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Quantum mechanics

This is a discussion on Quantum mechanics within the General Discussions forums, part of the Community Boards category; Originally Posted by MutantJohn I'm not sure if there are any mathematical uncertainties in Fourier analysis considering you can integrate ...

  1. #16
    Guest Sebastiani's Avatar
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    Quote Originally Posted by MutantJohn
    I'm not sure if there are any mathematical uncertainties in Fourier analysis considering you can integrate it over a continuous domain of any range. I would like to hear more about this.
    What I mean is that that Fourier analysis is simply a tool of human convention. Using trigonometric functions, we try to break up a waveform into it's spectral components. Analyzing duals (such as position and momentum) yields ambiguous results due to the simple fact that they are complementary and thus non-commutative. Heisenberg saw this as the *source* of uncertainty in the universe! Who are we to say that the universe depends on Fourier transforms though? Perhaps some other mathematical construct may arise which effectively eliminates the need for Fourier analysis altogether (just as quaternions eliminate the singularities encountered when working with Euler angles).

    Quote Originally Posted by MutantJohn
    I would also like to say that, Sebastiani, I'm not sure your post explains measuring position and momentum simultaneously. I'm talking about operators, not people in a lab.
    Exactly - the latter is what we should *really* be talking about here! Mathematical operators are mere human constructs that have no place in the formulation of a physical law. It's a procedural issue, nothing more...

  2. #17
    Registered User MutantJohn's Avatar
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    I know this is probably going to make you very angry because you seem very adamant but just reading the intro on the wiki, it seems to disagree with what you're talking about.

    Historically, the uncertainty principle has been confused[6][7] with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.[8] It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[4] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.[9] It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.[10]
    And I do trust the physicists to maintain the page.

    I'm confused though. What is it exactly that you're arguing because most of your argument seems to stem from people physically measuring things when I keep referring to the theoretical equations which as far as I can tell have not yet been disproven.

    Edit : I guess the real thing you seem to be arguing is that momentum states are conjugate variables to position because we made them that way but I think they just happen to be conjugate to each other and it's for that reason the uncertainty is intrinsic.

    Edit edit : I think this should help : http://physics.stackexchange.com/que...iable-of-posit
    Last edited by MutantJohn; 08-06-2014 at 02:59 PM.

  3. #18
    Guest Sebastiani's Avatar
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    Quote Originally Posted by MutantJohn
    I know this is probably going to make you very angry because you seem very adamant but just reading the intro on the wiki, it seems to disagree with what you're talking about.

    And I do trust the physicists to maintain the page.

    I'm confused though. What is it exactly that you're arguing because most of your argument seems to stem from people physically measuring things when I keep referring to the theoretical equations which as far as I can tell have not yet been disproven.
    Right, so in essence it states that Heisenberg produced a flawed theory that was only later refined for rigor (remember, the "observer effect" was Heisenberg's first formulation of the Uncertainty Principle).

    Quote Originally Posted by MutantJohn
    Edit : I guess the real thing you seem to be arguing is that momentum states are conjugate variables to position because we made them that way but I think they just happen to be conjugate to each other and it's for that reason the uncertainty is intrinsic.
    No! Remember, the underlying question here is about the uncertainty involved in measurement. Right? Position and momentum are conjugates - so what? Does that mean that we can never-ever-ever measure both at once? Who knows, clever techniques may one day make it possible. Why rule it out on the grounds that our *current* math yields an ambiguous duality? My point is simply that HUP is no law of physics; it's merely a law of the limitations of certain analytical methods, pure and simple...

  4. #19
    (?<!re)tired Mario F.'s Avatar
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    Quote Originally Posted by Sebastiani View Post
    Remember, the underlying question here is about the uncertainty involved in measurement. Right? Position and momentum are conjugates - so what? Does that mean that we can never-ever-ever measure both at once? Who knows, clever techniques may one day make it possible. Why rule it out on the grounds that our *current* math yields an ambiguous duality? My point is simply that HUP is no law of physics; it's merely a law of the limitations of certain analytical methods, pure and simple...
    Your point is not being accepted simply because you have no basis for saying it. Just your futuristic hunch. The Uncertainty Principle, particularly the observer effect, is a theory. Like any theory, it exists in a state of constant scientific review. No one in here or anywhere else holds it as a law. It's unclear why you think it is being held as a law. And being a student of sciences, you should know better. But you commit the exact same sin, in reverse, by stating it is not an acceptable theory when we have been observing it in effect more often than we have been able to witness the exact opposite of hat Heisenberg initially stated.

