Pilot Wave
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In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, the pilot wave theory, also known as Bohmian mechanics, was the first known example of a
hidden-variable theory In physics, a hidden-variable theory is a Determinism, deterministic model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional, possibly inaccessible, variables. The mathematical formulation of quantum ...
, presented by
Louis de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (15 August 1892 – 19 March 1987) was a French theoretical physicist and aristocrat known for his contributions to quantum theory. In his 1924 PhD thesis, he postulated the wave nature of elec ...
in 1927. Its more modern version, the
de Broglie–Bohm theory The de Broglie–Bohm theory is an interpretation of quantum mechanics which postulates that, in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all ...
, interprets
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
as a
deterministic Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...
theory, and avoids issues such as
wave function collapse In various interpretations of quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to in ...
, and the paradox of
Schrödinger's cat In quantum mechanics, Schrödinger's cat is a thought experiment concerning quantum superposition. In the thought experiment, a hypothetical cat in a closed box may be considered to be simultaneously both alive and dead while it is unobserved, ...
by being inherently nonlocal. The de Broglie–Bohm pilot wave theory is one of several interpretations of (non-relativistic) quantum mechanics.


History

Louis de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (15 August 1892 – 19 March 1987) was a French theoretical physicist and aristocrat known for his contributions to quantum theory. In his 1924 PhD thesis, he postulated the wave nature of elec ...
's early results on the pilot wave theory were presented in his thesis (1924) in the context of atomic orbitals where the waves are stationary. Early attempts to develop a general formulation for the dynamics of these guiding waves in terms of a relativistic wave equation were unsuccessful until in 1926 Schrödinger developed his non-relativistic wave equation. He further suggested that since the equation described waves in configuration space, the particle model should be abandoned. Shortly thereafter,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German-British theoretical physicist who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics, and supervised the work of a ...
suggested that the wave function of Schrödinger's wave equation represents the probability density of finding a particle. Following these results, de Broglie developed the dynamical equations for his pilot wave theory. Initially, de Broglie proposed a ''double solution'' approach, in which the quantum object consists of a physical wave (''u''-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. De Broglie presented the pilot wave theory at the 1927
Solvay Conference The Solvay Conferences () have been devoted to preeminent unsolved problems in both physics and chemistry. They began with the historic invitation-only 1911 Solvay Conference on Physics, considered a turning point in the world of physics, and ar ...
. However,
Wolfgang Pauli Wolfgang Ernst Pauli ( ; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and a pioneer of quantum mechanics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics "for the ...
raised an objection to it at the conference, saying that it did not deal properly with the case of
inelastic scattering In chemistry, nuclear physics, and particle physics, inelastic scattering is a process in which the internal states of a particle or a system of particles change after a collision. Often, this means the kinetic energy of the incident particle is n ...
. De Broglie was not able to find a response to this objection, and he abandoned the pilot-wave approach. Unlike
David Bohm David Joseph Bohm (; 20 December 1917 – 27 October 1992) was an American scientist who has been described as one of the most significant Theoretical physics, theoretical physicists of the 20th centuryDavid Peat Who's Afraid of Schrödinger' ...
years later, de Broglie did not complete his theory to encompass the many-particle case. The many-particle case shows mathematically that the energy dissipation in inelastic scattering could be distributed to the surrounding field structure by a yet-unknown mechanism of the theory of hidden variables. In 1932,
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
published a book, part of which claimed to prove that all hidden variable theories were impossible. This result was found to be flawed by
Grete Hermann Grete Hermann (2 March 1901 – 15 April 1984) was a German mathematician and philosopher noted for her work in mathematics, physics, philosophy and education. She is noted for her early philosophical work on the foundations of quantum mechanics ...
three years later, though for a variety of reasons this went unnoticed by the physics community for over fifty years. In 1952,
David Bohm David Joseph Bohm (; 20 December 1917 – 27 October 1992) was an American scientist who has been described as one of the most significant Theoretical physics, theoretical physicists of the 20th centuryDavid Peat Who's Afraid of Schrödinger' ...
, 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 is an interpretation of quantum mechanics which postulates that, in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all ...
. 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 only showed that the pilot wave theory did not have
locality Locality may refer to: * Locality, a historical named location or place in Canada * Locality (association), an association of community regeneration organizations in England * Locality (linguistics) * Locality (settlement) * Suburbs and localitie ...
.


