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DBB for momentum

  1. Jul 4, 2010 #1
    Hello all,

    I have a question about the de Broglie-Bohm theory. I hope there are some dBB fanatics among you to answer my question.

    As you all know, the guiding equation (or pilot wave, or whatever ontology you prefer), is derived from the Schr\"odinger Equation by `completing' the $\psi$ with $Re^{Si/\hbar}$

    This results in the two first order differential equations that fix the possible paths.

    But why is position the preferred variable?
    Isn't it possible to take the Schr\"odinger equation, put it through the Fourier Machinery, and fill in the $\psi$ in the SE for momentum.
    Solve, split the real and imaginary, and you're left with a similar set of equations, but with momentum instead of position as the preferred variable.

    I did not try this (lazy), but I just imagined it to be possible.
    If it is, it is also possible to find a fitting interpretation of the equations. The biggest problem would be to make macroscopic measurements in terms of momentum. Next to that, I don't see any big problems. What does this mean for the claims the dBB holds concerning reality.

    If someone is convinced that the dBB interpretation is the `real' interpretation, and the choice of the variable is arbitrary, what does this mean for the ontological status of the objects Bohm posited? Is this ontology not just as arbitrary as the choices made in the beginning?
    (I take configuration space as a `real' existing space, or else the existence of the objects is already debateable.)

    Regards,

    Vincent
     
  2. jcsd
  3. Jul 5, 2010 #2

    Demystifier

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    VinGwurf, good question! :approve:

    In fact, you have almost answered it by yourself:
    Exactly! ALL macroscopic measurements eventually reduce to measurements of positions. That is exactly the reason why dBB works only when the positions are preferred variables.

    After all, dBB is constructed to fit experiments!
     
  4. Jul 5, 2010 #3
    Thank you Demystifier,

    I figured something like this would be the case. So position is already a preferred variable in the macroscopic realm. and to create a meaningful theory, it should at least be reducible to the realm in which we can observe objects in an unambiguous way.

    I can't think of any other variables that are good for recording measurement outcomes.

    If I find one, I'll construct a new theory.
     
  5. Jul 5, 2010 #4
    Hi VinGwurf,

    Another interesting point in dBB theory is that when you calculate/measure what is conventionally referred to as 'the momentum' you're not actually calculating/measuring the momentum of the Bohmian particle at all. This is because the usual momentum operator is derived by 'quantizing' the classical quantity mv, and there is no reason why this should correspond to the momentum of anything that actually exists (except perhaps in the classical limit).

    In non-relativistic dBB theory, both particles and the 'wave field' represented mathematically by the Schroedinger wave function are supposed to objectively exist, with the latter apparently exerting a 'force' on the former (hence no classical trajectories and the momentum operator no longer gives the correct momentum of the particle). It turns out that if you analyze it carefully, the expectation value of the usual momentum operator is one component of the stress tensor of the wave field. Hmmm..

    So what, you might say. But look at the amount of argumentation in the literature - usually based on Heisenberg's uncertainty principle - that implicitly assumes that 'momentum' means momentum and concludes that e.g. particle trajectories are impossible, or the kinetic energy cannot go to zero, or whatever. This ceases to make sense when the [tex]\Delta p[/tex] in the uncertainty principle is understood not to be the momentum of the particles at all.

    Cheers,
    Zenith
     
  6. Jul 7, 2010 #5
    Hi Zenith,

    what you say, is that what in the orthodox, or Copenhagen or whatever you want to call it-, interpretation of QM, is called momentum, is not equal to the property momentum in the classical realm. Because of quantization. In dBB, the property momentum is equal to the classical property momentum, it's just [tex]\nabla S[\tex].

    But every interpretation has its problems in translating properties from the quantum to the classical realm. In the dBB interpretation, what is referred to as position, is position in configuration space. 3N dimensional configuration space. That is something different than classical position, I assume. (this might require another post)

    How do we interpret a particle's position in configuration space?
     
