In theoretical physics, the pilot wave theory was the first known example of a hidden variable
theory, presented by Louis de Broglie in 1927. Its more modern version, the Bohm interpretation,
remains a controversial attempt to interpret quantum mechanics as a deterministic theory,
avoiding troublesome notions such as instantaneous wave function collapse and the paradox of
Schrödinger's cat.
The pilot wave theory The pilot wave theory is one of several interpretations
of quantum mechanics. It uses the same mathematics as other interpretations of quantum mechanics;
consequently, it is also supported by the current experimental evidence to the same
extent as the other interpretations. Principles
The pilot wave theory is a hidden variable theory. Consequently:
the theory has realism; the theory has determinism.
The positions and momenta of the particles are considered to be the hidden variables.
However, the observer not only doesn't know the precise value of these variables, but
more importantly, cannot know them precisely because any measurement disturbs them – as
stipulated by the Heisenberg uncertainty principle. A collection of particles has an associated
matter wave, which evolves according to the Schrödinger equation. Each particle follows
a deterministic trajectory, which is guided by the wave function; collectively, the density
of the particles conforms to the magnitude of the wave function. The wave function is
not influenced by the particle and can exist also as an empty wave function.
The theory brings to light nonlocality that is implicit in the non-relativistic formulation
of quantum mechanics and uses it to satisfy Bell's theorem. Interestingly, these nonlocal
effects are compatible with the no-communication theorem, which prevents us from using them
for faster-than-light communication. Consequences
The pilot wave theory shows that it is possible to have a realistic and deterministic hidden
variable theory, which reproduces the experimental results of ordinary quantum mechanics. The
price which has to be paid for this is manifest nonlocality.
Mathematical foundations To derive the de Broglie–Bohm pilot-wave
for an electron, the quantum Lagrangian
where is the potential associated with the quantum force, is integrated along precisely
one path. This leads to the following formula for the Bohm propagator:
This propagator allows to track the electron precisely over time under the influence of
the quantum potential . Derivation of the Schrödinger equation
Pilot Wave theory is based on Hamilton–Jacobi dynamics rather than Lagrangian or Hamiltonian
dynamics. Using the Hamilton–Jacobi equation
it is possible to derive the Schrödinger equation:
Consider a classical particle — the position of which is not known with certainty. We must
deal with it statistically, so only the probability density is known. Probability must be conserved,
i.e. for each . Therefore it must satisfy the continuity equation
where is the velocity of the particle. In the Hamilton–Jacobi formulation of classical
mechanics, velocity is given by where is a solution of the Hamilton-Jacobi equation
We can combine and into a single complex equation by introducing the complex function , then
the two equations are equivalent to with
This is the time dependent Schrödinger equation with an extra potential, the quantum potential
, which is the potential of the quantum force, which is proportional to the curvature of
the amplitude of the wave function. Mathematical formulation for a single particle
The matter wave of de Broglie is described by the time-dependent Schrödinger equation:
The complex wave function can be represented as:
By plugging this into the Schrödinger equation, one can derive two new equations for the real
variables. The first is the continuity equation for the probability density:
where the velocity field is defined by the guidance equation
According to pilot wave theory, the point particle and the matter wave are both real
and distinct physical entities. The pilot wave guides the motion of the point particles
as described by the guidance equation. Ordinary quantum mechanics and pilot wave
theory are based on the same partial differential equation. The main difference is that in ordinary
quantum mechanics, the Schrödinger equation is connected to reality by the Born postulate,
which states that the probability density of the particle's position is given by . Pilot
wave theory considers the guidance equation to be the fundamental law, and sees the Born
rule as a derived concept. The second equation is a modified Hamilton–Jacobi
equation for the action :
where Q is the quantum potential defined by
By neglecting Q, our equation is reduced to the Hamilton–Jacobi equation of a classical
point particle. So, the quantum potential is responsible for all the mysterious effects
of quantum mechanics. One can also combine the modified Hamilton–Jacobi
equation with the guidance equation to derive a quasi-Newtonian equation of motion
where the hydrodynamic time derivative is defined as
Mathematical formulation for multiple particles The Schrödinger equation for the many-body
wave function is given by
The complex wave function can be represented as:
The pilot wave guides the motion of the particles. The guidance equation for the jth particle
is:
The velocity of the jth particle explicitly depends on the positions of the other particles.
This means that the theory is nonlocal. Empty wave function
Lucien Hardy and John Stewart Bell have emphasized that in the de Broglie–Bohm picture of quantum
mechanics there can exist empty waves, represented by wave functions propagating in space and
time but not carrying energy or momentum, and not associated with a particle. The same
concept was called ghost waves by Albert Einstein. The empty wave function notion has been discussed
controversially. In contrast, the many-worlds interpretation of quantum mechanics does not
call for empty wave functions. History
In his 1926 paper, Max Born suggested that the wave function of Schrödinger's wave equation
represents the probability density of finding a particle.
From this idea, de Broglie developed the pilot wave theory, and worked out a function for
the guiding wave. Initially, de Broglie proposed a double solution approach, in which the quantum
object consists of a physical wave in real space which has a spherical singular region
that gives rise to particle-like behaviour; in this initial form of his theory he did
not have to postulate the existence of a quantum particle. He later formulated it as a theory
in which a particle is accompanied by a pilot wave. He presented the pilot wave theory at
the 1927 Solvay Conference. However, Wolfgang Pauli raised an objection to it at the conference,
saying that it did not deal properly with the case of inelastic scattering. De Broglie
was not able to find a response to this objection, and he and Born abandoned the pilot-wave approach.
Unlike David Bohm, de Broglie did not complete his theory to encompass the many-particle
case. Later, in 1932, John von Neumann published
a paper claiming to prove that all hidden variable theories were impossible.. However,
in 1952, David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's
pilot wave theory. Bohm developed pilot wave theory into what is now called the de Broglie–Bohm
theory. The de Broglie–Bohm theory itself might
have gone unnoticed by most physicists, if it had not been championed by John Bell, who
also countered the objections to it. In 1987, John Bell rediscovered Grete Hermann's work,
and thus showed the physics community that Pauli's and von Neumann's objections really
only showed that the pilot wave theory did not have locality.
The de Broglie–Bohm theory is now considered by some to be a valid challenge to the prevailing
orthodoxy of the Copenhagen Interpretation, but it remains controversial.
Yves Couder and co-workers recently discovered a macroscopic pilot wave system in the form
of walking droplets. This system exhibits behaviour of a pilot wave, heretofore considered
to be reserved to microscopic phenomena. References
External links "Pilot waves, Bohmian metaphysics, and the
foundations of quantum mechanics", lecture course on pilot wave theory by Mike Towler,
Cambridge University. "Quantum mechanics writ large", article about
walking droplets by John W. M. Bush, MIT. "Hydrodynamic quantum analogues" articles
by John W. M. Bush and coworkers on hydrodynamic quantum analogues and hydrodynamic pilot-wave
theory. 2012, 2013. More complete HTML encyclopedic page about
the subject.