Pilot wave theory

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Couder's disputed experiments, purportedly "materializing" the pilot wave model.

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 quantum mechanics as a deterministic theory, and avoids issues such as wave–particle duality, instantaneous wave function collapse, and the paradox of Schrödinger's cat 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'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 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. 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 abandoned the pilot-wave approach. Unlike David Bohm 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 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 three years later, though for a variety of reasons this went unnoticed by the physics community for over fifty years.

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 only showed that the pilot wave theory did not have locality.

The pilot wave theory

Principles

(a) A walker in a circular corral. Trajectories of increasing length are colour-coded according to the droplet's local speed (b) The probability distribution of the walker's position corresponds roughly to the amplitude of the corral's Faraday wave mode.

The pilot wave theory is a hidden-variable theory. Consequently:

• the theory has realism (meaning that its concepts exist independently of the observer);
• the theory has determinism.

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. 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. These nonlocal effects can be shown to be compatible with the no-communication theorem, 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. Attempts to reproduce these experiments have shown that wall-droplet interactions rather than diffraction or interference of the pilot wave may be responsible for the observed hydrodynamic patterns, which are different from slit-induced interference patterns exhibited by quantum particles.

Mathematical foundations

To derive the de Broglie–Bohm pilot-wave for an electron, the quantum Lagrangian

${\displaystyle L(t)=\frac{1}{2}m{v}^2-(V+Q),}$

where ${\displaystyle V}$ is the potential energy, ${\displaystyle v}$ is the velocity and ${\displaystyle Q}$ 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:

${\displaystyle K^Q(X_1,t_1;X_0,t_0)=\frac{1}{J(t{)}^{\frac{1}{2}}}\exp \left[\frac{i}{\hslash }{\int }_{t_0}^{t_1}L(t)dt\right].}$

This propagator allows one to precisely track the electron over time under the influence of the quantum potential ${\displaystyle Q}$.

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

${\displaystyle H\left(\overrightarrow{x},{\overrightarrow{\mathrm{\nabla }}}_{x}S,t\right)+\frac{\mathrm{\partial }S}{\mathrm{\partial }t}\left(\overrightarrow{x},t\right)=0}$

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 ${\displaystyle \rho (\overrightarrow{x},t)}$ is known. Probability must be conserved, i.e. ${\displaystyle \int \rho {\mathrm{d}}^3\overrightarrow{x}=1}$ for each ${\displaystyle t}$. Therefore, it must satisfy the continuity equation

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}=-\overrightarrow{\mathrm{\nabla }}\cdot (\rho \overrightarrow{v})(1)}$

where ${\displaystyle \overrightarrow{v}(\overrightarrow{x},t)}$ is the velocity of the particle.

In the Hamilton–Jacobi formulation of classical mechanics, velocity is given by ${\displaystyle \overrightarrow{v}(\overrightarrow{x},t)=\frac{1}{m}{\overrightarrow{\mathrm{\nabla }}}_{x}S(\overrightarrow{x},t)}$ where ${\displaystyle S(\overrightarrow{x},t)}$ is a solution of the Hamilton-Jacobi equation

${\displaystyle -\frac{\mathrm{\partial }S}{\mathrm{\partial }t}=\frac{{\left|\mathrm{\nabla }S\right|}^2}{2m}+\tilde{V}(2)}$

${\displaystyle (1)}$ and ${\displaystyle (2)}$ can be combined into a single complex equation by introducing the complex function ${\displaystyle \psi =\sqrt{\rho }{e}^{\frac{iS}{\hslash }},}$ then the two equations are equivalent to

${\displaystyle i\hslash \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}=\left(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+\tilde{V}-Q\right)\psi }$

with

${\displaystyle Q=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\nabla }}^2\sqrt{\rho }}{\sqrt{\rho }}.}$

The time-dependent Schrödinger equation is obtained if we start with ${\displaystyle \tilde{V}=V+Q,}$ the usual potential with an extra quantum potential ${\displaystyle Q}$. The quantum potential is the potential of the quantum force, which is proportional (in approximation) to the curvature of the amplitude of the wave function.

Note this potential is the same one that appears in the Madelung equations, 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:

${\displaystyle i\hslash \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}=\left(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V\right)\psi }$

The complex wave function can be represented as:

${\displaystyle \psi =\sqrt{\rho }\exp \left(\frac{iS}{\hslash }\right)}$

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 ${\displaystyle \rho :}$

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\overrightarrow{\mathrm{\nabla }}\cdot \left(\rho \overrightarrow{v}\right)=0,}$

where the velocity field is determined by the “guidance equation”

${\displaystyle \overrightarrow{v}\left(\overrightarrow{r},t\right)=\frac{1}{m}\overrightarrow{\mathrm{\nabla }}S\left(\overrightarrow{r},t\right).}$

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 ${\displaystyle \rho =|\psi {|}^2.}$ 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 for the action S:

${\displaystyle -\frac{\mathrm{\partial }S}{\mathrm{\partial }t}=\frac{{\left|\overrightarrow{\mathrm{\nabla }}S\right|}^2}{2m}+V+Q,}$

where Q is the quantum potential defined by

${\displaystyle Q=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\nabla }}^2\sqrt{\rho }}{\sqrt{\rho }}.}$

If we choose to neglect 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

${\displaystyle m\frac{d}{dt}\overrightarrow{v}=-\overrightarrow{\mathrm{\nabla }}(V+Q),}$

where the hydrodynamic time derivative is defined as

${\displaystyle \frac{d}{dt}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\overrightarrow{v}\cdot \overrightarrow{\mathrm{\nabla }}.}$

Mathematical formulation for multiple particles

The Schrödinger equation for the many-body wave function ${\displaystyle \psi ({\overrightarrow{r}}_1,{\overrightarrow{r}}_2,\cdots ,t)}$ is given by

${\displaystyle i\hslash \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}=\left(-\frac{{\hslash }^2}{2}\sum _{i=1}^N\frac{{\mathrm{\nabla }}_i^2}{m_i}+V({\mathbf{r}}_1,{\mathbf{r}}_2,\cdots {\mathbf{r}}_N)\right)\psi }$

The complex wave function can be represented as:

${\displaystyle \psi =\sqrt{\rho }\exp \left(\frac{iS}{\hslash }\right)}$

The pilot wave guides the motion of the particles. The guidance equation for the jth particle is:

${\displaystyle {\overrightarrow{v}}_j=\frac{{\mathrm{\nabla }}_{j}S}{m_j}.}$

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 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 (or "Gespensterfelder", ghost fields) 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.

Hydrodynamic quantum analogs

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Hydrodynamic_quantum_analogs

In physics, the hydrodynamic quantum analogs refer to experimentally-observed phenomena involving bouncing fluid droplets over a vibrating fluid bath that behave analogously to several quantum-mechanical systems. The experimental evidence for diffraction through slits has been disputed, however, though the diffraction pattern of walking droplets is not exactly the same as in quantum physics, it does appear clearly in the high memory parameter regime (at high forcing of the bath) where all the quantum-like effects are strongest.

A droplet can be made to bounce indefinitely in a stationary position on a vibrating fluid surface. This is possible due to a pervading air layer that prevents the drop from coalescing into the bath. For certain combinations of bath surface acceleration, droplet size, and vibration frequency, a bouncing droplet will cease to stay in a stationary position, but instead “walk” in a rectilinear motion on top of the fluid bath. Walking droplet systems have been found to mimic several quantum mechanical phenomena including particle diffraction, quantum tunneling, quantized orbits, the Zeeman Effect, and the quantum corral.

Besides being an interesting means to visualise phenomena that are typical of the quantum-mechanical world, floating droplets on a vibrating bath have interesting analogies with the pilot wave theory, one of the many interpretations of quantum mechanics in its early stages of conception and development. The theory was initially proposed by Louis de Broglie in 1927. It suggests that all particles in motion are actually borne on a wave-like motion, similar to how an object moves on a tide. In this theory, it is the evolution of the carrier wave that is given by the Schrödinger equation. It is a deterministic theory and is entirely nonlocal. It is an example of a hidden variable theory, and all non-relativistic quantum mechanics can be accounted for in this theory. The theory was abandoned by de Broglie in 1932, gave way to the Copenhagen interpretation, but was revived by David Bohm in 1952 as De Broglie–Bohm theory. The Copenhagen interpretation does not use the concept of the carrier wave or that a particle moves in definite paths until a measurement is made.

Physics of bouncing and walking droplets

History

Floating droplets on a vibrating bath were first described in writing by Jearl Walker in a 1978 article in Scientific American.

In 2005, Yves Couder and his lab were the first to systematically study the dynamics of bouncing droplets and discovered most of the quantum mechanical analogs.

John Bush and his lab expanded upon Couder's work and studied the system in greater detail. In 2015 three separate groups, including John Bush, attempted to reproduce the effect and were unsuccessful.

Stationary bouncing droplet

A fluid droplet can float or bounce over a vibrating fluid bath because of the presence of an air layer between the droplet and the bath surface. The behavior of the droplet depends on the acceleration of the bath surface. Below a critical acceleration, the droplet will take successively smaller bounces before the intervening air layer eventually drains from underneath, causing the droplet to coalesce. Above the bouncing threshold, the intervening air layer replenishes during each bounce so the droplet never touches the bath surface. Near the bath surface, the droplet experiences equilibrium between inertial forces, gravity, and a reaction force due to the interaction with the air layer above the bath surface. This reaction force serves to launch the droplet back above the air like a trampoline. Molacek and Bush proposed two different models for the reaction force.

