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Dividing line between QM and CM?

  1. Feb 12, 2014 #1
    Hey guys,

    I study biophysics, so maybe this is a sophomoric question. I've done a cursory, albeit fruitless search of the forums and the web, I think I don't know how to properly ask the question...

    I was wondering if there is some sort of limit -- I would assume a given mass of an object -- where we would expect to see the quantum mechanical effects on the object begin to diminish, and classical mechanics begin to make more sense?

    I would imagine the QM laws apply irregardless of mass, but I suspect/know this is not true for observed effects. IE: I have to worry about the superposition of a very small particle, but not a massive one.

    Can anyone enlighten me? I'm not scared of math.

  2. jcsd
  3. Feb 12, 2014 #2


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    The answer is often given via the formula [itex] \lambda=\frac{h}{p} [/itex] which gives the wavelength associated to a particle with momentum p. As the momentum of the particle gets larger, its wavelength(Maybe its better to say the mid-wavelength of its wavelength spectrum or any other representative of its wavelength spectrum) gets smaller and so the quantum effects become harder to observe and the particle becomes more classical.But its not like Lorentz transformations that you have an equation for which you can take the limit and get the classical formulas.Its somewhat qualitative in this way.
    But if you really want to see the transition,you should understand Ehrenfest theorem.
    Last edited: Feb 12, 2014
  4. Feb 12, 2014 #3
    Makes sense. So in practice, is there a rough dividing between QM and CM in terms of λ?

    Obviously if I tried to use a beam of coconuts for the double-slit experiment, their λ would be too small and we would expect classical results.

    I guess I'm looking for a ballpark, qualitative sort of idea.
  5. Feb 12, 2014 #4


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    The point is that nowhere in physics you can find such "rough dividing"s!
    It all depends on the desired accuracy. A problem can be encountered by two groups working in different fields.One of them may need a low enough accuracy to let them use classical mechanics but the other may want a level of accuracy which only using QM can give them.Its the same about other theories and also numerical and analytical approximations.
    So the only thing I can say is, in this respect,you can divide physical phenomena into three groups.Group 1:Phenomena for which everyone uses classical physics because quantum effects are practically unobservable for them.Group 2:Phenomena for which everyone uses QM because classical physics is too wrong for them! Group 3:Phenomena which lie between the first two groups and both classical and quantum physics can be used for them and only the application and desired accuracy tells which one should be used.Even the range of such phenomena isn't sharp!
  6. Feb 12, 2014 #5


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    Conceptually it's much more complicated than what's being implied here. I don't know if you care about all the conceptual subtleties involved in the classical limit of QM (think Schrodinger's cat) so I won't mention anything further, in avoidance of undesired complication.

    That being said, the general idea is that the center of mass motion of a macroscopic body can be described by a Gaussian wave packet sufficiently localized in both position and momentum (since the spreading of the wave packet due to Schrodinger's equation will be negligible for macroscopic bodies we can on these length scales find approximate simultaneous eigenstates of both position and momentum on the order of said length scales) so that the classical phase space trajectory of the center of mass can then be described by the Ehrenfrest relations for the time evolution of the ensemble average for both the position and momentum operators in formal analogy with Hamilton's equations. But as noted this is not without its conceptual issues.

    EDIT: Chapter 6 of Shankar "Principles of Quantum Mechanics" has a very detailed discussion of the classical limit in terms of the Ehrenfest relations for a Gaussian wave packet but again the conceptual issues are completely sidestepped.
    Last edited: Feb 12, 2014
  7. Feb 12, 2014 #6
  8. Feb 12, 2014 #7

    “Neither the statement 'this diamond is vibrating' nor 'this diamond is not vibrating' is true.”

    Logic fail.

    But i give them credit for trying to circumvent saying that the diamons is both vibrating and not vibrating.
  9. Feb 12, 2014 #8
    Well I put it to you, seeing you seem to be questioning a lot of Logic, how to describe the diamonds when they're entangled.
  10. Feb 12, 2014 #9


    Staff: Mentor

    Its actually got nothing to do with size - all objects show quantum effects. There is a technological limit on detecting it depending on size - but that is a moving target depending on exactly where we are technologically - however as a matter of principle its there always regardless of size.

