Bohm interpretation:

The Bohm interpretation of quantum mechanics, sometimes called Bohmian mechanics, the ontological interpretation, or the causal interpretation, is an interpretation postulated by David Bohm in 1952 as an extension of Louis de Broglie's pilot-wave theory of 1927 . Consequently it is sometimes called the de Broglie-Bohm theory.

The Bohm interpretation is an interpretation of quantum mechanics. In other words, it has not been disproven, but there are other schemes (such as the Copenhagen interpretation) that give the same theoretical predictions, so are equally confirmed by the experimental results.

Bohm's interpretation is an example of a hidden variables theory. Historically, it was hoped that hidden variables could provide a local, causal, objective description that would resolve or eliminate many of the paradoxes of quantum mechanics, such as Schrödinger's cat, the measurement problem, the collapse of the wavefunction, and similar concerns. However, Bell's theorem complicates this hope, as it demonstrates that there is no locally causal hidden variable theory that is compatible with quantum mechanics. Thus, one is left with choosing between two options: discarding locality, or discarding causality (the assumption that measurement settings are free variables). The Bohmian interpretation opts for keeping causality and accepting nonlocality.

The theory

Principles

The Bohm interpretation is an interpretation of quantum mechanics; it was originally developed as an objective and deterministic alternative to the Copenhagen interpretation. It says that the state of the universe evolves smoothly through time, with no collapsing of wavefunctions.

The Bohm interpretation is a hidden variables theory. In other words, there is a precisely defined history of the universe; however, some of the variables that define the history are not (and cannot be) known to the observer. For that reason, there is uncertainty in what we know about the universe.

Results

The Bohm interpretation demonstrates that some features of the Copenhagen interpretation are not essential to Quantum mechanics, but are a feature of the interpretation. Some specific features are:

Examination of the Bohm interpretation has shown that nonlocality is a general feature of Quantum mechanics interpretations, including the Copenhagen interpretation where it was not originally obvious.

Quantum potential

We can formulate the guiding wave's influence using a wavefunction-derived potential called the quantum potential, which acts upon the particles in a manner somewhat analogous to the interaction of particles and fields in classical physics. Thus, Bohm called this field the quantum potential force.

The mathematics behind the quantum potential are not specific to the Bohm Interpretation, but apply generally to Schrodinger's wave equation. They are described here because the concepts are used later in the article.

where the wavefunction $\psi(\mathbf{x},t)$ is a complex function of the spatial coordinate $\mathbf{x}$ and time t.

The probability density $\rho (\mathbf{x},t)$ is a real function defined as the magnitude of the wave function:

The magnitude R is $|\psi (\mathbf{x},t)|$ , so that

R 2

corresponds to the probability density $\rho (\mathbf{x},t) = |\psi (\mathbf{x},t)|^2$ , and the phase S is the complex phase so that:

The Schrödinger equation can then be split into two coupled equations by separating the magnitude and phase;

One-particle formalism

and the particle's energy as $E = - \partial S / \partial t$ ; the particle's position is not yet defined. Equation (1) is interpreted as simply the continuity equation for probability with

and equation (2) is a statement that total energy is the sum of potential energy, quantum potential and the kinetic energy. It is by no means accidental that S has the units and typical variable name of the action.

Many-particle formalism

The many-particle Schrödinger equation is a straightforward generalisation of the one-particle example:

where the i-th particle has mass $m_i\,$ and position coordinate $\mathbf{x_i}$ at time t. The wavefunction $\psi(\mathbf{x_1, x_2,...},t)$ is a complex function of the $\mathbf{x_i}$ and time t. $\nabla_i$ is the grad operator with respect to $\mathbf{x_i}$ , i.e. of the i-th particle's position coordinate. As before the probability density $\rho(\mathbf{x_1, x_2,...},t)$ is a real function defined by

The complex phase depends on the real variable $S(\mathbf{x_1, x_2,...},t)$ so that we can define the same relationship as in the 1-particle example:

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

and the particles' total energy as $E = - \partial S / \partial t$ ; equation (1) is the continuity equation for probability with

and equation (2) is a statement that total energy is the sum of the potential energy, quantum potential and the kinetic energies.