Walking droplet

For a small range of frequencies and drop sizes, a fluid droplet on a vibrating bath can be made to “walk” on the surface if the surface acceleration is sufficiently high (but still below the Faraday instability). That is, the droplet does not simply bounce in a stationary position but instead wanders in a straight line or in a chaotic trajectory. When a droplet interacts with the surface, it creates a transient wave that propagates from the point of impact. These waves usually decay, and stabilizing forces keep the droplet from drifting. However, when the surface acceleration is high, the transient waves created upon impact do not decay as quickly, deforming the surface such that the stabilizing forces are not enough to keep the droplet stationary. Thus, the droplet begins to “walk.”

Quantum phenomena on a macroscopic scale

A walking droplet on a vibrating fluid bath was found to behave analogously to several different quantum mechanical systems, namely particle diffraction, quantum tunneling, quantized orbits, the Zeeman effect, and the quantum corral.

Single and double slit diffraction

It has been known since the early 19th century that when light is shone through one or two small slits, a diffraction pattern appears on a screen far from the slits. Light has wave-like behavior, and interferes with itself through the slits, creating a pattern of alternating high and low intensity. Single electrons also exhibit wave-like behavior as a result of wave-particle duality. When electrons are fired through small slits, the probability of the electron striking the screen at a specific point shows an interference pattern as well.

In 2006, Couder and Fort demonstrated that walking droplets passing through one or two slits exhibit similar interference behavior. They used a square shaped vibrating fluid bath with a constant depth (aside from the walls). The “walls” were regions of much lower depth, where the droplets would be stopped or reflected away. When the droplets were placed in the same initial location, they would pass through the slits and be scattered, seemingly randomly. However, by plotting a histogram of the droplets based on scattering angle, the researchers found that the scattering angle was not random, but droplets had preferred directions that followed the same pattern as light or electrons. In this way, the droplet may mimic the behavior of a quantum particle as it passes through the slit.

Despite that research, in 2015 three teams: Bohr and Andersen's group in Denmark, Bush's team at MIT, and a team led by the quantum physicist Herman Batelaan at the University of Nebraska set out to repeat the Couder and Fort's bouncing-droplet double-slit experiment. Having their experimental setups perfected, none of the teams saw the interference-like pattern reported by Couder and Fort. Droplets went through the slits in almost straight lines, and no stripes appeared.

It has since been shown that droplet trajectories are sensitive to interactions with container boundaries, air currents, and other parameters. Though the diffraction pattern of walking droplets is not exactly the same as in quantum physics, and is not expected to show a Fraunhofer-like dependence of the number of peaks on the slit width, the diffraction pattern does appear clearly in the high memory regime (at high forcing of the bath).

Quantum tunneling

Quantum tunneling is the quantum mechanical phenomenon where a quantum particle passes through a potential barrier. In classical mechanics, a classical particle could not pass through a potential barrier if the particle does not have enough energy, so the tunneling effect is confined to the quantum realm. For example, a rolling ball would not reach the top of a steep hill without adequate energy. However, a quantum particle, acting as a wave, can undergo both reflection and transmission at a potential barrier. This can be shown as a solution to the time dependent Schrödinger Equation. There is a finite, but usually small, probability to find the electron at a location past the barrier. This probability decreases exponentially with increasing barrier width.

The macroscopic analogy using fluid droplets was first demonstrated in 2009. Researchers set up a square vibrating bath surrounded by walls on its perimeter. These “walls” were regions of lower depth, where a walking droplet may be reflected away. When the walking droplets were allowed to move around in the domain, they usually were reflected away from the barriers. However, surprisingly, sometimes the walking droplet would bounce past the barrier, similar to a quantum particle undergoing tunneling. In fact, the crossing probability was also found to decrease exponentially with increasing width of the barrier, exactly analogous to a quantum tunneling particle.

Quantized orbits

When two atomic particles interact and form a bound state, such the hydrogen atom, the energy spectrum is discrete. That is, the energy levels of the bound state are not continuous and only exist in discrete quantities, forming “quantized orbits.” In the case of a hydrogen atom, the quantized orbits are characterized by atomic orbitals, whose shapes are functions of discrete quantum numbers.

On the macroscopic level, two walking fluid droplets can interact on a vibrating surface. It was found that the droplets would orbit each other in a stable configuration with a fixed distance apart. The stable distances came in discrete values. The stable orbiting droplets analogously represent a bound state in the quantum mechanical system. The discrete values of the distance between droplets are analogous to discrete energy levels as well.

Zeeman effect

When an external magnetic field is applied to a hydrogen atom, for example, the energy levels are shifted to values slightly above or below the original level. The direction of shift depends on the sign of the z-component of the total angular momentum. This phenomenon is known as the Zeeman Effect.

