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Discrete Wave Functions

  1. Jan 28, 2014 #1
    Does anything connect the discrete wave functions? I thought they were suppose to be connected.
     
  2. jcsd
  3. Jan 28, 2014 #2

    bhobba

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    I cant quite grasp what you mean so will take a guess.

    You are asking how is it possible to jump between stationary discreet states.

    The answer is when they are perturbed they are no longer stationary so have a probability of changing states. This is the phenomena of stimulated emission and absorption. Spontaneous emission requires QFT to explain it - intuitively quantum vacuum fluctuation perturb it.

    If that isn't it please clarify what you are asking and myself or others may be able to help.

    Thanks
    Bill
     
  4. Jan 28, 2014 #3
    I know this might not seem accurate and for all I know it might not be, but the unified field should be connecting those waves somehow. Isn't it made up of all the same stuff? From what I gathered those wave function are just discrete places where a particle could be. But isn't everything a wave? Maybe it's just a measurement of where they could be?
     
  5. Jan 28, 2014 #4

    bhobba

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    Ok before proceeding I think I need to re-post something I posted in another thread to explain exactly what QM is about. Once you grasp that then we can proceed to answer your queries - if you still have them.

    QM is a theory about what happens when something is observed, not about when it's not being observed. What's going on when not observed the theory is silent about. We have speculations on that (called interpretations) but until experiment can decide they are just that - conjectures

    Here is the conceptual core of QM - forget about stuff you have read elsewhere - this is its core:
    http://www.scottaaronson.com/democritus/lec9.html
    'So, what is quantum mechanics? Even though it was discovered by physicists, it's not a physical theory in the same sense as electromagnetism or general relativity. In the usual "hierarchy of sciences" -- with biology at the top, then chemistry, then physics, then math -- quantum mechanics sits at a level between math and physics that I don't know a good name for. Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn't yet been successfully ported to this particular OS). There's even a word for taking a physical theory and porting it to this OS: "to quantize."

    You probably have heard of the double slit experiment. Normally it's used to motivate the QM formalism, but really it should be the other way around - QM should explain it - and its a good application of that conceptual core to explain it:
    http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

    Of course the answer is expressed in the language of mathematics - sorry but physics is about mathematical models. Explaining this stuff without math is unfortunately well nigh impossible. I often answer queries on this sort of stuff and mostly it ends up with - the jig is up - you cant go further until you study and understand the technical details and the math.

    Basically QM is a variant on standard probability theory that allows continuous transformations between so called pure states:
    http://arxiv.org/pdf/quant-ph/0101012.pdf

    Consider flipping a coin. Probability theory describes the frequency of outcomes - but not what causes each outcome. Same with QM - it describes the frequency of outcomes - but not what causes any outcome. We simply do not know what that is - or even if there is a cause - nature may simply be like that. But regardless the QM formalism is silent about it. We have interpretations that speculate about it - but until there is some way to decide experimentally they are simply conjectures.

    The 'waves' of QM are not waves like water waves - they are not physical and composed of actual 'stuff', they are waves of probability.

    Thanks
    Bill
     
    Last edited: Jan 28, 2014
  6. Jan 28, 2014 #5
    Thank you that helped a lot!
     
  7. Jan 28, 2014 #6

    ChrisVer

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    OP: what do you mean by unified field?
    The wavefunctions are the solutions of your Hamiltonian eigenvalue problem (let's say Schrodinger's equation). Depending on the basis that you choose to work, they are interpreted as probability functions. In the position basis, it gives you the probability of a particle being in a region in space...
    Not everything is a wave- there is the duality. We can "see" for example the electrons and atoms, although we need the QM wavefunctions to describe them.

    PS. Well personally I find the quote of what is QM was a disappointment I guess....
    I don't understand how someone could put QM somewhere between physics and mathematics. Of course it contains mathematics, because in that language physicists speak in, but it's a theory based and built on experimental data (we were observing something and we needed to describe it).
    In my interaction with it, I find it more physical than any other physical theory...
     
  8. Jan 28, 2014 #7

    bhobba

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    Its basically a variant of probability theory.

    Probability theory is like that as well. One can treat it purely mathematically via the Kolmogorov axioms but when you apply it you need to make certain reasonableness assumptions (without detailing what they are) so you get sensible results such as something with probability 1 always occurs.

    That's why its halfway between physics and math. One can apply probability to say finance as well and in doing that its halfway between math and finance and so on.

    Euclidean geometry is similar as well - as its usually presented points have position and no size etc so its not entirely mathematical, nor is it entirely physical - its sort of halfway. Hilbert showed how to make it entirely mathematical, but then applying it is not so clear.

    Another way to look at it is via mathematical models.

    Mathematical models usually contain all sorts of mathematical 'ingredients' - probability, geometry etc etc. Those ingredients are sort of halfway between mathematics and whatever you apply it to.

    Thanks
    Bill
     
    Last edited: Jan 28, 2014
  9. Jan 28, 2014 #8

    ChrisVer

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    This would go out of the topic, it's also a matter of how someone defines the things he has. In that view though, I guess all physics are somewhere between physics and maths, because not only QM but also classical mechanics before it were using mathematics to be expressed and described. I distinguish between them with which describes somehow the world, and which is not. Again it's a matter of taste, so there's no need to continue...^_^
     
  10. Jan 28, 2014 #9

    bhobba

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    You hit it in one.

    It's simply how you view such things - take it or leave it - it doesn't really matter either way.

    Often though viewing something slightly differently is illuminating.

    In this case, we know that QM to a large extent is determined by symmetry (eg Schrodinger's equation etc etc). The fundamental core of QM (the two axioms as detailed in Ballentine, or the 5 reasonable axioms in the paper I linked to) is the thing that symmetry is applied to - like the principle of least action is the thing symmetry is applied to in Classical mechanics.

