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Measurement operation

  1. Sep 1, 2009 #1
    Like all beginners to QM I'm really confused about the measurement operation. I understand that measurement is simply a dot product with an "operator" and the result is one of the operator's eigenvalues.

    Now my question is what exactly is an operator? If someone could explain what physical entity is an operator in the following situations that would help me understand this better.

    1. A particle moving at some velocity hits a wall/detector. Till it hits the detector its position is described by a combination of position eigenstates. Once it hits the detector its position becomes a single eigenvalue. Is the wall an operator here?
    2. A particle moves through an SG apparatus. Till it passes through the apparatus its spin state is a combination of two eigenstates (in some axis). Once it moves through the magnetic field its spin state becomes one of the eigenstates. Is the magnetic field an operator here?
     
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  3. Sep 2, 2009 #2

    tiny-tim

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    Hi svnaras! Welcome to PF! :smile:
    Nooooo …

    an operator is maths rather than physics.

    For example, momentum is an operator, and if you make a measurement, then you convert the wave function into an eigenstate (eigenvector), and its eigenvalue is the actual momentum measured.

    (in other words, the operator and the value measured tend to have the same name, which is confusing! :confused:)

    No.

    In terms of matrices and vectors, an operator is a matrix, and the eigenstate is an eigenvector.

    And measurement is the ordinary action of a matrix on a vector, giving another vector (while a dot product is a combination of two vectors to give a scalar). :wink:
     
  4. Sep 2, 2009 #3
    Re: measurement

    This is the part of quantum mechanics that no one has worked out yet. It's called the "measurement problem" because there isn't really any precise definition of what a measurement is, when it happens, and what takes place when a measurement happens.

    My personal view is that a measurement is an irreversible, macroscopic record of a property of a quantum system. Every time a measurement happens, the collapsed version of the wave function must be used to predict the result of subsequent measurements, but this is a mathematical process, not an actual, physical change to the quantum system. Thus:

    The detector is taking a measurement here. The operator is a projection matrix of the position of the particle on the portion of the wall covered by the detector.

    No measurement, since no irreversible, macroscopic record.

    Just my 2 cents.
     
  5. Sep 2, 2009 #4
    Re: measurement

    There is only a measurement problem in collapse theories. Bohr, Schrodinger, Einstein, Bohm, etc, did not have a problem with measurement.

    I agree that the OP seems to be confusing math with reality. Operators are symbolic. Physical things are not operators. We model the physical world with math (realistically or instrumentally depending on who you ask), but the two are separate. At best certain theoretical values may correspond to certain real properties, but operators are not properties or values, they are pure math.
     
  6. Sep 2, 2009 #5
  7. Sep 2, 2009 #6
    Re: measurement

    Thanks for the replies. That seems to clarify some of my misunderstandings.
    A couple more questions. If an operator is to be associated with any measurable quantity then there should be an infinite number of operators right?

    For example could we not define operators for velocity, acceleration or any combination of the fundamental units or are there operators for only some fundamental quantities like position, momentum, energy, etc?
     
  8. Sep 2, 2009 #7
    Re: measurement

    The paper seems to say exactly what you just said, and not much more. It takes as a hypothesis that the universe could be pure math, and draws some conclusions about what this might mean. I suppose as long as there are mathematicians there will be platonists. I am also not impressed when a mathematician tells me that supposing the universe is pure math is minimally hubristic :smile:.

    Even if the universe were pure math, how would you know that our current mathematical formulations are accurate representations of it? You end up at the same place we are without a mathematical universe. The underlying structure is irrelevant. Human theories about nature can never be proven, so human formalism at best arbitrarily corresponds to nature in some aspects. We can propose "math is real" or "reality is math," but we can't ever say a physical event is addition (without redefining addition a posteriori).

    Spinoza worked out many of the details of a logic based universe and came to these conclusions. Even if math is basic, our formalism is not.
     
  9. Sep 2, 2009 #8

    Fredrik

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    Re: measurement

    :confused: What are you talking about here? Are you saying that there's something in Bohmian mechanics that ensures that it doesn't have a measurement problem? (I don't know that theory, so I can't rule it out). And what alternatives to QM are you suggesting that Bohr, Schrödinger and Einstein used? :confused:
     
  10. Sep 2, 2009 #9
    Re: measurement

    I guess I'd have to ask you where the confusion is supposed to be in their views :smile:. I can't seem to find any confusion that any of them had with regard to measurement or observers or anything like that. It seems that von Neumann and opponents of CI brought about all of the collapse and measurement issues.

    For Bohm it was easy. The world is deterministic. Nothing special happens during measurement, you're simply taking a reading. No one thought that measurement actually had a profound effect on reality. It only revealed it, with some necessary perturbation by the instrument. Measurement problems arise when we pretend that measuring actually does something special like force a superposition of alive and dead cat to become one or the other. No one used to think that a cat could be both alive and dead.

