Bohm interpretation

From Free net encyclopedia

The Bohm interpretation of quantum mechanics, sometimes called the Causal interpretation, the Ontological interpretation or the de Broglie-Bohm theory, is an interpretation postulated by David Bohm in which the existence of a non-local universal wavefunction 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 'guides' the motion of the particle, and evolves according to the Schrödinger equation. The interpretation 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 number of hidden variables, namely the positions of all the particles in the universe, which, like probability amplitudes in other interpretations, can never be measured directly.

Contents

[edit]

Mathematical foundation

The Schrödinger equation is

<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 complex phase that depends on the real variable S, both of which are also functions of position and time, as

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

The Schrödinger equation can then be split into two coupled equations by taking the real and imaginary terms;

<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>

where

<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> is called the quantum potential.

If we identify the momentum as <math>\mathbf{p} = \nabla S</math> and the energy as <math>E = - \partial S / \partial t</math>, then equation (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 equation (2) is a statement that total energy is the sum of potential energy, kinetic energy and 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 also re-interpreted the equations. It can also be extended to include spin, although extension to relativistic conditions has not yet been successful.

[edit]

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 now less so, 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 Paul Dirac) differs in crucial points: ontological vs. epistemological; quantum potential or active information vs. ordinary wave-particle and probability waves; nonlocality vs. locality wholeness vs. regular segmentary approach. Standard QM is also non-local; see EPR paradox. 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 the 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 keep in mind that the Bohmian approach is not new with regard to mathematical formalism, but is a reinterpretation of the ordinary quantum mechanical Schrödinger equation (which under a certain approximation is the same as the classical Hamilton-Jacobi equation), that 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. Therefore, Bohm's is not an original mathematical formalism (it's just a wave function in radial form, with the Schrödinger equation applied to it) but an 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), and no epistemological approach (i.e., quantum realism and ontology).
  • Perhaps the most interesting part about Bohm's approach is its formalism: it gives a new version of the 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 a complementarity principle) - one is left with the impression that what Bohm offers is perhaps a new paradigm and absolutely a boldly 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.
[edit]

Benefits

For supporters, Bohm's interpretation is the better formulation of Quantum Mechanics, because it is defined more precisely than the Copenhagen interpretation which is based on theorems which are not expressed in precise mathematical terms but in natural words, like "when measuring".

Indeed, Bohm's interpretation subsumes the quantum concepts of measurement, complementarity, decoherence, and entanglement into mathematically precise guidance conditions and position variables.

The minimum benefit of Bohm's interpretation - independently from the debate whether it is the preferable formulation - is a disproof of the claim that quantum mechanics implies that particles cannot exist before being measured.

Bohm's interpretation gives non-mystical explanations of famous experiments of Quantum Mechanics. For example, in the Double-slit experiment for electrons, each electron just travels through only one slit, but the wave function causes the interference pattern. Not only the wave function, but also the trajectory of each electron can be calculated back when knowing the position where the electron hit the screen.

Bohm's interpretation gives natural answers to such philosophical questions. For example, every particle exists all the time and has an unique position, also when not being measured at the moment.

[edit]

Criticisms

The main points of critics, together with the responses of Bohm-interpretation advocates, are summarized in the following points:

  • The wavefunction must "disappear" after the measurement, and this process seems highly unnatural in the Bohmian models.
Response: The von Neumann theory of quantum measurement combined with the Bohmian interpretation explains why physical systems behave as if the wavefunction "disappeared", despite the fact that there is no true "disappearance". This is called decoherence.
  • The theory artificially picks privileged observables: while 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.
Response: Every physical theory can be rewritten based on different fundamental variables without being different empirically. The Hamilton-Jacobi equation formulation of the classical mechanics is an example. Positions may be considered as a natural choice for the selection because positions are most directly measurable.
Response: Non-locality and Lorentz invariance are not in contradiction. An example of a non-local Lorenz-invariant theory is the Feynman-Wheeler theory of electromagnetism.
Furthermore, it is questionable whether other interpretations of quantum theory are in fact local, or simply less explicit about non-locality. Recent tests of Bell's Theorem add weight to the latter belief.
That said, it is true that finding a Lorentz-invariant expression of the Bohm interpretation (or any similar nonlocal hidden-variable theory) has proved very difficult, and it remains an open question for phyicists today whether such a theory is possible and how it would be achieved.
  • The Bohmian interpretation has subtle problems to incorporate spin and other concepts of quantum physics: the eigenvalues of the spin are discrete, and therefore contradict rotational invariance unless the probabilistic interpretation is accepted.
Response: This criticism is based on the wrong assumption that the particle position variables in Bohm's equations must carry spin. There are natural variants of the Bohm interpretation in which such problems do not appear: Spin is only a property of the wave function as in the Schrödinger equation, but the particle variables itself have no spin in the mathematical formulation, spin being a measurable result of the wave function.
  • The Bohmian interpretation also seems incompatible with 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.
Response: When the Bohm interpretation is treated together with the von Neumann theory of quantum measurement, no incompatibility with the insights about decoherence remains. On the contrary, the Bohm interpretation may be viewed as a completion of the decoherence theory, because it provides an answer to the question that decoherence by itself cannot answer: What causes the system to pick up a single definite value of the measured observable?
  • The Bohm interpretation does not lead to new measurable predictions, so it is not really a scientific theory.
Response: In the domain in which the predictions of the conventional interpretation of quantum mechanics are unambiguous, the predictions of the Bohm interpretation are identical to those of the conventional interpretation. However, in the domain in which the conventional interpretation is ambiguous, such as the question of the time-observable in non-relativistic quantum mechanics and the position-observable in relativistic quantum mechanics, the Bohm interpretation leads to new unambiguous measurable predictions.
Another possible route to new measureable predictions is opened up by current developments in quantum chaos. In this theory, there exist quantum wave functions that are fractal and thus differentiable nowhere. While such wave functions can be solutions of the Schrödinger equation, taken in its entirety, they would not be solutions of Bohm's coupled equations for the polar decompsition of ψ into ρ and S, given above. The breakdown occurs when expressions involving ρ or S become infinite (due to the non-differentiability), even though the average energy of the system stays finite, and the time-evolution operator stays unitary. As of 2005, it does not appear that experimental tests of this nature have been performed.
  • The Bohm interpretation involves reverse-engineering of quantum potentials and trajectories from standard QM. Diagrams in Bohm's book are constructed by forming contours on standard QM interference patterns and are not calculated from his "mathematical" formulation. Recent experiments with photons [arXiv:quant-ph/0206196 v1 28 Jun 2002] favor standard QM over Bohm's trajectories.
[edit]

See also

[edit]

References

Retrieved from "http://www.netipedia.com/index.php/Bohm_interpretation"