    In fact I suspect its the reason the continuous transformations between pure states in those 5 reasonable axioms is so important, and why standard probability theory cant really be used - the existence of such transformations allows much richer symmetries.

    Thanks
    Bill
     
    Last edited: Jan 28, 2014
  11. Jan 28, 2014 #10
    Thanks! That also gives rise to the next question I had. From what I've learned (from one guy at least) that everything is just a hierarchy based off waves and every level of that hierarchy has different laws. Never got how things are waves and are also particles. From what I guess, no one knows. But it would be interesting to know. Never got the duality yet unity part.
     
  12. Jan 28, 2014 #11

    atyy

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    To say so, does one have to assume that all variables are discrete? For example, can it include continuous variables like position and momentum of non-relativistic quantum mechanics?
     
  13. Jan 28, 2014 #12

    bhobba

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    Hey Atyy you got me.

    When discussing foundational issues in QM, and even in probability best to stick with finite discreet outcomes.

    This can be extended in a number of ways to continuous cases by, for example, the Rigged Hilbert space formalism where you introduce, for mathematical convenience, the dual of those finite discreet cases, which is much richer and includes weird stuff like the Dirac measure.

    Thanks
    Bill
     
  14. Jan 28, 2014 #13

    atyy

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    Actually, I was hoping you'd say we can throw away rigged Hilbert spaces, and that it's ok to think of position and momentum as discrete! To argue for the general applicability of Hardy's scheme, perhaps one could argue that a large but finite volume lattice gauge theory is consistent with all quantum mechanical observations? I think that main problem for such a view is that chiral interactions so far seem problematic in lattice gauge theory.
     
  15. Jan 28, 2014 #14

    bhobba

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    If he is telling you that then he is wrong.

    Sure things are different depending on your level.

    We have non relativistic QM where everything is modeled as a particle, but particles obeying the peculiar rules of QM. Waves are not waves in the usual sense - they are waves of 'quantum' probability of the position of the particle if you were to measure it.

    Next we have relativistic QM. This fixes up an issue with non relativistic QM. Relativity treats space and time on the same footing - but non relativistic QM has position as an operator and time as a parameter. To fix this up relativistic QM treats everything as a field so that both time and position are parameters. Particles are shown to be excitations in this field.

    Then we have string theory which treats everything as a relativistic quantum string.

    But at each level QM is the overarching paradigm ie the conceptual core applies at all levels.

    As far as we can tell today QM applies at all levels.

    Thanks
    Bill
     
  16. Jan 28, 2014 #15

    bhobba

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    In both QM and probability, finite discreet outcomes only ever occur in practice.

    Formally, without going into the terms I am about to bandy about, if you understand them great, if you don't, don't worry, you will get the drift; the following gives an overview.

    The dual of a vector space of finite vectors, but of any size, is in fact the limit of such elements under the weak topology of such a space. They can then be viewed as the idealization of a very large number of finite cases introduced for mathematical convenience.

    It would seem there is no way around it if you want to analyse cases that are really discreet, but of a large but unknown size.

    In your lattice example that would be when the spacing is so small, and the number of discreet 'points' so large, you can model it as a continuous function, which is an element of the dual - along with tons of other stuff - some of them with weird mathematical properties like the Dirac delta function.

    Thanks
    Bill
     
    Last edited: Jan 28, 2014
  17. Jan 28, 2014 #16

    ChrisVer

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    I don't know. That's the reason I find the double slit experiment illustrating- people can just see that. Or even the photon as a photon and as Electromagnetic wave in the photoelectric phenomenon.
    As for the hierarchy- hierarchy over what?
    It's the first time I hear of that... The main methods of QM always remain the same, what changes is the formalism or the problem. For example in field theories you again do the quantization by imposing the canonical commutation relations (the [itex] [p , x] [/itex] ) you had from QM- although now you have the canonical coordinates... or you deal with second quantization that is similar approach as the one someone gets by solving the QM harmonic oscillator (introducing creation/annihilation operators).
    In principle the main ideas/methods still remain- of course as I already mentioned, a lot of things also differ. Also the way you look at the results still remains - you are speaking about probabilities (maybe for a scattering process or transitions probabilities) etc...

    Even in string theory (if it really exists- but even if it doesn't it's a nice toy) you can again do the same things - canonical quantization, second quantization and stuff...

    Someone could correct me for what I'm saying, but I guess I am not wrong. At least from what I've seen so far, I keep dealing with the same and same procedures (maybe at each step you have to generalize more your way of thinking) of QM...
     
  18. Jan 29, 2014 #17

    bhobba

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    It exists all right - if it applies to our universe is the question - although there are hints it leads to some hummus number of possible universes, and the hope is one of them is ours.

    Another interesting thing about string theory, at least I have read this is the case, I certainly am no expert, is you can certainly second quantize it, but it doesn't seem to really buy you anything.

    Is this important? Who knows. Only time will tell.

    Thanks
    Bull
     
  19. Jan 29, 2014 #18

    ChrisVer

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    Well, I tend not to say that something exists until it's capable of giving predictions that can be verified. Unfortunately for many beautiful theories (apart from string theory, another example are the axions), this does not hold and I (and I think a lot of people) say that "if it's valid". Mathematically it's correct :P

    Second quantization on the strings allows you to create states out of the vacuum by acting with the creation and annihilation operators. More importantly it allows you to count states (which is helpful when you want to comprehend them as particles). More information about how this works, you can find in "Barton Zwiebach -A First Course in String Theory, Second Edition 2009" at least that was the book I used as an introduction.
     
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