    Einstein and Schrodinger thought it was absurd that something could be and not be at the same time, so they devised macroscopic thought experiments to show the absurdity. Bohr also did not believe that a cat was both alive and dead at the same time - measurement simply reveals whether or not you've got an alive cat or a dead cat, and you can't meaningfully ask about whether it was alive or dead before you looked because only observables are real.

    Measurement problems require "the wave function is real and observation causes it to collapse."
     
    Last edited: Sep 2, 2009
  11. Sep 2, 2009 #10
    Re: measurement

    Not only does it not have a measurement problem, but as Bell said, 'all the usual quantum paradoxes are disposed of by the 1952 theory of Bohm'. The classical limit emerges out of the theory rather than having to be presupposed, one can derive the Born Rule, etc. etc.

    In Bohm theory, measurements are just perfectly ordinary many-body interactions, special only because the interaction leaves the wave field in a particular state (an eigenfunction of a Hermitian operator). Hermitian operators are important not just for the usual reasons but because their eigenvalues are spacetime constants (i.e. you can know the value of the observable without knowing the position of the particles).

    The usual problem is that if the wave function splits into branches following Schroedinger evolution all of these branches continue to exist, despite the fact that we only see one of them. In Bohm theory, things are made out of particles (which are 'guided' by the objectively-existing wave field represented mathematically by the wave function) and they just randomly but deterministically end up in one of the branches (with the usual quantum probabilities).

    Then the establishment of correlations between the quantum system and its macroscopic environment quickly reduces the magnitude of the interference terms between the different branches. This means that once the particles end up in the support of one branch, they stay there (trajectories can't pass through nodes in the wave field). [This is just decoherence of course - nothing mysterious about it - it's just ordinary Schroedinger evolution. It was first described by Bohm - for just this purpose].

    Note that decoherence alone does not solve the measurement problem - it merely provides a mechanism for the different branches of the wave field to stop interfering. All branches continue to exist. It requires something else such as the introduction of 'hidden variables' or some appropriate interpretation of the wave function to explain why one branch is what one sees. Bohm theory does this very simply.

    Lecture 4 of the http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken] is the thing to read here - a very good summary of both the measurement problem and its resolution in the Bohm theory..
     
    Last edited by a moderator: May 4, 2017
  12. Sep 2, 2009 #11
    Re: measurement

    To put it another way, Bohr believed that when you ask the question "is the cat alive or dead?" what you really mean is "when we look in the box will there be a dead or living cat?" He never believed that reality could be in a superposition of states. He did not believe that the wave function represented reality directly. Science, for Bohr, is about experiments and real observables, and if it's theoretically not an observable, then it's not real. Superpositions are theoretically not observable, so they are not real.

    The confusion comes from thinking "if it's not theoretically observable, it's not real" is equivalent to "observers create reality."
     
  13. Sep 2, 2009 #12
    Re: measurement

    Indeed, but unfortunately all the experimental evidence says otherwise. For example, the dynamical object represented by the wave function (let's call it the wave field) can interfere with itself. How can the terms of a quantum superposition interfere with each other, producing an observable interference pattern, if such a superposition is just an expression of our ignorance?

    As I have pointed out here before, we find that in matter-wave optics experiments - for example - it is also possible to diffract, reflect, focus, and do stimulated emission with the wave field. This is clear experimental evidence for the objective existence of the wave. If the wave can be subject to and utilized in such a process, it logically follows that the wave field must exist in order to act and be acted upon.
    Sorry, I had forgotten Bohr's pronouncements were holy scripture. I withdraw my previous remarks, and will go and flagellate myself immediately.
     
  14. Sep 2, 2009 #13
    Re: measurement

    Bohr's thoughts are scripture? Since when? I certainly didn't make that claim. Thanks for the constructive comment though :bugeye:?

    Bohr was also well aware of interference. The problem is that there is at least as much evidence for the objective existence of particles. QM has proven that we can't have both at the same time. One solution to this, which cannot be disproven experimentally without disproving all of QM, is to treat experiments as black boxes and only refer to the outputs as real. Trying to visualize what goes on inside the black box is meaningless and impossible. This was Bohr's view and is how he was uniquely able to restore classical entity realism.

    It's not the only option, but it doesn't have a measurement problem. You'll notice (or not?) that I was the one who brought up Bohm as another option. QM is paradoxical and there are plenty of issues with any interpretation, Bohm's included. The point is that measurement doesn't have to be one of those problems, and it wasn't for many important physicists.
     