Comparison with experimental data

An important prediction of the Bohm theory, made in Bohm's original 1952 paper, is that the electron in the ground state of a hydrogen atom is in rest (cf. equation (3) above - s states have spherical symmetry and thus have constant phase), as the quantum force introduced by Bohm balances the classical electromagnetic potential.

Any measurement of the momentum of a ground state electron will give a non-zero result as predicted by quantum mechanics, but Bohm's theory argues that the act of measuring the momentum disturbs the electron at rest, resulting in a non-zero expection value. ^[1]

Experimental observation of the decay rates of muons bound in exotic atoms has shown, however, that ground state electrons are in fact in motion. Because of their mass, muons captured in higher states rapidly cascade to the ground state, and about 99 percent of the bound muons decay from the 1S state. If the atomic number of the hydrogen-like atom is high enough, the muon motion will be relativistic, and subject to time dilation. The data show that for moderate Z atoms, the observed lengthening of the muon decay time can be attributed to the relativistic time dilation.^[2]

Since the motion of the muon has been demonstrated without any disturbance of the atom, it cannot be explained by disturbances related to the measurement process as with conventional measurements of the momentum of ground state electrons. An important conceptual prediction of the Bohm model, namely that ground state electrons are at rest, would seem to be contradicted by experimental evidence. It might be argued that Bohm's prediction of the lack of motion of the lepton in the s state is only a non-relativistic prediction, since the Schrödinger equation is non-relativistic. Therefore the muon result, which relies on time dilation, would not be at variance with the accepted theoretical demonstration that Bohm's theory reproduces all the non-relativistic results of the other conventional quantum interpretations.

Understanding quantum mechanics

Heisenberg's uncertainty principle

The Heisenberg uncertainty principle states that when two complementary measurements are made, there is a limit to the product of their accuracy. As an example, if one measures the position with an accuracy of

Δ x

, and the momentum with an accuracy of

Δ p

, then $\Delta x\Delta p\gtrsim h.$ If we make further measurements in order to get more information, we disturb the system and change the trajectory into a new one depending on the measurement set up; therefore, the measurement results are still subject to Heisenberg's uncertainty relation.

In Bohm's interpretation, there is no uncertainty in position and momentum of a particle; therefore a well defined trajectory is possible, but we have limited knowledge of what this trajectory is (and thus of the position and momentum). It is our knowledge of the particle's trajectory that is subject to the uncertainty relation. What we know about the particle is described by the same wave function that other interpretations use, so the uncertainty relation can be derived in the same way as for other interpretations of quantum mechanics.

To put the statement differently, the particles' positions are only known statistically. As in classical mechanics, successive observations of the particles' positions refine the initial conditions. Thus, with succeeding observations, the initial conditions become more and more restricted. This formalism is consistent with normal use of the Schrödinger equation. It is the underlying chaotic behaviour of the hidden variables that allows the defined positions of the Bohm theory to generate the apparent indeterminacy associated with each measurement, and hence recover the Heisenberg uncertainty principle.

Two-slit experiment

The double-slit experiment is an illustration of wave-particle duality. In it, a beam of particles (such as photons) travels through a barrier with two slits removed. If one puts a detector screen on the other side, the pattern of detected particles shows interference fringes characteristic of waves; however, the detector screen responds to particles. The photons must exhibit behaviour of both waves and particles.

The Copenhagen interpretation requires that the photons are not localised in space until they are detected, and travel through both slits.

In the Bohm interpretation, the guiding wave travels through both slits and sets up a ψ-field; the initial positions of the photons are not known to the observer, but they travel under the guidance of the ψ-field. Each photon that is detected has travelled through one of the slits; together, their statistics show the probability distribution of the ψ-field, which gives interference fringes that are characteristic of waves. The photons are localized particles at all times, and hence are able to activate the detector screen.