  7. Jul 7, 2010 #6
    This is because we (humans) are localized in space and we think in terms of space. There is no surprise that measurement devices we build work that way. But it has no value if we think about the fundamental level of physics
     
  8. Jul 7, 2010 #7
    Or, funnily enough, in the quantum realm.
    Not so. In deBB, there is an extra force over and above the classical force (charged particles repelling each other and so forth) which is due to the wave field 'pushing on the particles'. The particles thus follow trajectories which are not the classical ones. If the classical trajectory is a straight line, the quantum trajectory might be a wavy line (etc.). The usual momentum operator based on quantizing mv therefore does not give the momentum of the particle which is following the wavy trajectory, unless the force due to the wave field is zero, which it almost never is.

    So the point is, what we call 'momentum' in orthodox QM cannot be interpreted as the momentum of anything which any interpretation postulates to exist. Just for the record, in deBB it just happens to be one component of the stress tensor of the wave field (though essentially no-one realizes this) - I've only ever seen this mentioned in Holland's deBB text book.

    Remember also that deBB adds nothing to orthodox QM - it's just what you end up with you if you stop claiming that things cease to exist when no-one is looking at them (as the 1920s logical positivism movement tried to convince everyone was inevitable, which it isn't).
    No, it's just perfectly ordinary position. The fact that the wave function is defined on configuration space just implies the introduction of forces between the particles (in this case, the non-local instantaneous forces apparently supported by Bell et al). Though let's not get into that. The latest thread on that is over a thousand posts by now - don't these people have lives? :smile:
     
  9. Jul 7, 2010 #8

    dx

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    The interpretation of momentum in 'orthodox' qm is exactly the same as in classical mechanics. If an electron is accelerated through a potential V, then its momentum is (2meV)1/2.
     
  10. Jul 8, 2010 #9

    Demystifier

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    The momentum is indeed equal to [tex]\nabla S[\tex], but it is not classical because here S does not satisfy the classical Hamilton-Jacobi equation.

    In the same way as we do it in CLASSICAL mechanics.
     
  11. Jul 8, 2010 #10

    Demystifier

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    ... provided that we know what the fundamental level is, and that we know that it is not given by particle positions. But who can say that he knows that for sure?
     
  12. Jul 8, 2010 #11
    No it isn't - it's just the expectation value of some arbitrary operator defined on the basis of a (false) analogy with classical mechanics. Now, to give that a 'meaning', you have to say exactly what it is that 2meV1/2 is the momentum of - i.e. postulate something that exists.

    So, define an 'electron' for us, please..
     
  13. Jul 8, 2010 #12


    It's what you measure(a detection event), but what is a 'particle' in Debb?
     
  14. Jul 8, 2010 #13
    That's precisely my point. Bell wrote an article called 'Against measurement' for just that reason. The word 'measurement' implies that you are revealing a pre-existing property of something that objectively exists. So I'll ask again, what is that thing, and what physical property of that thing is referred to by the word 'momentum'?

    It's perfectly OK to say 'I don't know' in response to that, but I just want to make it clear this is the correct response in orthodox QM.

    [in non-relativistic QM, a deBB particle is a continuously existing thing with properties like mass and charge that follows a trajectory defined by the Schroedinger current, and whose momentum is not given by the usual prescription.]
     
    Last edited: Jul 8, 2010
  15. Jul 8, 2010 #14

    dx

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    It is the momentum of the electron (a charged particle). In classical physics, such a particle is endowed with an immediate 'reality', but in quantum mechanics it is not that simple, since the very foundation of qm is the realization that ordinary mechanical pictures, no matter how cleverly constructed they are, cannot incroporate things such as stationary states and other 'discontinouous' processes. The existance of the quantum of action implies a fundamental limitation in the use of customary attributes of the electron like position and momentum.

    The formal apparatus of quantum mechanics consists of various symbolic procedures which talk about 'momentum operators' and so on, whose relationship with the ordinary concept of momentum is indirect. The Schrodiner rule p → i(∂/∂q) and the commutation rule pq - qp = ih refer to such symbolic procedures. However, in the description of actual experiments, one must in the end always refer to the ordinary concept of momentum. In fact, the electron was discovered within the classical frame of concepts, before quantum mechanics was understood. The only thing that qm changes is that various classical ideas like momentum and position cannot be used at the same time in the description of atomic systems.