    Last edited: Sep 2, 2009
  15. Sep 2, 2009 #14
    Re: measurement

    Well done! In which case you should realize that your following remark is absolute nonsense:
    Unforunately the point of the Bohm theory is that both particles and waves have an objective existence. Given that it's predictions match QM entirely (and how could it not - it is QM - one can derive the entire thing just by changing the usual meaning of a single word) then QM cannot possibly have 'proven that we can't have both at the same time'.

    In fact, what on earth are you talking about?
     
  16. Sep 2, 2009 #15
    Re: measurement

    I'll assume that you're referring to the flavor of Bohmian Mechanics that takes positions to be the basic properties of particles. Particles with no basic properties besides position are not photons, electrons, etc. They are completely ineffectual by themselves and you wouldn't even know if they were there or not. This flavor of the Bohmian Interpretation does not have real classical entities. Bohr's interpretation does. Heisenberg sums up the problems with this version of the Bohmian Interpretation:

    What does it mean to call waves in configuration space “real”? This space is a very abstract space. The word “real” goes back to the Latin word “res,” which means “thing;” but things are in the ordinary three-dimensional space, not in an abstract configuration space... Bohm considers himself able to assert: “We do not need to abandon the precise, rational, and objective description of individual systems in the realm of quantum theory.” This objective description, however, reveals itself as a kind of “ideological superstructure,” which has little to do with immediate physical reality.

    But maybe you were referring to Bohm's interpretation as of Wholeness and the Implicate Order, in which particles are given added complexity and free will and in part “determine themselves independently of the infinitely complex fluctuations inside the associated regions of space.” Self-determining particles are also not photons or electrons.

    Or were you referring to the infinitely complex particles Bohm proposes that are guided by mind-like active information, which is explained by deeper physical levels, which are guided by deeper mind-like levels, ad infinitum? This can't be it though, since apparently you don't agree with Bohm when he says, "The deeper reality is something beyond either mind or matter, both of which are only aspects that serve as terms for analysis." Bohm believed that "deeper reality" was inherently unknowable and that none of his theoretical constructs were basic.

    Bohm had some great thoughts and a viable view, but let's not pretend any variety of Bohm's interpretation gives us a reality that is in any way classical. Photons, electrons, and three dimensional objects do not exist as basic entities for any flavor of Bohm. He showed us a viable way to restore determinism, but Bohmian realism is not located in space-time and does not involve classical entities.
     
    Last edited: Sep 2, 2009
  17. Sep 2, 2009 #16
    Re: measurement

    One last Bohm quote for you:

    ...space and time are no longer the dominant factors determining the relationships of dependence or independence of different elements. Rather, an entirely different sort of basic connection of elements is possible, from which our ordinary notions of space and time, along with those of separately existent material particles, are abstracted as forms derived from the deeper order.
     
  18. Sep 2, 2009 #17
    Re: measurement

    Here is how John Bell characterized one aspect of the measurement problem, from his paper Quantum Mechanics for Cosmologists:

    It would seem that the theory is exclusively concerned with 'results of measurement' and has nothing to say about anything else. When the 'system' in question is the whole world where is the 'measurer' to be found? Inside, rather than outside, presumably. What exactly qualifies some subsystems to play this role? Was the world wave function waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer for some more highly qualified measurer - with a Ph.D.? If the theory is to apply to anything but idealized laboratory operations, are we not obliged to admit that more or less 'measurement-like' processes are going on more or less all the time more or less everywhere? Is there ever then a moment when there is no jumping and the Schrodinger equations applies?

    The concept of 'measurement' becomes so fuzzy on reflection that it is quite surprising to have it appearing in physical theory at the most fundamental level.
    ...

    The problem is this: quantum mechanics is fundamentally about 'observations'. It necessarily divides the world into two parts, a part which is observed and a part which does the observing. The results depend in detail on just how this division is made, but no definite prescription for it is given. All that we have is a recipe which, because of practical human limitations, is sufficiently unambiguous for practical purposes. So we may ask with Stapp: 'How can a theory which is fundamentally a procedure by which gross macroscopic creatures, such as human beings, calculate predicted probabilities of what they will observe under macroscopically specified circumstances ever be claimed to be a complete description of physical reality?'. Rosenfeld makes the point with equal eloquence: '... the human observer, whom we have been at pains to keep out of the picture, seems irresistibly to intrude into it, since after all the macroscopic character of the measuring apparatus is imposed by the macroscopic structure of the sense organs and the brain. It thus looks as if the mode of description of quantum theory would indeed fall short of ideal perfection to the extend that it is cut to the measure of man.'​
    I daresay Bell did not find the "statistical/ensemble" interpretation to solve much of anything.
     
  19. Sep 2, 2009 #18
    Re: measurement

    Which is not true. You can formally write down an observable corresponding to some macroscopic superposition. It can be shown that such observables can be measured in principle, but not in practice.
     