Measuring spin and polarization

It is not possible to measure the spin or polarization of a particle directly; instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that the particle is aligned with the measuring apparatus, or -1, meaning that it is aligned the opposite way. For an ensemble of particles, if we expect the particles to be aligned, the results are all 1. If we expect them to be aligned oppositely, the results are all -1. For other alignments, we expect some results to be 1 and some to be -1 with a probability that depends on the expected alignment. For a full explanation of this, see the Stern-Gerlach Experiment.

In the Bohm interpretation, we calculate the wave function that corresponds to the setup of the apparatus (including the orientation of the detectors); the particles are guided by this function, so we can calculate the probabilities that the actual particles will reach each part of the detector apparatus.

The Einstein-Podolsky-Rosen paradox

In the Einstein-Podolsky-Rosen paradox^[3], the authors point out that it is possible to create pairs of particles with quantum states that are mirror-images of each other; these particles are now described as entangled. They describe a thought-experiment showing that either quantum mechanics is an incomplete theory or that it has nonlocality.

John Bell then described Bell's theorem (see p.14 in^[4]), in which he shows that all hidden-variable theories (including the Bohm interpretation) have nonlocality. Bell went further, showing that quantum mechanics itself is nonlocal and that this cannot be avoided by appealing to any alternative interpretation (p. 196 in^[5]): "It is known that with Bohm's example of EPR correlations, involving particles with spin, there is an irreducible nonlocality."

In Bell's description of the experiment, entangled pairs of particles are created; the particles are separated, travelling to remote measuring apparatus. The orientation of the measuring apparatus can be changed while the particles are in flight, demonstrating the non-locality of the effect. The apparatus makes a statistical detection of the orientation of the particles (see Measuring spin and polarization for a description of this). Bell's Theorem shows that the results at each detector depend on the orientation of both detectors.

The Bohm interpretation describes this experiment as follows: to understand the evolution of these particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wave function. The particles in the experiment follow the guidance of the wave function; we don't know the initial conditions of these particles, but we can predict the statistical outcome of the experiment from the wave function. It is the wave function that carries the faster-than-light effect of changing the orientation of the apparatus.

History

Bohm became dissatisfied with the conventional interpretation of quantum mechanics, pointing out that, although it requires one to give up "the possibility of even conceiving what might determine the behaviour of an individual system at a quantum level", it doesn't prove that this requirement is necessary.

To highlight this, Bohm published 'A Suggested Interpretation of Quantum Mechanics in Terms of "Hidden" Variables' [Bohm 1952]. In this paper, Bohm presented his alternative.

While preparing this paper, Bohm became aware of de Broglie's Pilot wave theory. This was a hidden-variable theory which de Broglie presented at the 1927 Solvay Conference^[6]. At the conference, Wolfgang Pauli pointed out that it did not deal properly with the case of inelastic scattering. De Broglie was persuaded by this argument, and abandoned this theory. Later, in 1932, John von Neumann published a paper^[7], claiming to prove that all hidden-variable theories are impossible. This clearly applied to both de Broglie's and Bohm's theories.

Bohm's paper was largely ignored by other physicists; surprisingly, it was not supported by Albert Einstein (who was also dissatisfied with the prevailing orthodoxy and had discussed Bohm's ideas with him before publication). So Bohm lost interest in it.

The cause was taken up by John Bell. In "Speakable and Unspeakable in Quantum Mechanics" [Bell 1987], several of the papers refer to hidden variables theories (which include Bohm's). Bell showed that Pauli's and von Neumann's objections amounted to showing that hidden variables theories are nonlocal, and that nonlocality is a feature of all quantum mechanical systems.

The Bohm interpretation is now considered by some to be a valid challenge to the prevailing orthodoxy of the Copenhagen Interpretation, but it remains controversial.