In the context of walking droplets, an analogous Zeeman Effect can be demonstrated by observing orbiting droplets in a vibrating fluid bath. The bath is also brought to rotate at a constant angular velocity. In the rotating bath, the equilibrium distance between droplets shifts slightly farther or closer. The direction of shift depends on whether the orbiting drops rotate in the same direction as the bath or in opposite directions. The analogy to the quantum effect is clear. The bath rotation is analogous to an externally applied magnetic field, and the distance between droplets is analogous to energy levels. The distance shifts under an applied bath rotation, just as the energy levels shift under an applied magnetic field.

Quantum corral

Researchers have found that a walking droplet placed in a circular bath does not wander randomly, but rather there are specific locations the droplet is more likely to be found. Specifically, the probability of finding the walking droplet as a function of the distance from the center is non-uniform and there are several peaks of higher probability. This probability distribution mimics that of an electron confined to a quantum corral.

Double-slit experiment

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https://en.wikipedia.org/wiki/Double-slit_experiment

Photons or matter (like electrons) produce an interference pattern when two slits are used

 

Light from a green laser passing through two slits 0.4 mm wide and 0.1 mm apart

In modern physics, the double-slit experiment demonstrates that light and matter can exhibit behavior of both classical particles and classical waves. This type of experiment was first performed by Thomas Young in 1801, as a demonstration of the wave behavior of visible light. In 1927, Davisson and Germer and, independently, George Paget Thomson and his research student Alexander Reid demonstrated that electrons show the same behavior, which was later extended to atoms and molecules. Thomas Young's experiment with light was part of classical physics long before the development of quantum mechanics and the concept of wave–particle duality. He believed it demonstrated that the Christiaan Huygens' wave theory of light was correct, and his experiment is sometimes referred to as Young's experiment or Young's slits.

The experiment belongs to a general class of "double path" experiments, in which a wave is split into two separate waves (the wave is typically made of many photons and better referred to as a wave front, not to be confused with the wave properties of the individual photon) that later combine into a single wave. Changes in the path-lengths of both waves result in a phase shift, creating an interference pattern. Another version is the Mach–Zehnder interferometer, which splits the beam with a beam splitter.

In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles (not waves); the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. These results demonstrate the principle of wave–particle duality.

Other atomic-scale entities, such as electrons, are found to exhibit the same behavior when fired towards a double slit. Additionally, the detection of individual discrete impacts is observed to be inherently probabilistic, which is inexplicable using classical mechanics.

The experiment can be done with entities much larger than electrons and photons, although it becomes more difficult as size increases. The largest entities for which the double-slit experiment has been performed were molecules that each comprised 2000 atoms (whose total mass was 25,000 atomic mass units).

The double-slit experiment (and its variations) has become a classic for its clarity in expressing the central puzzles of quantum mechanics. Richard Feynman called it "a phenomenon which is impossible […] to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]."

Overview

Same double-slit assembly (0.7 mm between slits); in top image, one slit is closed. In the single-slit image, a diffraction pattern (the faint spots on either side of the main band) forms due to the nonzero width of the slit. This diffraction pattern is also seen in the double-slit image, but with many smaller interference fringes.

If light consisted strictly of ordinary or classical particles, and these particles were fired in a straight line through a slit and allowed to strike a screen on the other side, we would expect to see a pattern corresponding to the size and shape of the slit. However, when this "single-slit experiment" is actually performed, the pattern on the screen is a diffraction pattern in which the light is spread out. The smaller the slit, the greater the angle of spread. The top portion of the image shows the central portion of the pattern formed when a red laser illuminates a slit and, if one looks carefully, two faint side bands. More bands can be seen with a more highly refined apparatus. Diffraction explains the pattern as being the result of the interference of light waves from the slit.

If one illuminates two parallel slits, the light from the two slits again interferes. Here the interference is a more pronounced pattern with a series of alternating light and dark bands. The width of the bands is a property of the frequency of the illuminating light. (See the bottom photograph to the right.)

Young's drawing of diffraction

When Thomas Young (1773–1829) first demonstrated this phenomenon, it indicated that light consists of waves, as the distribution of brightness can be explained by the alternately additive and subtractive interference of wavefronts. Young's experiment, performed in the early 1800s, played a crucial role in the understanding of the wave theory of light, vanquishing the corpuscular theory of light proposed by Isaac Newton, which had been the accepted model of light propagation in the 17th and 18th centuries.

However, the later discovery of the photoelectric effect demonstrated that under different circumstances, light can behave as if it is composed of discrete particles. These seemingly contradictory discoveries made it necessary to go beyond classical physics and take into account the quantum nature of light.

Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment. He also proposed (as a thought experiment) that if detectors were placed before each slit, the interference pattern would disappear.

The Englert–Greenberger duality relation provides a detailed treatment of the mathematics of double-slit interference in the context of quantum mechanics.

