non-zero to experimental precision. One non-trivial issue was, indeed, to identify the reason for such a splitting as being, indeed, the result of quantum fluctuations of the electromagnetic field, produced by the electron, even as it was bound to the proton-then to calculate it-which meant generalizing quantum mechanics to take into account the quantum fluctuations of fields, not, just, particles-and that led to the correct formulation of quantum electrodynamics, that had started, already, in the 1920s: Bethe showed how to get the ``right'' answer, but it wasn't clear why his approximations were justified. Feynman, Schwinger and Tomonaga, independently, found formulations that provided the justification for Bethe's calculation and the framework for generalizing it. And Dyson showed that the three, apparently different, formulations were, indeed, equivalent. Since then experiments and calculations show that it is possible to perform and understand measurements of systems subject to quantum fluctuations, that describe all known properties of matter, if spacetime can be assumed to be flat.(G. 't Hooft has made decisive contributions in this subject.) When spacetime remains classical, but can be curved and matter is subject to known quantum fluctuations, then there are problems, that remain to be solved-i.e. finding the right questions to ask is an ongoing process (to which G. 't Hooft, also, has and is contributing in a substantial way-by first advancing the idea that combined the so-called ``holographic'' picture for describing gravity with the relevance of a finite dimensional space of states for quantum systems, when gravitational effects are relevant, to cite two issues.) The experiments in this context have to do with black holes and/or cosmology, which makes them correspondingly harder to do, however. That's how physics works-by providing a well-defined calculational framework that is the basis of imagining what kind of experiments can be done.

The discussion was about whether or not the expression “the wave function collapses” is meaningful and by extension, whether or not there exist issues related to the interpretation of quantum mechanics. In this respect, it is not bad to know what other people say about that, especially those who successfully work with quantum mechanics, and certainly this does not hint at any kind of appeal to “authorities”.

The wave function is not a physically measurable quantity, but it is used to calculate probabilities, which are physically measurable; hence, it also belongs to the theory (quantum mechanics), as does any other quantity that is used by the theory; for instance, the Schrödinger equation. It is clear that a physical theory does not contain only physically measurable quantities and issues that are related to physically non-measurable quantities of a theory could well have meaning inside the theory and impact on the physically measurable quantities of the theory.

The expression “the wave function collapses” describes intuitively what happens to the state of a system, according to the Copenhagen interpretation, when a measurement is performed on it; i.e., it conveys the meaning of the respective postulate; i.e., that the state of a system from a linear combination, whether finite or infinite, of eigenstates of the Hermitian operator describing the quantity to be measured changes – “collapses” – after the measurement, to some eigenstate. The fact that the said expression has been part of the quantum mechanics parlance for a century now demonstrates its usefulness; otherwise, it would have been “deprecated”.

PS: It may be useful to note that in mathematics, there does not exist such a notion as “the collapse of a function”. The expression “the wave function such and such collapses” is not a mathematical statement.

The answer, once more, is that the statement ``the wavefunction collapses''

doesn't mean anything and can't mean anything, because the wavefunction isn't a physical quantity.

Non-physical quantities only serve to construct physical quantities, nothing more and nothing less. More precisely the difference is between quantities that are not invariant under the transformations, that define the symmetries of the theory and quantities that are invariant under these transformations. Non-invariant quantities don't have any deeper physical meaning, beyond that of being building blocks for the invariant quantities, while invariants do. That's what matters: The symmetries define the theory and are part of the process of discovery about natural phenomena. That took many years to be understood, but, now, it's background.

There are many expressions that have some sort of legacy status, but that's just a statement about history, not of technical content. People were confused; many people remain confused. In the first years of quanum mechanics, they may have had some excuse, now they don't. History of physics is an interesting subject, but it's a different subject from physics.

Once upon a time people were troubled with the appearance of probability distributions for physical quantities, because at the same time that they were trying to understand quantum mechanics, they didn't really understand statistical mechanics and thermodynamics, either. (Not to speak about deterministic chaos.) So there was a reluctance in dealing with probability distributions. Now all this is understood (and the questions that are of interest have shifted correspondingly)-but, unfortunately, there's a lag and many textbooks, still, contain too much history, without spelling out what's history and what isn't.. What matters is what is understood, not the confusion that reigned before understanding it.

The following are only a few, randomly chosen, papers that have been published recently and are related to the wave-function collapse.

(i) Models of wave-function collapse, underlying theories, and experimental tests.

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.85.471

(ii) Epistemology of Wave Function Collapse in Quantum Physics.

https://www.journals.uchicago.edu/doi/abs/10.1093/bjps/axu038?journalCode=bjps

(iii) Detecting wave function collapse without prior knowledge.

https://pubs.aip.org/aip/jmp/article-abstract/56/8/082103/381064/Detecting-wave-function-collapse-without-prior

(iv) Nonlinearity of Quantum Mechanics and Solution of the Problem of Wave Function Collapse.

https://iopscience.iop.org/article/10.1088/0253-6102/64/1/47/meta

(v) Gravity induced wave function collapse.

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.96.104013

More can be easily found. Clearly, there is no point in repeating oneself again and again in the same discussion.