Extensions

Nonlocality

Bohm's pilot wave theory has been generalized by Antony Valentini of the Perimeter Institute to include signal nonlocality that would allow entanglement to be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" the message encoded in the entanglement. This violates orthodox quantum theory but it has the virtue that it makes the parallel universes of the cosmic landscape of eternal chaotic inflation observable in principle. This is good news for string theorists although the price to be paid may be too high for many. Valentini cites earlier work that shows that orthodox quantum theory corresponds to "sub-quantal thermal equilibrium" for the hidden variables. The larger theory is a non-equilibrium theory. Still another way to look at this is to recognize that orthodox quantum theory in Bohm's ontology is a test particle approximation like in general relativity. That is, the hidden variable "particles" and "field configurations" receive their marching orders from the pilot quantum information wave but do not directly react back on the pilot wave. Bohm and Hiley briefly discuss this in Ch 14 of the "Undivided Universe." This direct feedback of particle on its guiding pilot wave introduces an instability, a feedback loop that pushes the hidden variables out of thermal equilibrium "sub-quantal heat death" (Valentini). The resulting theory becomes nonlinear and non-unitary. The Born probability interpretation can no longer be sustained in this emergence of new order in complex systems that P.W. Anderson has called "More is different."

Seen as isomorphic to many worlds

Computational Bohmian Dynamics

Most of the discussion in this page focuses upon interpretation and ontological consequences of Bohm's formalism. However, it is mathematically valid and provides a potentially novel way to perform quantum mechanical calculations. Work by Robert Wyatt in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows the actual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantum wave function with a mesh of quadrature points. One then evolves the quadrature points in time according to the Bohm equations of motion written above. At each time-step, one then re-synthesizes the wave function from the points, recomputes the quantum forces, and continues the calculation. Quick-time movies of this for H+H₂ reactive scattering can be found on the Wyatt group web-site at UT Austin.

This approach has been adapted, extended, and used by a number of researchers in the Chemical Physics community as a way to compute semi-classical and quasi-classical molecular dynamics. A recent issue of the Journal of Physical Chemistry A was dedicated to Prof. Wyatt and his work on "Computational Bohmian Dynamics". Eric Bittner's group at the University of Houston has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points. This technique was recently used to estimate quantum effects in the heat-capacity of small clusters Ne_n for n~100.

There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in the quantum potential due to nodes in the quantum wave function. In general, nodes forming due to interference effects lead to the case where $\frac{1}{R}\nabla^2R\rightarrow\infty.$ This results in an infinite force on the sample particles forcing them to move away from the node and often crossing the path of other sample points (which violates single-valuedness). Various schemes have been developed to overcome this; however, no general solution has yet emerged.

There has also been recent work in developing Complex Bohmian trajectories which satisfy isochronal relations.

Other extensions

The theory can also easily be extended to include spin, as well as the relativistic Dirac theory. In the latter case, such relativistic extensions inherently involve an experimentally unobservable preferred frame, and this is considered by many to be in tension with the "spirit" of special relativity.

As probability density and probability current can be defined for any set of commuting operators, the Bohmian formalism is not limited to the position operator.^[10]

Frequently asked questions

Q: Quantum field theory has been verified experimentally remarkably well. Since locality is essential to quantum field theory, how can it be reconciled with Bohm's interpretation?

A: A Bohmian interpretation of quantum field theory also exists (see [1], [2], and references therein) and reproduces all measurable predictions of quantum field theory. Although this Bohmian theory is not local at the fundamental level, all local properties of quantum field theory are reproduced.

Q: Why should we consider this to be a separate theory when it looks contrived, and gives the same measurable predictions as conventional quantum mechanics?

A: Bohm's original aim was to demonstrate that hidden-variables theories are possible, contrary to the belief (due to von Neumann) that they are not. To accomplish this aim it was necessary that the predictions of the two theories be the same. However, Bohm's hope was that this demonstration could lead to new insights and experiments. Bohm's hope was that his theory could be extended in ways that conventional theory would not suggest, and these extensions may lead to new measurable predictions.^[11]

There is an argument that, because both theories predict the same experimental results, that it is the conventional theory that is not really a separate theory [3].

Finally, as shown recently in [4] and [5], there is even an experimental motivation for introducing Bohmian velocities because a weak measurement of a particle velocity necessarily yields the Bohmian velocity.

Q: The wavefunction must "disappear" or "collapse" after a measurement, and this process seems highly unnatural in the Bohmian models.