A low-intensity double-slit experiment was first performed by G. I. Taylor in 1909, by reducing the level of incident light until photon emission/absorption events were mostly non-overlapping. A slit interference experiment was not performed with anything other than light until 1961, when Claus Jönsson of the University of Tübingen performed it with coherent electron beams and multiple slits. In 1974, the Italian physicists Pier Giorgio Merli, Gian Franco Missiroli, and Giulio Pozzi performed a related experiment using single electrons from a coherent source and a biprism beam splitter, showing the statistical nature of the buildup of the interference pattern, as predicted by quantum theory. In 2002, the single-electron version of the experiment was voted "the most beautiful experiment" by readers of Physics World. Since that time a number of related experiments have been published, with a little controversy.

In 2012, Stefano Frabboni and co-workers sent single electrons onto nanofabricated slits (about 100 nm wide) and, by detecting the transmitted electrons with a single-electron detector, they could show the build-up of a double-slit interference pattern. Many related experiments involving the coherent interference have been performed; they are the basis of modern electron diffraction, microscopy and high resolution imaging.

In 2018, single particle interference was demonstrated for antimatter in the Positron Laboratory (L-NESS, Politecnico di Milano) of Rafael Ferragut in Como (Italy), by a group led by Marco Giammarchi.

Variations of the experiment

Interference from individual particles

An important version of this experiment involves single particle detection. Illuminating the double-slit with a low intensity results in single particles being detected as white dots on the screen. Remarkably, however, an interference pattern emerges when these particles are allowed to build up one by one (see the image below).

 

Experimental electron double slit diffraction pattern. Across the middle of the image at the top, the intensity alternates from high to low, showing interference in the signal from the two slits. Bottom: movie of the pattern being built up dot-by-dot.

This demonstrates the wave–particle duality, which states that all matter exhibits both wave and particle properties: The particle is measured as a single pulse at a single position, while the modulus squared of the wave describes the probability of detecting the particle at a specific place on the screen giving a statistical interference pattern. This phenomenon has been shown to occur with photons, electrons, atoms, and even some molecules: with buckminsterfullerene (C
₆₀) in 2001, with 2 molecules of 430 atoms (C
₆₀(C
₁₂F
₂₅)
₁₀ and C
₁₆₈H
₉₄F
₁₅₂O
₈N
₄S
₄) in 2011, and with molecules of up to 2000 atoms in 2019. In addition to interference patterns built up from single particles, up to 4 entangled photons can also show interference patterns.

Mach-Zehnder interferometer

Main article: Mach–Zehnder interferometer

Light in Mach–Zehnder interferometer produces interference (wave-like behavior) even when being detected one photon at a time (particle-like behavior)

The Mach–Zehnder interferometer can be seen as a simplified version of the double-slit experiment. Instead of propagating through free space after the two slits, and hitting any position in an extended screen, in the interferometer the photons can only propagate via two paths, and hit two discrete photodetectors. This makes it possible to describe it via simple linear algebra in dimension 2, rather than differential equations.

A photon emitted by the laser hits the first beam splitter and is then in a superposition between the two possible paths. In the second beam splitter these paths interfere, causing the photon to hit the photodetector on the right with probability one, and the photodetector on the bottom with probability zero. Blocking one of the paths, or equivalently detecting the presence of a photon on a path eliminates interference between the paths: both photodetectors will be hit with probability 1/2. This indicates that after the first beam splitter the photon does not take one path or another, but rather exists in a quantum superposition of the two paths.

"Which-way" experiments and the principle of complementarity

A well-known thought experiment predicts that if particle detectors are positioned at the slits, showing through which slit a photon goes, the interference pattern will disappear. This which-way experiment illustrates the complementarity principle that photons can behave as either particles or waves, but cannot be observed as both at the same time. Despite the importance of this thought experiment in the history of quantum mechanics (for example, see the discussion on Einstein's version of this experiment), technically feasible realizations of this experiment were not proposed until the 1970s. (Naive implementations of the textbook thought experiment are not possible because photons cannot be detected without absorbing the photon.) Currently, multiple experiments have been performed illustrating various aspects of complementarity.

An experiment performed in 1987 produced results that demonstrated that partial information could be obtained regarding which path a particle had taken without destroying the interference altogether. This "wave-particle trade-off" takes the form of an inequality relating the visibility of the interference pattern and the distinguishability of the which-way paths.

Delayed choice and quantum eraser variations

Main article: Delayed-choice quantum eraser

A diagram of Wheeler's delayed choice experiment, showing the principle of determining the path of the photon after it passes through the slit

Wheeler's delayed-choice experiments demonstrate that extracting "which path" information after a particle passes through the slits can seem to retroactively alter its previous behavior at the slits.

Quantum eraser experiments demonstrate that wave behavior can be restored by erasing or otherwise making permanently unavailable the "which path" information.