    What its got to do with is interaction with the environment which via decoherence converts superpositions to improper mixed states giving the appearance of definite properties, and is how the classical world emerges.

    Depending on your background the books to get are:

    Omnes book is at the lay or beginning QM level.
    Schlosshauer is THE book on the issue if you have a good background in QM.

    Last edited by a moderator: May 6, 2017
  11. Feb 12, 2014 #10
  12. Feb 13, 2014 #11

    Well, it's been customary to assume that quantum states of elementary particles reflect indefinite knowledge on part of the observer(i.e. states are not real in any sense) but this is about states of classical diamonds, i.e. classical matter? While it may make some sense to some that a quantum particle has no definite position outside the measurement context(i.e. does not exist prior to measurement) , it makes no sense to sense to say that a diamond cannot be said to be vibrating while it's vibrating(here we must acknowldge some reality to diamonds being classical - we separate the world into classical and quantum stuff to make the theory work anyway).
  13. Feb 13, 2014 #12
    So you're saying that there is a divide between QM and CM, that macroscopic objects don't obey Quantum Theory?
  14. Feb 13, 2014 #13
    I don't think that is the case. It's a bit like trying to make a distinction between relativistic and non relativistic objects. Relativity applies to all of them it's just that the effects are not very noticeable for the non relativistic ones. In QM if the energies (and characteristic frequencies) are very high it is difficult to observe interference effects on a macroscopic scale.
  15. Feb 13, 2014 #14

    That's the standard treatment for getting definite outcomes and it obviously leaves a lot to be desired.

    Mass seems to be a fundamental factor in 'classicality' and it's visible in the relationship between momentum and wavelength.
    Last edited: Feb 13, 2014
  16. Feb 13, 2014 #15


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    There's a huge conceptual difference between the two so the analogy isn't really fair. Relativity and Newtonian mechanics are both classical theories so the formal mathematical limit isn't divorced from the conceptual aspects of the limit. The same is obviously not true of the classical limit of QM wherein the classical limit via Ehrenfest's theorem engenders a divide between mathematics and concept. While we can easily set up Gaussian wave packets whose centroids mathematically satisfy an average form of Hamilton's equations through the Ehrenfest relations so as to yield an approximately classical definite phase space trajectory of the centroid or in other words a set of centroid trajectories associated with a classical ensemble, and make sense of this in the limit as characteristic length scales get large, it's clearly not a sufficient limiting scheme because we can easily have macroscopic systems exhibiting quantum behavior by means of entanglement. The degree of entanglement and the potentiality to decohere through environmental interactions provides a better characteristic scale of the quantum/classical tendencies of a system.
  17. Feb 13, 2014 #16
    Indeed. The smaller and faster objects have a longer wavelength and exhibit more quantum effects than the heavier slower ones. There is probably a really useful formula as to what the relationship has to be between between the rest mass and the momentum so that quantum effects become important.

    Any one? ........
  18. Feb 13, 2014 #17
    It's p=m.v
  19. Feb 13, 2014 #18
    Wouldn't h need to be in there somewhere too?
  20. Feb 13, 2014 #19

    Yes indeed, p=m.v is useful only for massive particles.
    Last edited: Feb 13, 2014
  21. Feb 13, 2014 #20
    Definitely. My stat mech teacher made a big deal about this, but unfortunately I don't remember the specifics. We were usually interested in ##\lambda## vs particle separation, but he also gave a very nice (not always rigorous) way of testing for "quantumness". It had something to do with a limit as ##\hbar## goes to zero, but I don't really remember since it is usually fairly obvious where the line is at the Undergrad level. Since ##\hbar## is an action, I want to say that it had to do with a Lagrangian limit, but I really don't remember (it made so much sense that I didn't write it down!)

    logic isn't as easy when the real world is involved.
  22. Feb 13, 2014 #21
    If there are no takers... I'll give it a go!
    Kx -wt > pi.
    Px-Et >h (approx.)

    In the non-relativistic limit:
    Mvx - mc^2 x/v> h
    Mc^2 x/v (v^2/c^2 - 1)>h

    Where x is the scale on which the effects are being observed. Apologies I don't have Latex.
    I'm braced to be shot down in flames, but it looks about right. It can be massively improved of course by making the distance along the axis of motion much longer than the transverse axis.
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