    Likewise, Heisenberg initial formulation of the theory was reviewed by himself too. If you know anything about the early 20th century, you should know that Quantum Physics was an emerging branch, still infant, going through a rapid cycle of discoveries and lacking much of the technology required to produce empirical evidence of the mathematical formulations and where discussions often resorted to philosophy constructs like thought experiments. Saying that Heisenberg produced a flawed theory is like saying that Newton produced a flawed theory because someone later realized it couldn't be applied to very large and massive objects. Or that Einstein produced a flawed theory because it invariably breaks at the quantum level.
    Last edited by Mario F.; 08-07-2014 at 08:37 AM.
    The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
    The programmer comes home with 12 loaves of bread.


    Originally Posted by brewbuck:
    Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.

  5. #20
    Guest Sebastiani's Avatar
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    Quote Originally Posted by Mario F. View Post
    Your point is not being accepted simply because you have no basis for saying it. Just your futuristic hunch. The Uncertainty Principle, particularly the observer effect, is a theory. Like any theory, it exists in a state of constant scientific review. No one in here or anywhere else holds it as a law. It's unclear why you think it is being held as a law. And being a student of sciences, you should know better. But you commit the exact same sin, in reverse, by stating it is not an acceptable theory when we have been observing it in effect more often than we have been able to witness the exact opposite of hat Heisenberg initially stated.

    Likewise, Heisenberg initial formulation of the theory was reviewed by himself too. If you know anything about the early 20th century, you should know that Quantum Physics was an emerging branch, still infant, going through a rapid cycle of discoveries and lacking much of the technology required to produce empirical evidence of the mathematical formulations and where discussions often resorted to philosophy constructs like thought experiments. Saying that Heisenberg produced a flawed theory is like saying that Newton produced a flawed theory because someone later realized it couldn't be applied to very large and massive objects. Or that Einstein produced a flawed theory because it invariably breaks at the quantum level.
    I'm not going to argue the semantic differences between "law" and "theory" here, they are often enough interchangeable, so I'll just leave it at that. Anyway, your comparison is way off. All physical laws are approximations. There will never be such thing as "a complete law of physics" simply because it would require an infinite level of refinement to achieve such a thing (that is to say, the laws of physics are infinite). Einstein's laws didn't invalidate Newton's, they just refined and extended them to a larger domain. HUP, on the other hand, says nothing about the fundamental rules governing the universe. Rather, it is nothing more than a statement about the technology and methods used by us human beings to make sense of what we observe.

    Funny thing is, we sit here arguing nonsensically with eachother just as others have in the past. I sincerely doubt we're going to be able to reconcile the two camps in this contentious debate here, so maybe we should just "agree to disagree"?
    Code:
    #include <cmath>
    #include <complex>
    bool euler_flip(bool value)
    {
        return std::pow
        (
            std::complex<float>(std::exp(1.0)), 
            std::complex<float>(0, 1) 
            * std::complex<float>(std::atan(1.0)
            *(1 << (value + 2)))
        ).real() < 0;
    }

  6. #21
    Registered User MutantJohn's Avatar
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    Sebastiani, you seem too dogmatic for me to consider you a good scientist which is a shame because you seem incredibly intelligent and well-versed in this.

    I only say this because you're just saying these things and you're so adamant about it. This has to lead me to ask, do you have any proof? The scientific method would require that at some point, these claims are made valid otherwise they are just mere conjecture. If you think the wave equation is wrong then I would like some counter-evidence because saying the HUP is wrong is logically equivalent to saying the wave equation itself is wrong.

  7. #22
    Guest Sebastiani's Avatar
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    Quote Originally Posted by MutantJohn
    This has to lead me to ask, do you have any proof? The scientific method would require that at some point, these claims are made valid otherwise they are just mere conjecture. If you think the wave equation is wrong then I would like some counter-evidence because saying the HUP is wrong is logically equivalent to saying the wave equation itself is wrong.
    You've misunderstood me. Does HUP describe a limitation found in the various QM methods in use today? Yes! Does it describe a natural limitation of the universe? I don't know, but I'm not going to be so arrogant as to assume that it does or that some new QM method won't come along someday that effectively mitigates HUP.