A simple do-it-at-home illustration of the quantum eraser phenomenon was given in an article in Scientific American. If one sets polarizers before each slit with their axes orthogonal to each other, the interference pattern will be eliminated. The polarizers can be considered as introducing which-path information to each beam. Introducing a third polarizer in front of the detector with an axis of 45° relative to the other polarizers "erases" this information, allowing the interference pattern to reappear. This can also be accounted for by considering the light to be a classical wave, and also when using circular polarizers and single photons. Implementations of the polarizers using entangled photon pairs have no classical explanation.

Weak measurement

Main article: Weak measurement

In a highly publicized experiment in 2012, researchers claimed to have identified the path each particle had taken without any adverse effects at all on the interference pattern generated by the particles. In order to do this, they used a setup such that particles coming to the screen were not from a point-like source, but from a source with two intensity maxima. However, commentators such as Svensson have pointed out that there is in fact no conflict between the weak measurements performed in this variant of the double-slit experiment and the Heisenberg uncertainty principle. Weak measurement followed by post-selection did not allow simultaneous position and momentum measurements for each individual particle, but rather allowed measurement of the average trajectory of the particles that arrived at different positions. In other words, the experimenters were creating a statistical map of the full trajectory landscape.

Other variations

A laboratory double-slit assembly; distance between top posts is approximately 2.5 cm (one inch).

Near-field intensity distribution patterns for plasmonic slits with equal widths (A) and non-equal widths (B).

In 1967, Pfleegor and Mandel demonstrated two-source interference using two separate lasers as light sources.

It was shown experimentally in 1972 that in a double-slit system where only one slit was open at any time, interference was nonetheless observed provided the path difference was such that the detected photon could have come from either slit. The experimental conditions were such that the photon density in the system was much less than 1.

In 1991, Carnal and Mlynek performed the classic Young's double slit experiment with metastable helium atoms passing through micrometer-scale slits in gold foil.

In 1999, a quantum interference experiment (using a diffraction grating, rather than two slits) was successfully performed with buckyball molecules (each of which comprises 60 carbon atoms). A buckyball is large enough (diameter about 0.7 nm, nearly half a million times larger than a proton) to be seen in an electron microscope.

In 2002, an electron field emission source was used to demonstrate the double-slit experiment. In this experiment, a coherent electron wave was emitted from two closely located emission sites on the needle apex, which acted as double slits, splitting the wave into two coherent electron waves in a vacuum. The interference pattern between the two electron waves could then be observed. In 2017, researchers performed the double-slit experiment using light-induced field electron emitters. With this technique, emission sites can be optically selected on a scale of ten nanometers. By selectively deactivating (closing) one of the two emissions (slits), researchers were able to show that the interference pattern disappeared.

In 2005, E. R. Eliel presented an experimental and theoretical study of the optical transmission of a thin metal screen perforated by two subwavelength slits, separated by many optical wavelengths. The total intensity of the far-field double-slit pattern is shown to be reduced or enhanced as a function of the wavelength of the incident light beam.

In 2012, researchers at the University of Nebraska–Lincoln performed the double-slit experiment with electrons as described by Richard Feynman, using new instruments that allowed control of the transmission of the two slits and the monitoring of single-electron detection events. Electrons were fired by an electron gun and passed through one or two slits of 62 nm wide × 4 μm tall.

In 2013, a quantum interference experiment (using diffraction gratings, rather than two slits) was successfully performed with molecules that each comprised 810 atoms (whose total mass was over 10,000 atomic mass units). The record was raised to 2000 atoms (25,000 amu) in 2019.

Hydrodynamic pilot wave analogs

Hydrodynamic analogs have been developed that can recreate various aspects of quantum mechanical systems, including single-particle interference through a double-slit. A silicone oil droplet, bouncing along the surface of a liquid, self-propels via resonant interactions with its own wave field. The droplet gently sloshes the liquid with every bounce. At the same time, ripples from past bounces affect its course. The droplet's interaction with its own ripples, which form what is known as a pilot wave, causes it to exhibit behaviors previously thought to be peculiar to elementary particles – including behaviors customarily taken as evidence that elementary particles are spread through space like waves, without any specific location, until they are measured.

Behaviors mimicked via this hydrodynamic pilot-wave system include quantum single particle diffraction, tunneling, quantized orbits, orbital level splitting, spin, and multimodal statistics. It is also possible to infer uncertainty relations and exclusion principles. Videos are available illustrating various features of this system. 

However, more complicated systems that involve two or more particles in superposition are not amenable to such a simple, classically intuitive explanation. Accordingly, no hydrodynamic analog of entanglement has been developed. Nevertheless, optical analogs are possible.

Double-slit experiment on time

In 2023, an experiment was reported recreating an interference pattern in time by shining a pump laser pulse at a screen coated in indium tin oxide (ITO) which would alter the properties of the electrons within the material due to the Kerr effect, changing it from transparent to reflective for around 200 femtoseconds long where a subsequent probe laser beam hitting the ITO screen would then see this temporary change in optical properties as a slit in time and two of them as a double slit with a phase difference adding up destructively or constructively on each frequency component resulting in an interference pattern. Similar results have been obtained classically on water waves.

