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THE DE BROGLIE - BOHM PILOT WAVE THEORY

Introduction

As a physical theory, Quantum Mechanics has been very successful in predicting experimental results. However, philosophical problems arise when it is mathematically demonstrated that it cannot have all of the properties that one would intuitively expect for it to have.

In 1927, Louis de Broglie effectively discovered what we now call Bohmian Mechanics (other names include the "Pilot Wave model" and the "causal" or "ontological" interpretation of quantum mechanics), and although de Broglie later abandoned this idea, it was rediscovered in 1952 by David Bohm who was the first person genuinely to understand its significance and implications. However, the roots of the pilot-wave theory trace back to Albert Einstein, even before the discovery of quantum mechanics itself. Einstein hoped that interference phenomena involving particle-like photons could be explained if the motion of the photons were somehow guided by the electromagnetic field, which would thus play the role of the guiding field (something I will explain later). It is due to de Broglie's original conception and Bohm's rediscovery of the idea that this interpretation is known today as the de Broglie - Bohm Pilot Wave Theory.

Hidden Variables

The Pilot Wave Theory is the earliest example of what is often called a hidden-variable theory of Quantum Mechanics. Let me first explain what is meant by a hidden-variable theory. A hidden-variable theory is used by a minority of physicists who argue that the statistical nature of Quantum Mechanics implies that Quantum Mechanics is incomplete. It is really applicable only to ensembles of particles; new physical phenomena beyond Quantum Mechanics are needed to explain an individual event. The fact that Quantum Mechanics generally does not predict measurement outcomes with certainty leads us to the conclusion that Quantum Mechanics itself is not deterministic, and it only provides us with the probabilities of the possible outcomes. Now this leads to the bizarre situation, where measurements of a particular parameter done on two identical systems can yield different results. It then follows that there might be a more fundamental theory, hidden deep beneath the floor of Quantum Mechanics, that can always predict the outcome of each measurement with certainty - a hidden variable theory.

There are important considerations preventing the construction of a simple hidden-variable theory, notably Bell's theorem, which states that no local hidden-variable theory can make predictions in agreement with those of Quantum Mechanics. The only types of hidden-variable theories that appear not to have been ruled out are nonlocal hidden-variable theories , which maintain the existence of instantaneous causal relations between entities, physically separated by arbitrarily distant regions. Einstein referred to this instantaneous action as "spooky action at a distance", and we shall find out why this is a key idea in the analysis of the de Broglie - Bohm Pilot Wave Theory.

So let's now look at the fundamentals of the Pilot Wave Theory. In this interpretation, particles are particles at all times, not just when they are observed, and they always possess a real position and velocity. The wavefunction exists and evolves according to Schrödinger's equation, but it does not provide a complete description or representation of a quantum system. Rather, it governs the motion of the fundamental variables, the positions of the particles. More specifically, the wavefunction is an unusual field or wave, consisting of both classical versions of forces such as electromagnetism, and an entirely new force responsible for non-classical effects - something Bohm dubbed the "quantum potential". Thus, in Bohmian mechanics the configuration of a system of particles evolves via a deterministic motion choreographed by the wave function, i.e. the pilot wave. This pilot wave guides the particles in such a way that their statistical properties are just those predicted by standard quantum mechanics.

Mathematical Details

By looking at the mathematics of the theory, we can see how this works. We begin with the (nonrelativistic) three-dimensional time-dependent Schrödinger equation, regulating the motion of a particle under the influence of a potential V(x):

(1)

We introduce two real functions, R(x,t) and S(x,t), such that

(2)

By substituting (1) into (2), cancelling the common factor exp(iS/ħ), and separating the real and imaginary parts, we obtain two equations:

(3)

(4)

where the quantum potential, Q, is defined by

(5)

and the dell operator,, is given by

We can now begin to see how the manipulation of the Schrödinger equation leads to the fundamental concepts of the de Broglie - Bohm theory, and how they relate to standard quantum mechanics. Let's take the Born postulate, P = R 2 = |ψ|2, which relates the squared modulus of the wavefunction to the position probability density. Equation (3) can be rewritten as

(6)

It then follows that if we identify a velocity field, v, as

(7)

then equation (6) is just the quantum continuity equation, and it follows that position probability distribution will evolve in time exactly the way standard quantum mechanics predicts it will. Equation (7) is often referred to as the 'guidance' condition in Bohm's theory, and this defines the possible trajectories for the particles. Identifying the momentum as p = m v, and taking the guidance condition (7) for v, we can show that equation (4) can be written as

so that d p/d t = , and F is the gradient of the potential energy (V + Q). We are now able to understand how Bohm has taken the Schrödinger equation, and by means of a mathematical transformation, rewritten it in an equivalent form similar to Newton's second law of motion F = ma, where F is now determined by both the classical potential V and the quantum potential Q.