    At the end of the day, you still have to deal with HUP either way, whether you consider it a natural law or just a mere mathematical constraint. All of this philosophical debate is really only useful in the context of interpretations and the ramifications thereof on the field of physics.
    Code:
    #include <cmath>
    #include <complex>
    bool euler_flip(bool value)
    {
        return std::pow
        (
            std::complex<float>(std::exp(1.0)), 
            std::complex<float>(0, 1) 
            * std::complex<float>(std::atan(1.0)
            *(1 << (value + 2)))
        ).real() < 0;
    }

  8. #23
    Guest Sebastiani's Avatar
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    By the way, here's a statement made by Niels Bohr (himself a proponent of HUP, of course) that might help you understand where I'm coming from:

    "Heisenberg concluded that these uncertainties or imprecisions in the measurements were not the fault of the experimenter, but fundamental in nature and are inherent mathematical properties of operators in quantum mechanics arising from definitions of these operators."

    Anyway, I think Einstein sums up my argument quite well:

    "Concepts that have proven useful in ordering things easily achieve such an authority over us that we forget their earthly origins and accept them as unalterable givens. Thus they come to be stamped as 'necessities of thought,' 'a priori givens,' etc."

    Amen, brother!
    Last edited by Sebastiani; 08-07-2014 at 02:13 PM.
    Code:
    #include <cmath>
    #include <complex>
    bool euler_flip(bool value)
    {
        return std::pow
        (
            std::complex<float>(std::exp(1.0)), 
            std::complex<float>(0, 1) 
            * std::complex<float>(std::atan(1.0)
            *(1 << (value + 2)))
        ).real() < 0;
    }

  9. #24
    Registered User MutantJohn's Avatar
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    Yeah, and Newton believed in alchemy.
    Mario F. likes this.

  10. #25
    (?<!re)tired Mario F.'s Avatar
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    Quote Originally Posted by Sebastiani View Post
    By the way, here's a statement made by Niels Bohr (himself a proponent of HUP, of course) that might help you understand where I'm coming from:

    "Heisenberg concluded that these uncertainties or imprecisions in the measurements were not the fault of the experimenter, but fundamental in nature and are inherent mathematical properties of operators in quantum mechanics arising from definitions of these operators."

    Anyway, I think Einstein sums up my argument quite well:

    "Concepts that have proven useful in ordering things easily achieve such an authority over us that we forget their earthly origins and accept them as unalterable givens. Thus they come to be stamped as 'necessities of thought,' 'a priori givens,' etc."

    Amen, brother!
    As chance decided, a couple of months ago I finished reading The complete Road to Reality, by Roger Penrose (ISBN 0224044478). There's an important chapter in the book that addresses your concerns, that I would I'd like to put here:

    "In the development of mathematical ideas, one important initial driving force has always been to findnd mathematical structures that accurately mirror the behavior of the physical world. But it is normally not possible to examine the physical world itself in such precise detail that appropriately clear-cut mathematical notions can be abstracted directly from it. Instead, progress is made because mathematical notions tend to have a ‘momentum’ of their own that appears to spring almost entirely from within the subject itself. Mathematical ideas develop, and various kinds of problem seem to arise naturally. Some of these (as was the case with the problem of finding the length of the diagonal of a square) can lead to an essential extension of the
    original mathematical concepts in terms of which the problem had been formulated. Such extensions may seem to be forced upon us, or they may arise in ways that appear to be matters of convenience, consistency, or mathematical elegance. Accordingly, the development of mathematics may seem to diverge from what it had been set up to achieve, namely simply to reflect physical behaviour. Yet, in many instances, this drive for mathematical consistency and elegance takes us to mathematical structures and concepts which turn out to mirror the physical world in a much deeper and more broad-ranging way than those that we started with. It is as though Nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought."
    He then proceeds to give examples in the context of the real-number system. A few paragraphs later he briefly discusses this thought in the context of quantum mechanics:

    Yet we may still ask whether the real-number system is really ‘correct’ for the description of physical reality at its deepest levels. When quantum mechanical ideas were beginning to be introduced early in the 20th century, there was the feeling that perhaps we were now beginning to witness a discrete or granular nature to the physical world at its smallest scales. Energy could apparently exist only in discrete bundles —or ‘quanta’— and the physical quantities of ‘action’ and ‘spin’ seemed to occur only in discrete multiples of a fundamental unit. However, as we now understand quantum mechanics, that theory does not force us (nor even lead us) to the view that there is a discrete or granular nature to space, time, or energy at its tiniest levels. [...] Nevertheless, the idea has remained with us that there may indeed be, at root, such a fundamental discreteness to Nature, despite the fact that quantum mechanics, in its standard formulation, certainly does not imply this. For example, the great quantum physicist Erwin Schroedinger was among the first to propose that a change to some form of fundamental spatial discreteness might actually be necessary:

    "The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is accessible to us" -- Schroedinger
    Finally, I would like to expose you to an Einstein quote, since you seem so eager to refer to him in this discussion:

    "One can give good reasons why reality cannot be represented as a continuous field. Quantum phenomena must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory."



    ...

    Your argument against an unknown mathematical construct or property isn't acceptable, Sebastiani. No one since Heisenberg or before him, for whom science has been their cradle, will ever attribute to magic or little goblins the unknowns they tend to discover along their path. Your faith in mathematics should, it too, be tempered. We have been expanding our mathematical knowledge along our history exactly because we have been faced with realities that forced us to create new formal systems that better describe the physical world. Other times (and more often) we just "stumble" upon a new formal system naturally as a consequence of new discoveries.

    To think that Heisenberg left to small mathematical goblins his discoveries is wrong. Quantum physics emerged almost entirely out of purely mathematical extrapolations, as there was little in the way of technology to produce empirical evidence. He was first and foremost a mathematician. And he stumbled upon a problem that even his contemporaries, among them Einstein, clearly understood to be something new and potentially changing in the mathematical field. He was awarded much more respect by physicists to this day than what you have done here. Even on the matter of the observation effect.

    And even though our understanding of quantum mechanics has evolved in the 90 years since, there's still much left to do that does not remove the idea that indeed we may need a new formal system to describe reality at the quantum level. Heisenberg mathematical goblins haven't been discarded. And it is this thought, this way to look at science, this complete refusal to close our eyes to new thoughts and ideas that makes science move forward. Not close our eyes and mock the past. Your beloved Einstein would agree to this.
    The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
    The programmer comes home with 12 loaves of bread.


    Originally Posted by brewbuck:
    Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.

  11. #26
    Guest Sebastiani's Avatar
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    Quote Originally Posted by Mario F. View Post
    As chance decided, a couple of months ago I finished reading The complete Road to Reality, by Roger Penrose (ISBN 0224044478). There's an important chapter in the book that addresses your concerns, that I would I'd like to put here:



    He then proceeds to give examples in the context of the real-number system. A few paragraphs later he briefly discusses this thought in the context of quantum mechanics:



    Finally, I would like to expose you to an Einstein quote, since you seem so eager to refer to him in this discussion:

    "One can give good reasons why reality cannot be represented as a continuous field. Quantum phenomena must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory."



    ...

    Your argument against an unknown mathematical construct or property isn't acceptable, Sebastiani. No one since Heisenberg or before him, for whom science has been their cradle, will ever attribute to magic or little goblins the unknowns they tend to discover along their path. Your faith in mathematics should, it too, be tempered. We have been expanding our mathematical knowledge along our history exactly because we have been faced with realities that forced us to create new formal systems that better describe the physical world. Other times (and more often) we just "stumble" upon a new formal system naturally as a consequence of new discoveries.

    To think that Heisenberg left to small mathematical goblins his discoveries is wrong. Quantum physics emerged almost entirely out of purely mathematical extrapolations, as there was little in the way of technology to produce empirical evidence. He was first and foremost a mathematician. And he stumbled upon a problem that even his contemporaries, among them Einstein, clearly understood to be something new and potentially changing in the mathematical field. He was awarded much more respect by physicists to this day than what you have done here. Even on the matter of the observation effect.