The pilot wave theory


Principles

The pilot wave theory is a
hidden-variable theory In physics, a hidden-variable theory is a Determinism, deterministic model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional, possibly inaccessible, variables. The mathematical formulation of quantum ...
. Consequently: * the theory has realism (meaning that its concepts exist independently of the observer); * the theory has
determinism Determinism is the Metaphysics, metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes ov ...
. The positions of the particles are considered to be the hidden variables. The observer doesn't know the precise values of these variables; they cannot know them precisely because any measurement disturbs them. On the other hand, the observer is defined not by the wave function of their own atoms but by the atoms' positions. So what one sees around oneself are also the positions of nearby things, not their wave functions. A collection of particles has an associated matter wave which evolves according to the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
. 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 In theoretical physics, the pilot wave theory, also known as Bohmian mechanics, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets quan ...
. 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 Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measuremen ...
. These nonlocal effects can be shown to be compatible with the
no-communication theorem In physics, the no-communication theorem (also referred to as the no-signaling principle) is a no-go theorem in quantum information theory. It asserts that during the measurement of an entangled quantum state, it is impossible for one observer ...
, which prevents use of them for faster-than-light communication, and so is empirically compatible with relativity.


Macroscopic analog

Couder, Fort, ''et al.'' claimed that macroscopic oil droplets on a vibrating fluid bath can be used as an analogue model of pilot waves; a localized droplet creates a periodical wave field around itself. They proposed that resonant interaction between the droplet and its own wave field exhibits behaviour analogous to quantum particles: interference in double-slit experiment, unpredictable tunneling (depending in a complicated way on a practically hidden state of field), orbit quantization (that a particle has to 'find a resonance' with field perturbations it creates—after one orbit, its internal phase has to return to the initial state) and
Zeeman effect The Zeeman effect () is the splitting of a spectral line into several components in the presence of a static magnetic field. It is caused by the interaction of the magnetic field with the magnetic moment of the atomic electron associated with ...
. While attempts to reproduce these experiments have shown some aspects to be questionable and the interpretation with respect to quantum mechanics has been challenged, work on the concept has continued with some success.


Mathematical foundations

To derive the de Broglie–Bohm pilot-wave for an electron, the quantum
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
:L(t)=mv^2-(V+Q), where V is the potential energy, v is the velocity and Q is the potential associated with the quantum force (the particle being pushed by the wave function), is integrated along precisely one path (the one the electron actually follows). This leads to the following formula for the Bohm
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
: :K^Q(X_1, t_1; X_0, t_0) = \frac \exp\left frac\int_^L(t)\,dt\right This
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
allows one to precisely track the electron over time under the influence of the quantum potential Q.