So far we have looked at the relevant mathematics of the de Broglie Bohm theory, and how is has been derived from the Schrödinger equation. What I intend to do now is compare and contrast the understanding of physical phenomena that comes from the Copenhagen Interpretation and the Bohm version of Quantum Mechanics. A good way to do this is to look at the famous double slit experiment, however, first of all I feel it is necessary to give an overview of the Copenhagen Interpretation.

Comparison with the Copenhagen Interpretation

The Copenhagen Interpretation is an interpretation of Quantum Mechanics formulated around 1927 by Niels Bohr and Werner Heisenberg, while collaborating in Copenhagen. There are several fundamental ideas associated with this view. Firstly, the wavefunction (which evolves according to the Schrödinger equation) is purely an auxiliary mathematical tool (not a physical entity) whose only physical meaning is our ability to calculate probabilities and hence make statistical predictions about the results of experiments. In this interpretation, there is the assumption that there are two processes which influence the wavefunction: the unitary evolution governed by the Schrödinger equation, and the aspect of measurement. The latter is very important because in the Copenhagen Interpretation it is incorrect to consider the quantum-mechanical system as separate from the measuring apparatus.

Another very important concept which arises in the Copenhagen Interpretation is the idea of complementarity (for example wave-particle duality). Problems arise from the Heisenberg Uncertainty Principle when complementary pairs are formed between certain properties, such as position and momentum, as it is an intrinsic property of nature that you cannot define the two properties with absolute certainty. For example, if you know the exact position of a particle, the uncertainty principle states that it is impossible to define the momentum with the same amount of certainty.

So if we look at the double slit experiment, where a stream of photons is allowed to pass through a screen containing two slits. On a second screen, the placement of the incoming particles is recorded, and interference patterns are observed, which is expected given the photon's dual particle and wave characteristics. When propagating through the slits, the photons behave as a wave, the slits causing both constructive and destructive interference, resulting in the observed interference pattern consisting of regions of high and low intensity. The experiment poses the following questions:

What are the rules to determine where a single particle will be observed, seeing as Quantum Mechanics can only predict statistically where the photons are likely to be observed?

What happens to the particle in between the times it is emitted and observed?

The photon appears to travel through both slits at the same time, but when observed, it is a point particle. What causes the particle to apparently switch between statistical and non-statistical behaviour?

The Copenhagen Interpretation answers these questions in the following way: Physics is the science of outcomes of measurement processes, and measurement outcomes are fundamentally indeterministic. This means that the question "where was the particle before its position was measured?" is meaningless, as particle trajectories do not exist in space-time. The wavefunction is a superposition of eigenstates, which, upon measurement, collapses into a single eigenstate (eigenvector), associated with the eigenvalue of the given operator, which is the result of that particular measurement. Or, in other words, the measurement process randomly picks out exactly one of the many possibilities allowed for by the state's wavefunction, and the wavefunction changes instantaneously to reflect that pick.

The Pilot Wave Theory resolves the double slit wave-particle dilemma in quite a straightforward but elegant way. Instead of the particle apparently interacting with both slits at the same time (a fundamental problem in the explanation of the Copenhagen Interpretation), the particle travels through only one of the slits while the wave travels through both. The slit through which the particle passes and where it arrives on the second screen are completely determined by its initial position and wave function. The particle is influenced such that it does not go where the waves cancel out, but is attracted to where they cooperate. This hereby develops an interference profile in the wave, which in turn generates a similar pattern in the trajectories guided by the wave.

The de Broglie - Bohm Theory and the Copenhagen Interpretation are thus different in crucial points: It is ontology (nature of being) versus epistemology (theory of knowledge); quantum potential and active information versus ordinary wave-particle and probability waves. According to the Copenhagen Interpretation, there is not a single bit of Quantum Theory in the description of the 'classical world', unlike in Bohmian Mechanics. The Pilot Wave Theory avoids collapsing wavefunctions, which are a fundamental aspect of the Copenhagen Interpretation, as well as the idea of complementarity.

One thing the Copenhagen Interpretation does not confront is the problem of what constitutes as a quantum measurement (known as the measurement problem). As stated earlier, it is wrong to regard the measuring apparatus (macroscopic) as separate from the quantum system (microscopic), and thus the Copenhagen Interpretation doesn't seem to distinguish between decoherence (the process whereby the macroscopic systems evolve into a state where interference is for all purposes forbidden) and collapse (a process whereby the outcome is unpredictable, although the statistical properties of a large number of similar experiments can be calculated) of the wavefunction. In short, Quantum Theory implies that measurements typically fail to have outcomes of the sort the theory was created to explain. If, as in the Bohmian Interpretation, the description provided by the wavefunction is considered incomplete, then the description of the after-measurement situation includes, as well as the wavefunction, at least the values of the variables that register the result. Hence, there is no measurement problem.