    And even though our understanding of quantum mechanics has evolved in the 90 years since, there's still much left to do that does not remove the idea that indeed we may need a new formal system to describe reality at the quantum level. Heisenberg mathematical goblins haven't been discarded. And it is this thought, this way to look at science, this complete refusal to close our eyes to new thoughts and ideas that makes science move forward. Not close our eyes and mock the past. Your beloved Einstein would agree to this.
    I'm starting to get the feeling that noone here actually understands the crux of my argument. Well, we've all made our points, let's just move on then...

    Looks like a good book, by the way.
    Code:
    #include <cmath>
    #include <complex>
    bool euler_flip(bool value)
    {
        return std::pow
        (
            std::complex<float>(std::exp(1.0)), 
            std::complex<float>(0, 1) 
            * std::complex<float>(std::atan(1.0)
            *(1 << (value + 2)))
        ).real() < 0;
    }

  12. #27
    Registered User MutantJohn's Avatar
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    Wait, what was the crux of your argument again? Something about math being a human invention and physics being this thing we can never fully understand?

  13. #28
    Guest Sebastiani's Avatar
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    Quote Originally Posted by MutantJohn View Post
    Wait, what was the crux of your argument again? Something about math being a human invention and physics being this thing we can never fully understand?
    You don't agree with me, fine, but the least you could do is actually formulate an intelligent counterargument, Jackass.
    Code:
    #include <cmath>
    #include <complex>
    bool euler_flip(bool value)
    {
        return std::pow
        (
            std::complex<float>(std::exp(1.0)), 
            std::complex<float>(0, 1) 
            * std::complex<float>(std::atan(1.0)
            *(1 << (value + 2)))
        ).real() < 0;
    }

  14. #29
    Registered User MutantJohn's Avatar
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    That actually wasn't meant to be a counterargument lol. But I could see how it could be interpreted to be a smartass remark. That's the internet for you.

    But the thing is, I'd be careful about calling these operators fully man-made constructs because the position operator is relatively simple. And the momentum one is a very basic one as well as it's just a derivative. The only reason these operators don't commute is because of the structure of the wave function itself.

    I wasn't ever trying to be a d***. I'm just saying, let's slow it all down and read some literature. What do other people in the community think? Has there been any recent literature on this?

    Recently, people have attempted to create formal proofs for the Born rule which I think would be worth investigating if I could actually understand what they were talking about lol.

  15. #30
    Guest Sebastiani's Avatar
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    Quote Originally Posted by MutantJohn
    That actually wasn't meant to be a counterargument lol. But I could see how it could be interpreted to be a smartass remark. That's the internet for you.
    Sorry, yeah I've been crabby all day.

    Quote Originally Posted by MutantJohn
    But the thing is, I'd be careful about calling these operators fully man-made constructs because the position operator is relatively simple. And the momentum one is a very basic one as well as it's just a derivative. The only reason these operators don't commute is because of the structure of the wave function itself.
    Quantum operators are constructs of QM, and non-commutativity is most certainly an artifact of the mathematics involved.

    Quote Originally Posted by MutantJohn
    I wasn't ever trying to be a d***. I'm just saying, let's slow it all down and read some literature. What do other people in the community think? Has there been any recent literature on this?
    First of all, let me just reiterate that Heisenberg's original formulation, the Measurement-Disturbance Relationship (MDR), appears to be false as articulated by Heisenberg (see the work of Rozema, et al, Ozawa, and others).

    Second, you aren't going to find much in the way of criticisms of HUP (which is really due to Earle Kennard, not Heisenberg) simply because it is 100% valid in terms of QM methods; without it, the math breaks down.

    Quote Originally Posted by MutantJohn
    Recently, people have attempted to create formal proofs for the Born rule which I think would be worth investigating if I could actually understand what they were talking about lol.
    What's there to prove? It's been verified experimentally, so why bother? The rule is pretty simple, anyway. It really just boils down to this: given some wave-function (basically just a vector in Hilbert space), the probability of obtaining a particular eigenvalue (result) from a measurement is the amplitude (associated with that eigenvalue) squared.
    Code:
    #include <cmath>
    #include <complex>
    bool euler_flip(bool value)
    {
        return std::pow
        (
            std::complex<float>(std::exp(1.0)), 
            std::complex<float>(0, 1) 
            * std::complex<float>(std::atan(1.0)
            *(1 << (value + 2)))
        ).real() < 0;
    }

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