A: Collapse is a main feature of von Neumann's theory of quantum measurement. In the Bohm interpretation, a wave does not collapse. Instead, a measurement produces what Bohm called "empty channels" consisting of portions of the wave that no longer affect the particle. This explains why physical systems behave as if parts of the wavefunction had disappeared, despite the fact that there is no true disappearance or collapse. (The process whereby a quantum system interacts with its environment to give the appearance of wavefunction collapse is called decoherence.)

Q: 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. How can this be justified, when there is no experimental reason to think that some observables are fundamentally different from others?

A: Positions may be considered as a natural choice for the selection because positions are most directly measurable. For example, one does not actually measure the "spin" of a particle in the Stern–Gerlach experiment, but instead measures the position of the light flashes on a detector. Often the observed quantities are positions, e.g. of a measuring needle or of the particles making up a computer display. And so there is justification for making position privileged.

Q: The Bohmian models are nonlocal; how can this be reconciled with the principles of special relativity? they make it highly nontrivial to reconcile the Bohmian models with up-to-date models of particle physics, such as quantum field theory or string theory, and with some very accurate experimental tests of special relativity, without some additional explanation. On the other hand, other interpretations of quantum mechanics, such as consistent histories or the many-worlds interpretation, allow us to explain the experimental tests of quantum entanglement without any nonlocality whatsoever.

A: It is questionable whether other interpretations of quantum theory are local or are simply less explicit about nonlocality. See for example the EPR type of nonlocality. And recent tests of Bell's Theorem add weight to the belief that all quantum theories must abandon either the principle of locality or counterfactual definiteness (the ability to speak meaningfully about the definiteness of the results of measurements, even if they were not performed).

Finding a Lorentz-invariant expression of the Bohm interpretation (or any nonlocal hidden-variable theory) has proved difficult, and it remains an open question for physicists today whether such a theory is possible and how it would be achieved.

Q: The Bohmian interpretation has subtle problems with incorporating 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.

A: This criticism is based on the false 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. In these, spin is a property of the wave only; the particle variables have no spin in the mathematical formulation.

Q: 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.

A: 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. 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? And the fact that the theory does not postulate the existence of any boundary between two "worlds" can be viewed as one of its major advantages.

Another possible route to new 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 decomposition 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.

Q: 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.

A: The Bohm interpretation takes the Schrödinger equation even more seriously than does the conventional interpretation. In the Bohm interpretation, the quantum potential is a quantity derived from the Schrödinger equation, not a fundamental quantity. Thus, the interference patterns in the Bohm interpretation are identical to those in the conventional interpretation. As shown in [6] and [7], the experiments cited above only disprove a misinterpretation of the Bohm interpretation, not the Bohm interpretation itself.

Q: Hugh Everett says that Bohm's particles are not observable entities, but surely they are - what hits the detectors and causes flashes?

A: Both Everett and Bohm both treat the wavefunction as a complex-valued but real field. Everett's Many-worlds interpretation is an attempt to demonstrate that the wavefunction alone is sufficient to account for all our observations. When we see the particle detectors flash or hear the click of a gieger counter or whatever then Everett interprets this as our wavefunction responding to changes in the detector's wavefunction, which is responding in turn to the passage of another wavefunction (which we think of as a "particle", but is actually just another wave-packet). But no particle in the Bohm sense of having a defined position and velocity is involved. For this reason Everett sometimes referred to his approach as the "wave interpretation". Talking of Bohm's approach, Everett says:

In this view, then, the Bohm particles are unobservable entities, similar to and equally as unnecessary as, for example the luminiferous ether was found to be unnecessary in special relativity. We can remove the particles from Bohm's theory and still account for all our observations. The unobservability of the "hidden particles" stems from an asymmetry in the causal structure of the theory; the wavefunction influences the position and velocity of the hidden variables (i.e. the particles feel for the "force" exerted by the wavefunction), but the hidden variables do not influence the time development of the wavefunction (i.e. there is no analogue of Newton's third law -- the particles do not react back onto the wavefunction) Thus the particles do not make their presence known in any way; as the theory says, they are hidden.

Bohm interpretation

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