Problems with the Pilot Wave Theory

There are a number of reasons as to why the de Broglie - Bohm Theory has not gained as much acceptance among scientists as other theories (namely the Copenhagen Interpretation and Everett's Many Worlds Interpretation). Firstly there is the proposal of the quantum potential. In physics, both classical and quantum potentials arise out of physical interactions between objects, but the quantum potential arises solely out of the mathematics of the Schrödinger equation and has no physical basis. Additionally, the forces associated with the theory violate Newton's third law which states that every action has an equal and opposite reaction. The wave influences the particle through the quantum potential but the particle does not react on the wave.

The Pilot Wave Theory has subtle problems when incorporating 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. Bohmian mechanics does not account for phenomena such as the pair creation and annihilation characteristic of Quantum Field Theory. It also appears to make unnecessary postulates, for example the existence of particle trajectories, which can never be observed more accurately than the uncertainty principle allows. Although the theory does not require the collapse of the wavefunction in order to explain quantum phenomena, once the measurement is made, the wavefunction must disappear, which seems highly unnatural.

A main reason the Pilot Wave Theory is not universally accepted is that it is essentially a nonlocal theory. This means that the particle may be influenced not only by the potential at the particle's location, but also by its values at other points in space. But if I am not mistaken, the Copenhagen Interpretation is also nonlocal. This leads me to question why the de Broglie - Bohm Theory has not taken pole position. According to James T. Cushing, historical contingency played a big part in the rooting of the Copenhagen Interpretation, as he declares:

In other words, the Copenhagen Interpretation has enjoyed more success than the de Broglie - Bohm Theory simply because it appeared on the scene first. I believe this could well be a contributing factor, but I do not think it to be the sole reason.

To many, this theory seems contrived and it is thought that it was deliberately designed to give predictions which are in all details identical to conventional Quantum Mechanics. The belief is that this theory was not intended as a serious counterproposal, but simply to demonstrate that hidden variable theories are indeed possible. In 1935, Einstein, Podolsky and Rosen argued that a nonlocal hidden variable theory was not only possible, but in fact necessary, proposing the EPR Paradox as proof. Additionally, in 1964, John Bell showed that nonlocality is implied by the predictions of Quantum Theory itself.

Nonlocality implies that entities can travel faster than the speed of light, thus violating Einstein's relativity principle and the Lorentz invariance. This statement is only true if you have real particle positions, as you do in the Pilot Wave Theory. The Copenhagen Interpretation does not deal with real particle positions and has thus been formulated in a Lorentz invariant manner. Furthermore, the quantum potential behaves in a nonrelativistic fashion, so Bohmian Mechanics is fine for explaining nonrelativistic Quantum Mechanics, but it hasn't been joined to special relativity in a consistent fashion.

So there we have it, Ladies and Gentlemen, the de Broglie - Bohm Pilot Wave Theory. It most certainly has the potential (no pun intended), but until it can be united with special relativity, much to my regret, the Copenhagen Interpretation reigns supreme.

References

  • 1. Rae, A. I. M. Quantum Mechanics (Fourth Edition) IOP Publishing Ltd. 2002
  • 2. Cushing, J. T. Philosophical Concepts in Physics Cambridge University Press 1998
  • 3. Bohm, D. Wholeness and the Implicate Order Routledge 1980
  • 4. Oxford A Dictionary of Physics (Fourth Edition) Oxford University Press 2000
  • 5. a href="http://encyclopedia.thefreedictionary.com/Bohm%20interpretation">http://encyclopedia.thefreedictionary.com/Bohm%20interpretation
  • 6. a href="http://plato.stanford.edu/entries/qm-bohm/">http://plato.stanford.edu/entries/qm-bohm/
  • 7. a href="http://www.math.rutgers.edu/~oldstein/papers/qts/node4.html">http://www.math.rutgers.edu/~oldstein/papers/qts/node4.html
  • 8. a href="http://en.wikipedia.org/wiki/Copenhagen_interpretation">http://en.wikipedia.org/wiki/Copenhagen_interpretation
  • 9. a href="http://pericson.com/writings/bohmindeterminism/">http://pericson.com/writings/bohmindeterminism/
  • 10. a href="http://www.uncletaz.com/library/scimath/pilotwave.html">http://www.uncletaz.com/library/scimath/pilotwave.html
  • 11. a href="http://en.wikipedia.org/wiki/EPR_paradox#Hidden_variables">http://en.wikipedia.org/wiki/EPR_paradox#Hidden_variables
 
 
 
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