    So before getting confused by the various abstract procedures, I suggest that one examine how the idea of momentum enters the quantum mechanical description of simple situations like the scattering of a charged particle by a Coulomb field (and how Rutherford's classical analysis must be modified; if 2|e1e2|/hv ~ 1, then classical mechanical pictures lose meaning). Since quantum mechanical systems do not have a directly observable 'state' like classical systems, quantum mechanical experiments always involve two measurements, and the first one determinates what kind of description and prediction is possible of the second. If we allow an electron to pass through a rigid diaphram with a hole, that would be the first measurement. The fact that the diaphragm is fixed implies that no control of exchange of momentum with it is possible. On the other hand, if we want to use such a description, the experimental arrangement must be modified, and the diaphragm must be considered to be part of the observed system, and we would then be dealing with a situation analogous to the compton effect, the appearance of which requires a latitude in the spacetime description of the collision sufficient to define the wavelength of the de Broglie waves.
     
  16. Jul 8, 2010 #15

    I think the right attitude should be "it's not the task of physics to describe nature as it is, but what we can say about nature".



    A "thing" that moves in a trajectory in time and space implies size/dimensions. Is it treated as a point-particle in debb, (does it have any spatial extension)? If it's spatially extended, i would naively ask about its structure?

    It's not at all clear to me what a debb particle is supposed to be. :shy:
     
    Last edited: Jul 8, 2010
  17. Jul 8, 2010 #16
    Look, dx, whenever I respond to a newbie thread about orthodox QM using de Broglie-Bohm arguments the moderators or Dr. Chinese or whoever always have a go at me. If you want to contribute to a deBB thread using orthodox QM arguments then you really should get your facts straight first.

    The whole point about deBB is that everything you say above is not true. Is this because it is 'cleverly constructed'? No. You just look at the equations and say that they refer to Nature as it 'is' rather than only when we measure it. From that point of view, the electrons must have trajectories, and they must naturally follow the streamlines of the probability current. Result: all mathematical predictions are the same, but we have changed the theory from a statistical theory of observation to a perfectly consistent dynamical theory of particle trajectories.

    You clearly believe that systems can only possess certain values of physical quantities corresponding to the spectra of Hermitian operators. In deBB theory all quantities are in fact well-defined and continuously variable for all quantum states - the values for the subset of eigenstates have no fundamental physical significance. From this point of view one of the characteristic features of QM - the existence of discrete energy levels - is due to the restriction of a basically continuous theory to motion associated with a subclass of eigenfunctions where e.g. the wave fits in with the boundary conditions of the bound state. Such states may possess particular physical importance in relation to the stability of matter, but particle momentum and energy are just as unambiguously defined when the wave is a superposition of eigenstates. There are no 'quantum jumps' in the sense of a process that is instantaneous or beyond analysis.

    The existence of the 'quantum of action' does not 'imply a fundamental limitation in the use of customary attributes of the electron like position and momentum' in deBB. The only reason you think this is because you believe that the standard definition of the momentum refers to the momentum of the electron, which it just doesn't. You're making precisely the mistake that I was trying to highlight above.

    Once you accept this, then we can get on with your other points.
     
    Last edited: Jul 8, 2010
  18. Jul 9, 2010 #17

    Demystifier

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    In principle, you are right.
    However, it is often easier to find out what you can say about nature if you have a clear picture of what nature is (even if the correctness of this picture cannot be strictly proved by experiments).

    For example, you cannot strictly prove that Moon is there when nobody looks. Yet, as every practical astronomer knows, it is much easier to predict measurable properties of the Moon if you imagine that it is.
     
  19. Jul 9, 2010 #18


    I agree with what you say and there is another point to be made in the same direction - that insanity starts knocking on your door once you do away with the realism assumption.
     
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