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Nonlocality of the quantum potential

  1. Sep 22, 2004 #1

    hellfire

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    I was reading about Bohms interpretation of QM and the Hamilton-Jacobi equation with the quantum potential:

    [tex] Q = \frac{- \hbar^2}{2 m} \frac{\nabla^2 R}{R}[/tex]

    and

    [tex] \Psi = R e^{i S / \hbar} [/tex]

    with R and S real valued functions.

    It is claimed that Q is the source of non-locality in QM. How can I prove that this potential leads to non-locality or instantaneous correlations (e.g. between EPR pairs)?
     
    Last edited: Sep 22, 2004
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  3. Sep 22, 2004 #2

    ttn

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    In a 2-particle situation, the wave function (and hence R and S) depend on the coordinates of both particles, x1 and x2. So when you plug into the guidance equation to find the velocity of particle 1, it is, in the general case (viz, for non-factorizable wf's) a function of both particle 1's position and particle 2's position. And this is true even if particle 2 is a million miles away.

    Another way to say the same thing: what each particle does depends on the configuration of the whole system of entangled particles. And since some of these particles might be far away, this is a kind of nonlocal effect.

    (Note that I didn't mention the quantum potential. This is actually an unnecessary and potentially misleading -- no pun intended -- concept in Bohmian mechanics. It is simpler to forget about the Q potential, give up on trying to make Bohmian mechanics look like Newtonian physics, and just write down the guidance formula for the particles: [tex] \vec{v} = \vec{j}/\rho[/tex] where [tex]\vec{j}[/tex] and [tex]\rho[/tex] are the usual quantum mechanical probability current and density, respectively.
     
  4. Sep 22, 2004 #3

    hellfire

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    Thanks. I understood the point about non-locality, but now I am confused with your statement:

    My understanding is that the Bohmian interpretation considers the Hamilton-Jacobi equation derived from the Schrödinger equation and interprets the new term as a real physical entity (the quantum potential Q). If one forgets about Q, what is then the characteristic of the Bohmian interpretation?
     
  5. Sep 22, 2004 #4

    ttn

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    Yes, that's how Bohm himself usually formulated the theory (e.g., in the originaly 1952 papers and in "The Undivided Universe"). In this formulation, the basic dynamics has the same form as Newtonian mechanics: F=ma. The only difference (or so it might seem) is that there is an extra force term -- a force exerted on the particles by the wave function, which force can be expressed in the usual way as the gradient of a potential (namely the quantum potential you mentioned). This formulation is nice in terms of making clear certain relationships between Bohmian mechanics and classical physics.

    But something very important tends to be suppressed (and hence completely missed) in this formulation: whereas in classical physics, one can specify arbitrary initial positions *and velocities* for the particles, this is not so in Bohmian mechanics. Once you specify the initial positions, the velocities are *fixed* by the theory. So although the Newtonian-like formulation in terms of the quantum potential makes it *look* like Bohm's theory is a 2nd order theory (i.e., a theory in which the acceleration is the quantity determined by forces or whatever) it actually isn't. It's a first order theory. If you specify the quantum state (i.e., the wf) and the positions of the particles, the velocities are *determined*. There is no extra freedom to choose the initial velocities.

    And given that, it is much simpler to just write down a guidance formula for the velocities (as a function of the configuration and the wave function). This has the form that one would naively guess if one were told that the wave function acts to guide particles, a form that shows up all over the place in physics (e.g., classical fluid flow):

    [tex]
    \vec{v} = \frac{\vec{j}}{\rho}
    [/tex]

    where [tex]\vec{v}[/tex] is the velocity of the particles, [tex]\vec{j}[/tex] is the usual quantum mechanical probability current (whose exact form depends on whether one is talking about spinless or spin 1/2 particles or whatever, but you can look up all the standard expressions in any QM text) and [tex]\rho[/tex] is the usual QM probability density ([tex]|\psi|^2[/tex] or its analogues for variously spinning particles...).

    For more details, I would strongly recommend the papers of Bell collected in "Speakable and Unspeakable..." and also the article on Bohmian Mechanics by Sheldon Goldstein at the Stanford Internet Encyclopedia:

    http://plato.stanford.edu/entries/qm-bohm/


    Oh yeah, one other thing. You suggested that, in the Bohm interpretation, Q is regarded as a real, physical entity. That's the right spirit, but not exactly precise I think. The world, if Bohm's theory is right, consists of two things: the wave function and the actual configuration (i.e., locations!) of the particles. Now, according to standard QM, the wf is supposed to be a "complete description of reality", so presumably that means orthodox QM takes the wave function as denoting something real. This it shares with Bohmian mechanics. Bohm's theory simply adds the idea that, in addition, there exist particles which have deifnite positions at all times, i.e., which follow definite trajectories. Note also that this makes most people refer to particle position in Bohmian mechanics as a "hidden variable". This is an unfortunate misnomer. In Bohm's theory, when you do an experiment to detect the location of the particle, you actually find the particle somewhere -- namely, where it was when you looked. So if *anything* is "hidden" in Bohm's theory, it certainly isn't the particle positions. And I guess that means it must be the wave functions. I'll leave the question of what this means for the Copenhagen interpretation for future discussion...
     
  6. Sep 23, 2004 #5

    hellfire

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    Thank you for your explanation and for the interesting link, it gives a very good insight.
     
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