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Bohm interpretation

From Academic Kids

The Bohm interpretation of quantum mechanics, sometimes called the Causal interpretation, or Ontological interpretation, is an interpretation postulated by David Bohm in which the existence of a non-local universal wavefunction (Schrödinger equation) allows distant particles to interact instantaneously.

The interpretation generalizes Louis de Broglie's pilot wave theory from 1927, which posits that both wave and particle are real. The wave function evolves according to the Schrödinger equation and somehow 'guides' the particle. It assumes a single, nonsplitting universe (unlike the Everett many-worlds interpretation) and is deterministic (unlike the Copenhagen interpretation). It says the state of the universe evolves smoothly through time, without the collapsing of wavefunctions when a measurement occurs, as in the Copenhagen interpretation. However, it does this by assuming a huge number of hidden variables, which can never be measured directly.

Contents

Mathematical foundation

In the Schrödinger equation

<math>\frac{-\hbar^2}{2 m} \nabla^2 \psi(\mathbf{r},t) + V(\mathbf{r}) \psi(\mathbf{r},t) = i \hbar \frac{\partial \psi(\mathbf{r},t)}{\partial t}<math>,

where the wavefunction ψ(r,t) is a complex function of position r and time t, the probability density ρ(r,t) is a real function defined by

<math>\rho(\mathbf{r},t) = R(\mathbf{r},t)^2 = |\psi(\mathbf{r},t)|^2 = \psi^*(\mathbf{r},t) \psi(\mathbf{r},t)<math>.

Without loss of generality, we can express the wavefunction ψ in terms of a real probability density ρ = |ψ|2 and a phase function of the real variable S that are both also functions of position and time

<math>\psi = \sqrt{\rho} e^{i S / \hbar}<math>.

When we do this, the Schrödinger equation separates into two coupled equations,

<math>-\frac{\partial \rho}{\partial t} = \nabla \cdot (\rho \frac{\nabla S}{m}) \qquad (1) <math>
<math>-\frac{\partial S}{\partial t} = V + \frac{1}{2m}(\nabla S)^2 + Q \qquad (2) <math>

with

<math>Q = -\frac{\hbar^2}{2 m} \frac{\nabla^2 R}{R}

= -\frac{\hbar^2}{2 m} \frac{\nabla^2 \sqrt{\rho}}{ \sqrt{\rho}} = -\frac{\hbar^2}{2 m} \left(

\frac{\nabla^2 \rho}{2 \rho}

-\left(

\frac{\nabla \rho}{2 \rho}

\right)^2

\right)

<math>.

If we identify the momentum as <math>\mathbf{p} = \nabla S<math> and the energy as <math>E = - \partial S / \partial t<math>, then (1) is simply the continuity equation for probability with

<math>\mathbf{j} = \rho \mathbf{v} = \rho \frac{\mathbf{p}}{m} = \rho \frac{\nabla S}{m}<math>,

and (2) is a statement that total energy is the sum of potential energy, kinetic energy, and some additional term Q, which has been called the quantum potential. It is by no means accidental that S has the units and typical variable name of the action.

The particle is viewed as having a definite position, with a probability distribution ρ that may be calculated from the wavefunction ψ. The wavefunction "guides" the particle by means of the quantum potential Q. Much of this formalism was developed by Louis de Broglie, Bohm extended it from the case of a single particle to that of many particles and re-interpreted the equations. It may also be extended to include spin, although extension to relativistic conditions has not yet been successful.

Commentary

The Bohm interpretation is not popular among physicists for a number of scientific and sociological reasons that would be fascinating but long to study, but perhaps we can at least say here it is considered very inelegant by some (it was considered as "unnecessary superstructure" even by Einstein who dreamed about a deterministic replacement for the Copenhagen interpretation). Presumably Einstein, and others, disliked the non-locality of most interpretations of quantum mechanics, as he tried to show its incompleteness in the EPR paradox. The Bohm theory is unavoidably non-local, which counted as a strike against it; but this is less so now, now that non-locality has become more compelling due to experimental verification of Bell's Inequality. However the theory was used by others as the basis of a number of books such as the Dancing Wu-li Masters, which purport to link modern physics with Eastern religions. This, as well as Bohm's long standing philosophical friendship with J. Krishnamurti, may have led some to discount it.