Classical wave-optics formulation

Two-slit diffraction pattern with an incident plane wave

Photo of the double-slit interference of sunlight.

Two slits are illuminated by a plane wave, showing the path difference.

Much of the behaviour of light can be modelled using classical wave theory. The Huygens–Fresnel principle is one such model; it states that each point on a wavefront generates a secondary wavelet, and that the disturbance at any subsequent point can be found by summing the contributions of the individual wavelets at that point. This summation needs to take into account the phase as well as the amplitude of the individual wavelets. Only the intensity of a light field can be measured—this is proportional to the square of the amplitude.

In the double-slit experiment, the two slits are illuminated by the quasi-monochromatic light of a single laser. If the width of the slits is small enough (much less than the wavelength of the laser light), the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts. The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.

If the viewing distance is large compared with the separation of the slits (the far field), the phase difference can be found using the geometry shown in the figure below right. The path difference between two waves travelling at an angle θ is given by:

${\displaystyle d\sin \theta \approx d\theta }$

Where d is the distance between the two slits. When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximum, and when they are in anti-phase, i.e. the path difference is equal to half a wavelength, one and a half wavelengths, etc., then the two waves cancel and the summed intensity is zero. This effect is known as interference. The interference fringe maxima occur at angles

${\displaystyle d{\theta }_n=n\lambda ,n=0,1,2,\dots }$

where λ is the wavelength of the light. The angular spacing of the fringes, θ_f, is given by

${\displaystyle {\theta }_f\approx \lambda /d}$

The spacing of the fringes at a distance z from the slits is given by

${\displaystyle w=z{\theta }_f=z\lambda /d}$

For example, if two slits are separated by 0.5 mm (d), and are illuminated with a 0.6 μm wavelength laser (λ), then at a distance of 1 m (z), the spacing of the fringes will be 1.2 mm.

If the width of the slits b is appreciable compared to the wavelength, the Fraunhofer diffraction equation is needed to determine the intensity of the diffracted light as follows:

${\displaystyle \begin{array}{rl}I(\theta )& \propto {\cos }^2\left[\frac{\pi d\sin \theta }{\lambda }\right]{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}}^2\left[\frac{\pi b\sin \theta }{\lambda }\right]\end{array}}$

where the sinc function is defined as sinc(x) = sin(x)/x for x ≠ 0, and sinc(0) = 1.

This is illustrated in the figure above, where the first pattern is the diffraction pattern of a single slit, given by the sinc function in this equation, and the second figure shows the combined intensity of the light diffracted from the two slits, where the cos function represents the fine structure, and the coarser structure represents diffraction by the individual slits as described by the sinc function.

Similar calculations for the near field can be made by applying the Fresnel diffraction equation, which implies that as the plane of observation gets closer to the plane in which the slits are located, the diffraction patterns associated with each slit decrease in size, so that the area in which interference occurs is reduced, and may vanish altogether when there is no overlap in the two diffracted patterns.

Path-integral formulation

One of an infinite number of equally likely paths used in the Feynman path integral (see also: Wiener process)

The double-slit experiment can illustrate the path integral formulation of quantum mechanics provided by Feynman. The path integral formulation replaces the classical notion of a single, unique trajectory for a system, with a sum over all possible trajectories. The trajectories are added together by using functional integration.

Each path is considered equally likely, and thus contributes the same amount. However, the phase of this contribution at any given point along the path is determined by the action along the path: $${\displaystyle A_{\text{path}}(x,y,z,t)=e^{iS(x,y,z,t)}}$$

All these contributions are then added together, and the magnitude of the final result is squared, to get the probability distribution for the position of a particle: $${\displaystyle p(x,y,z,t)\propto {\left|{\int }_{\text{all paths}}{e}^{iS(x,y,z,t)}\right|}^2}$$

As is always the case when calculating probability, the results must then be normalized by imposing: $${\displaystyle {\iiint }_{\text{all space}}p(x,y,z,t)dV=1}$$

The probability distribution of the outcome is the normalized square of the norm of the superposition, over all paths from the point of origin to the final point, of waves propagating proportionally to the action along each path. The differences in the cumulative action along the different paths (and thus the relative phases of the contributions) produces the interference pattern observed by the double-slit experiment. Feynman stressed that his formulation is merely a mathematical description, not an attempt to describe a real process that we can measure.

Interpretations of the experiment

Like the Schrödinger's cat thought experiment, the double-slit experiment is often used to highlight the differences and similarities between the various interpretations of quantum mechanics.

Standard quantum physics

The standard interpretation of the double slit experiment is that the pattern is a wave phenomenon, representing interference between two probability amplitudes, one for each slit. Low intensity experiments demonstrate that the pattern is filled in one particle detection at a time. Any change to the apparatus designed to detect a particle at a particular slit alters the probability amplitudes and the interference disappears. This interpretation is independent of any conscious observer.