Derivation of the Schrödinger equation

Pilot wave theory is based on Hamilton–Jacobi dynamics, rather than
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
or
Hamiltonian dynamics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
. Using the Hamilton–Jacobi equation : H\left(\,\vec\,, \;\vec_\, S\,, \;t \,\right) + \left(\,\vec,\, t\,\right) = 0 it is possible to derive the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
: Consider a classical particle – the position of which is not known with certainty. We must deal with it statistically, so only the probability density \rho (\vec,t) is known. Probability must be conserved, i.e. \int\rho\,\mathrm^3\vec = 1 for each t. Therefore, it must satisfy the continuity equation :\frac = - \vec \cdot (\rho \,\vec ) \qquad\qquad (1) where \,\vec(\vec,t)\, is the velocity of the particle. In the Hamilton–Jacobi formulation of
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, velocity is given by \; \vec(\vec,t) = \frac \, \vec_ S(\vec,\,t) \; where \, S(\vec,t) \, is a solution of the Hamilton-Jacobi equation :- \frac = \frac + \tilde \qquad\qquad (2) \,(1)\, and \,(2)\, can be combined into a single complex equation by introducing the complex function \; \psi = \sqrt \, e^\frac \;, then the two equations are equivalent to :i\, \hbar\, \frac = \left( - \frac \nabla^2 +\tilde - Q \right)\psi \quad with : \; Q = - \frac \frac~. The time-dependent Schrödinger equation is obtained if we start with \;\tilde = V + Q \;, the usual potential with an extra
quantum potential The quantum potential or quantum potentiality is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952. Initially presented under the name ''quantum-mechanical potential'', subsequently ''qu ...
Q. The quantum potential is the potential of the quantum force, which is proportional (in approximation) to the
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
of the amplitude of the wave function. Note this potential is the same one that appears in the
Madelung equations In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the Schrödinger equation for a spinless non relativistic particle, written in terms of hydrodynamical v ...
, a classical analog of the Schrödinger equation.


Mathematical formulation for a single particle

The matter wave of de Broglie is described by the time-dependent Schrödinger equation: : i\, \hbar\, \frac = \left( - \frac \nabla^2 + V \right)\psi \quad The complex wave function can be represented as: \psi = \sqrt \; \exp \left( \frac \right) ~ 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 \,\rho\,: :\frac + \vec \cdot \left( \rho\, \vec \right) = 0 ~ , where the
velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
is determined by the “guidance equation” :\vec\left(\,\vec,\,t\,\right) = \frac \, \vec S\left(\, \vec,\, t \,\right) ~ . According to pilot wave theory, the point particle and the matter wave are both real and distinct physical entities (unlike standard quantum mechanics, which postulates no physical particle or wave entities, only observed wave-particle duality). 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 \; \rho = , \psi, ^2 ~ . 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 In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
for the action : :- \frac = \frac + V + Q ~ , where is the quantum potential defined by : Q = - \frac \frac ~. If we choose to neglect , 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 :m \, \frac \, \vec = - \vec( V + Q ) ~ , where the hydrodynamic time derivative is defined as :\frac = \frac + \vec \cdot \vec ~ .


Mathematical formulation for multiple particles

The Schrödinger equation for the many-body wave function \psi(\vec_1, \vec_2, \cdots, t) is given by : i \hbar \frac =\left( -\frac \sum_^ \frac + V(\mathbf_1,\mathbf_2,\cdots\mathbf_N) \right) \psi The complex wave function can be represented as: :\psi = \sqrt \; \exp \left( \frac \right) The pilot wave guides the motion of the particles. The guidance equation for the jth particle is: : \vec_j = \frac\; . The velocity of the jth particle explicitly depends on the positions of the other particles. This means that the theory is nonlocal.


Relativity

An extension to the relativistic case with spin has been developed since the 1990s.


Empty wave function

Lucien Hardy Lucien Hardy (born 1966) is a British-Canadian theoretical physicist currently based at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. Hardy is best known for his work on the foundation of quantum physics, including the ...
and
John Stewart Bell John Stewart Bell (28 July 1928 – 1 October 1990) was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum mechanics, quantum physics regarding hidden-variable theory, hidden-variable theor ...
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'' (or "Gespensterfelder", ''ghost fields'') by
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
. The empty wave function notion has been discussed controversially. In contrast, the
many-worlds interpretation The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is Philosophical realism, objectively real, and that there is no wave function collapse. This implies that all Possible ...
of quantum mechanics does not call for empty wave functions.


See also

* Hydrodynamic quantum analogues *
Quantum potential The quantum potential or quantum potentiality is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952. Initially presented under the name ''quantum-mechanical potential'', subsequently ''qu ...


Notes


References


External links


"Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics"
, lecture course on pilot wave theory by Mike Towler, Cambridge University (2009). * *Klaus von Bloh'
Bohmian mechanics demonstrations
in

{{DEFAULTSORT:Pilot wave Hidden variable theory Interpretations of quantum mechanics Quantum measurement