I guess I still don't get the answer. To be more specific, focus on the energy of a particle as being the quantity to be measured. Textbooks say that an ideal measurement will result in the post-measurement particle state being one with a definite energy and with that energy equal to the measured value. No real measurement that I have learned about is anything like that. For example, some methods deduce the energy of an ionizing particle from the amount of ionization produced in some device, such as a surface barrier detector. The post-measurement state of the particle is a particle stopped in the detector. There are other measurement techniques but all of them that I have learned about have two characteristics. First, the measured energy is the energy the particle had before the measurement was performed, not the energy after the measurement was performed. Second, the post-measurement state of the particle is nothing like an energy state with energy equal to the measured value. So I am asking the same question again. How, in real life, do we measure the energy of a particle in such a way so that the particle state after the measurement is (approximately because no real measurement is perfect) an energy state with energy equal to the measured value?

L.D. Edmonds My favorite paper that uses mass equals energy over the speed of light squared, thus defining energy, is "Origins of mass" by Frank Wilczek.

(PDF) Origins of Mass (researchgate.net)

To go beyond this paper, I become the modern-day Boltzmann, who famously said. "If you can heat it up, then it's made out of atoms." Specifically, STP or Space-Time-Matter atoms that get hot in algebraic octonion physics. Using thermodynamics as my "theory-of-everything" I can be Boltzmann and add quantum pre-gravity to the minimal standard model and discover that zero-active-mass cosmology must be true in a condensed matter table-top experiment. Also, on research-gate.

(PDF) Using The Principle of Equivalence in Thrust Capacitor Technology (researchgate.net)

Particle Rest Mass energy is now gravitational binding energy, where the total particle energy equals rest-mass-energy plus free-energy. And momentum is conserved in free-energy and rest-mass inertia is a vacuum energy interaction. Specifically, when electron rest-mass goes to 0.56 times the electron effective rest-mass an equivalent amount of Larmor energy is scattered out of the Unruh vacuum. The Unruh/Larmor electrodynamic and mechanical physics is encoded into existing quantum and classical physics. Except now, accelerating electrons massive electron waves that are being prevented from localizing for up to 35.42 seconds, lose rest-mass to Larmor radiation, that is a measurement that nobody ever thought to make. The electron regains its normal mass and loses its binding-energy rest-mass when it localizes on the cathode.

L.D. Edmonds Equations speak better than words.

The total Unruh/Larmor vacuum energy that is produced, N [e] x EL [J], equals the tunneling electron fraction 0.56 [e] times gravitational binding energy of one tunneling electron Eb [J],

N [e] x EL [J] = 0.56 [e] x Eb [J]

where N[e] equals the number of electrons on the anode. Zero-active-mass cosmology is demonstrated by heterojunction tunneling!

N [e] x PL [J/s] = I [e/s] x Eb [J]

where I [e/s] is the tunneling current.

I don't understand any of that. I'm looking for instructions that can be given to a laboratory technician (not necessarily a theoretical genius) on how to measure the energy of a particle in such a way so that the particle energy after (not necessarily before) the measurement is (approximately because no real measurement is perfect) equal to the measured value. QM textbooks say that this is what an ideal measurement or observation does (often called wave function collapse). How do you do this ideal measurement or observation in the laboratory?

I haven't specified what kind of particle and under what conditions (e.g., an electron bound to an atom, a free electron, etc.) because you can take your pick, whatever case that makes the question easiest to answer.

L.D. Edmonds Hook a thrust capacitor to a battery and measure the energy lost by the battery to Larmor radiation and you can also measure the Larmor radiation directly. Compute the energy lost using quantum field theory, classical electrodynamics, and gravitational thermodynamics, and show that all three theoretical computations give the same answer. Simple.

Does that leave the particle (an electron? I don't know) in an energy eigenstate after the measurement was made. Elementary QM textbooks that don't talk about QM field theory, and certainly not gravitational thermodynamics, tell us what an ideal measurement or observation will do (sometimes called collapsing a wave function). How did the authors of those books do it? I don't think you understand my question. I'm not talking about any kind of measurement. I'm asking what laboratory procedure will collapse a wave function.

Let me ask the question this way. How do you collapse a wavefunction? The typical answer is "you make a measurement or an observation." My counter is that not just any kind of measurement will do, as I pointed out a few posts ago (e.g., measuring the energy of an ionizing particle by the amount of ionization produced in some device measures the energy the particle had before the measurement, not the energy after the measurement, and certainly does not make the post-measurement particle state an energy eigenstate with energy equal to the measured value). So let me phrase the question this way. What laboratory procedure produces the kind of measurement that collapses a wave function?

You can select any example (e.g., an electron stuck to an atom, a free electron, or whatever) that makes the answer easy because any example will make me happy. This is such a basic and fundamental concept of elementary QM that I am amazed that it is so difficult to get an answer.

L.D. Edmonds The state of the art, dated 22 February 2024, describes an experiment that in theory can make the measurement of a wavefunction localizing that you are requiring.

[2402.13980] Geometry-induced wavefunction collapse (arxiv.org)

Quoting this reference "We note that this feature of infinite oscillations was previously used to establish the atomic collapse states about a Coulomb impurity in graphene [30, 33]. From a classical point of view, the geometry-induced and Coulomb-impurity induced collapse states share a common feature: the particle appears..." I side-step these experimental difficulties by using 22 trillion electron wavefunctions producing the Coulomb field, but with only one tunneling electron localizing at one time. My geometry is produced by an electron acceleration of 2 x 10^19 [m/s^2], one electron at a time. And the QFT, Coulomb, and acceleration-gravity computations all give the same answer in the following reference.

Kajol Paithankar and Sanved Kolekar, “Role of the Unruh effect in Bremsstrahlung” Phys. Rev. D 101, 065012 (2020)

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.065012

Thank you George Soli .