Complementarity

Main article: Complementarity (physics)

Niels Bohr interpreted quantum experiments like the double-slit experiment using the concept of complementarity. In Bohr's view quantum systems are not classical, but measurements can only give classical results. Certain pairs of classical properties will never be observed in a quantum system simultaneously: the interference pattern of waves in the double slit experiment will disappear if particles are detected at the slits. Modern quantitative versions of the concept allow for a continuous tradeoff between the visibility of the interference fringes and the probability of particle detection at a slit.

Copenhagen interpretation

Main article: Copenhagen interpretation

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, stemming from the work of Niels Bohr, Werner Heisenberg, Max Born, and others. The term "Copenhagen interpretation" was apparently coined by Heisenberg during the 1950s to refer to ideas developed in the 1925–1927 period, glossing over his disagreements with Bohr. Consequently, there is no definitive historical statement of what the interpretation entails. Features common across versions of the Copenhagen interpretation include the idea that quantum mechanics is intrinsically indeterministic, with probabilities calculated using the Born rule, and some form of complementarity principle. Moreover, the act of "observing" or "measuring" an object is irreversible, and no truth can be attributed to an object, except according to the results of its measurement. In the Copenhagen interpretation, complementarity means a particular experiment can demonstrate particle behavior (passing through a definite slit) or wave behavior (interference), but not both at the same time. In a Copenhagen-type view, the question of which slit a particle travels through has no meaning when there is no detector.

Relational interpretation

According to the relational interpretation of quantum mechanics, first proposed by Carlo Rovelli, observations such as those in the double-slit experiment result specifically from the interaction between the observer (measuring device) and the object being observed (physically interacted with), not any absolute property possessed by the object. In the case of an electron, if it is initially "observed" at a particular slit, then the observer–particle (photon–electron) interaction includes information about the electron's position. This partially constrains the particle's eventual location at the screen. If it is "observed" (measured with a photon) not at a particular slit but rather at the screen, then there is no "which path" information as part of the interaction, so the electron's "observed" position on the screen is determined strictly by its probability function. This makes the resulting pattern on the screen the same as if each individual electron had passed through both slits.

Many-worlds interpretation

As with Copenhagen, there are multiple variants of the many-worlds interpretation. The unifying theme is that physical reality is identified with a wavefunction, and this wavefunction always evolves unitarily, i.e., following the Schrödinger equation with no collapses. Consequently, there are many parallel universes, which only interact with each other through interference. David Deutsch argues that the way to understand the double-slit experiment is that in each universe the particle travels through a specific slit, but its motion is affected by interference with particles in other universes, and this interference creates the observable fringes. David Wallace, another advocate of the many-worlds interpretation, writes that in the familiar setup of the double-slit experiment the two paths are not sufficiently separated for a description in terms of parallel universes to make sense.

De Broglie–Bohm theory

Main article: de Broglie–Bohm theory

An alternative to the standard understanding of quantum mechanics, the De Broglie–Bohm theory states that particles also have precise locations at all times, and that their velocities are defined by the wave-function. So while a single particle will travel through one particular slit in the double-slit experiment, the so-called "pilot wave" that influences it will travel through both. The two slit de Broglie-Bohm trajectories were first calculated by Chris Dewdney while working with Chris Philippidis and Basil Hiley at Birkbeck College (London). The de Broglie-Bohm theory produces the same statistical results as standard quantum mechanics, but dispenses with many of its conceptual difficulties by adding complexity through an ad hoc quantum potential to guide the particles.

While the model is in many ways similar to Schrödinger equation, it is known to fail for relativistic cases and does not account for features such as particle creation or annihilation in quantum field theory. Many authors such as nobel laureates Werner Heisenberg, Sir Anthony James Leggett and Sir Roger Penrose have criticized it for not adding anything new.

More complex variants of this type of approach have appeared, for instance the three wave hypothesis of Ryszard Horodecki as well as other complicated combinations of de Broglie and Compton waves. To date there is no evidence that these are useful.

Bohmian trajectories

Trajectories of particles in De Broglie–Bohm theory in the double-slit experiment.

 

100 trajectories guided by the wave function. In De Broglie-Bohm's theory, a particle is represented, at any time, by a wave function and a position (center of mass). This is a kind of augmented reality compared to the standard interpretation.

 

Numerical simulation of the double-slit experiment with electrons. Figure on the left: evolution (from left to right) of the intensity of the electron beam at the exit of the slits (left) up to the detection screen located 10 cm after the slits (right). The higher the intensity, the more the color is light blue – Figure in the center: impacts of the electrons observed on the screen – Figure on the right: intensity of the electrons in the far field approximation (on the screen). Numerical data from Claus Jönsson's experiment (1961). Photons, atoms and molecules follow a similar evolution.