  20. Sep 2, 2009 #19
    Re: measurement

    Don't get clever, mate. You know perfectly well that - unless stated otherwise - the use of phrases like "Bohm interpretation" or "Bohmian mechanics" refers to [the modern development of] his 1952 theory, which anyway in itself is just de Broglie's 1927 version with a proper theory of measurement bolted on. See Peter Holland's 1993 textbook for a full presentation of this.

    Bohm's later Indian guru musings, while interesting to some people, are of little interest to physics or even metaphysics without further qualification on his part. Which is difficult, with his being dead and all.

    And in the paragraph of mine that you quoted, I wasn't referring to Bohm, I was responding to your ludicrous assertion that 'QM has proved that particles and waves cannot both simultaneously exist' which, as you did not admit, is just wrong.
    Bohmian electrons have positions, momenta and trajectories just like classical ones. They only reason they don't behave exactly like classical particles is that they are acted upon by an additional force - the quantum force - which arises from being guided by the wave field.
    In the limiting case that the quantum force is zero (i.e. the effect of the quantum wave becomes negligible) then they believe like classical particles.
    No it doesn't. There are no entities at all in a Bohrian quantum system, or if there are, we are not permitted to know about them.

    If you are referring to the classical measuring apparatus which he presupposes exists, then that is usually thought to be a weakness of his view, as there is then no well-defined boundary between the microscopic and the macroscopic.

    In the Bohm case, the behaviour of macroscopic objects flows directly out of the theory rather than having to be presupposed, and measurements are just ordinary many-body interactions.

    Unfortunately almost all Heisenberg's pronouncements about 'hidden variable theories' are not only wrong but laughable. Do you remember this one:

    'The idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them, is impossible'.

    Your quote above is almost unique in being relatively sane. But only relatively. He's getting confused between the real thing and the mathematical object which represents it.

    The use of a wave function defined on a multi-dimensional configuration space does not imply that this space exists in the same sense that the physical three-dimensional space may be said to exist. (Remember even in classical mechanics we tend to use a configuration space description).

    In classical mechanics the config space representation is just a convenient summary of the positions of all the particles; in QM the situation is different because the physics is different - there is the possibility of entanglement due to non-local interactions. So a simply-connected 3d space alone cannot describe the holistic quantum connectiveness and nonlocality features of multi-particle quantum systems. Instead this is done formally by employment of the n-dimensional config space.

    The problem with such a space actually existing are considerable, and include:

    (1) needing at least 3 separate dimensions for every particle in the universe

    (2) the total number of dimensions in the universe varying from moment to moment along with the creation and annihilation of particles.

    (3) the extra dimensions always being completely unnoticeable at macroscopic scales, and

    (4) a complete lack of any experimental evidence for the existence of multi-dimensional physical spaces.

    An additional point is that we do not currently know the 'means' by which quantum non-local connections are actualized. This is not because of the non-relativistic context since non-locality is also present in relativistic versions of QM.

    Given the strong reasons against taking multi-dimensional space as real, the vast amount of evidence in favour of physically real wave fields, and the absence of information about the 'means' of non-local connections, it is a coherent position to take the wave function to be a mathematical representation of a real field in physical space.

    The notion of an n-particle system is described in Bohm theory by its trajectory which is traced out in 3n-dimensional config space. Even though this description is given using a multi-dimensional space, the motion of individual particles can be calculated since there is a natural mapping from the system's trajectory in 3n-dimensional space to trajectories in 3d space.

    Maybe when we have discovered (or developed a model of) the 'means' by which quantum non-local connections are actualized then we will be able to describe the wave field in physical 3d space. That'll be the day.
    Bugger Bohm. His original theory (which was de Broglie's anyway) lives without him, and is a living disproof of all the mystical nonsense that pervades quantum mechanics. By showing us that QM can be taken to be just classical statistical mechanics with an extra force (and therefore a quantum rather than a classical dynamics) he shows up the supposedly definitive pronouncements of Bohr and Heisenberg (particularly Heisenberg) as being simply due to a lack of imagination.

    And as for your statement - the opposite is the truth. Ordinary Bohmian mechanics is located in space-time. And it does involve classical particles, if by classical you mean 'objectively existing' (as I think you do) rather than 'particles following classical Newtonian trajectories' (which surely you don't).
     
    Last edited: Sep 2, 2009
  21. Sep 2, 2009 #20
    Re: measurement

    What quantum states are "superpositions" depends on which basis one chooses for the Hilbert space. A definite |x-spin = +1/2> is just the superposition of |z-spin = -1/2> and |z-spin = +1/2> states. And surely the |x-spin = +1/2> state is physically meaningful.
     
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