Bohm's interpretation vs. Copenhagen (or quasi-Copenhagen as defined by Von Neumann and Dirac) is different in crucial points: ontological vs. epistemological; quantum potential or active information vs. ordinary wave-particle and probability waves; nonlocality vs. locality (it should be noted that standard QM is also non-local, see EPR paradox); wholeness vs. regular segmentary approach. In his posthumous book The Undivided Universe, Bohm has (with Hiley, and, of course, in numerous previous papers) presented an elegant and complete description of the physical world. This description is in many aspects more satisfying than the prevailing one, at least to Bohm and Hiley. According to the Copenhagen interpretation, there is a classical realm of reality, of large objects and large quantum numbers, and a separate quantum realm. There is not a single bit of quantum theory in the description of "the classical world"- unlike the situation one encounters in Bohmian version of quantum mechanics. It also differs in a few matters that are experimentally tested with no consensus whether the Copenhagen, or other, interpretation has been proven inadequate; or the results are too vague to be interpreted unambiguously. The papers in question are listed at the bottom of the page, and their main contention is that quantum effects, as predicted by Bohm, are observed in classical world- something unthinkable in the dominant Copenhagen version.

The Bohmian interpretation of Quantum Mechanics is characterized by the following features:

  • it is based on concepts of non-local quantum potential and active information. Just as an aside-one should mention that Bohmian approach is not new with regard to mathematical formalism, but a reinterpretation of the ordinary quantum mechanical Schrödinger equation (which under a certain approximation is the same as the classical Hamilton-Jacobi equation), which simply, in the process of calculation, gives an additional term Bohm had interpreted as a quantum potential and developed a new view on quantum mechanics. So, Bohm's is (as anyone familiar with The Undivided Universe knows) not original mathematical formalism (it's just a wave function in radial form, and Schrodinger equation applied on it) -- but in interpretation that denies central features of ordinary quantum mechanics: no wave-particle dualism (electron is a real particle guided by a real quantum potential field); no epistemological approach (i.e., quantum realism and ontology).
  • maybe the most interesting part about Bohm's approach is its formalism: it gives a new version of microworld, not only a new (albeit radical) interpretation. It describes a world where concepts such as causality, position and trajectory have concrete physical meanings. Putting aside possible objections with regard to non-locality, and possible triumphs of Bohmian view (for instance, no need for anything like complementarity principle) - one is left with impression that what Bohm offers is perhaps a new paradigm and absolutely a boldy rephrased version of the old and established quantum mechanics.
  • Bohm emphasized that experiment and experimenter comprise an undivided whole. There is nothing separate from this undivided whole. The quantum potential Q does not go to zero at infinity.

Criticisms

The main points of critics may be summarized to the following points:

  • the wavefunction must "disappear" after the measurement, and this process seems highly unnatural in the Bohmian models
  • the theory artificially picks privileged observables: while the orthodox quantum mechanics admits many observables on the Hilbert space that are treated almost equivalently (much like the bases composed of their eigenvectors), Bohm's interpretation requires one to pick a set of "privileged" observables that are treated classically - namely the position. There is no experimental reason to think that some observables are fundamentally different from others.
  • the Bohmian interpretation has subtle problems to incorporate the spin and other concepts of quantum physics: the eigenvalues of the spin are discrete, and therefore contradict the rotational invariance unless the probabilistic interpretation is accepted
  • the Bohmian interpretation also seems incompatible with the modern insights about decoherence that allow one to calculate the "boundary" between the "quantum microworld" and the "classical macroworld"; according to decoherence, the observables that exhibit classical behavior are determined dynamically, not by an assumption

See also

External links

References

  • Holland, Peter R. The Quantum Theory of Motion : An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge: Cambridge U. Press, 1993. ISBN 0521485436 . An Amazon reviewer claims this is clearer than Bohm's own:
  • Bohm, David and B.J. Hiley. The Undivided Universe: An ontological interpretation of quantum theory. London: Routledge, 1993. ISBN 0-415-12185-X.
  • Albert, David Z. "Bohm's Alternative to Quantum Mechanics", Scientific American, May, 1994.

For a start on comparing the various interpretations of quantum mechanics see

  • Wheeler and Zurek, ed., Quantum Theory and Measurement, Princeton: Princeton University Press, 1984 or
  • Jammer, Max. The Philosophy of Quantum Mechanics.
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