User:Ldm1954/Sandbox/Duality

See also: User:Ldm1954/Sandbox/Duality Old and User:Ldm1954/Sandbox/Duality V2

Wave–particle duality is the concept in quantum mechanics that quantum entities can have both particle and a wave properties. Classical waves are familiar in many systems, for instance water on a lake. Classical particles are also common, for instance projectiles from a gun. Classical mechanics is determinism, in other words if we know the position, velocity and force on a particle at some given instant then we know where it will be later. The same holds for classical waves. In contrast, quantum systems can exhibit either or both wave or particle behavior. They not deterministic, and can only be understood through statistical probabilities. This can be a disturbing view, for instance the famous quote by Albert Einstein

God does not play dice.

to which Niels Bohr once responded

Einstein, stop telling God what to do.

This article will discuss part of the historical origins which led to the concept of duality, some of the key experimental results involving both matter waves and photons both as single entities and in terms of statistical probabilities, plus some aspects of interpretation. The philosophical implications of wave-particle duality remains a topic of discussion and classes.

Classical and quantum waves and particles

Before proceeding further it is critical to introduce some definitions of waves and particles both in a classical sense, in quantum mechanics and with a probabilistic interpretation.

Classical waves and particles

Wave interference in water due to two sources marked as red points on the left

Classical trajectories for a mass thrown at an angle of 70°, at different speeds

Line trace for a two-slit electron interference pattern

Curved arc shows a cloud chamber trajectory of a positron

Quantum interference and trajectories both observed

Classical waves obey a wave equation; they have continuous values at many points in space that vary with time; their spatial extent can vary with time due to diffraction, and they display wave interference.

Classical particle obey classical mechanics; they have some center and extent; they follow trajectories characterized by positions and velocities that vary over time; in the absence of forces their trajectories are straight lines; particles do not exhibit interference.

Quantum waves have wavefunctions which obey equations similar to the wave equation. The probability that they can be measured at any point in space and time behaves similar to a classical wave with both diffraction and interference. They can be due to single entities such as photons or electrons, or they can be collective excitations which are called quasiparticles.

Quantum particles have discrete values called quanta for properties such as spin, electric charge and magnetic moment.

The quantum interpretations are based upon probability, statistics and measurements. When any quantum particle is observed, it has particle-like characteristics, but a statistically large number will follow probabilities and can display wave-like properties.

History

Around the year 1900 it was understood that light was a wave, and electrons as well as atoms were particles. There were a few pieces of experimental evidence that hinted at something deeper. Over the next quarter of a century there was a major change in scientific thinking with acceptance of quantization of light as well as wave behavior of electrons, all of which led to the concept of wave-particle duality.

In the late 17th century Sir Isaac Newton had advocated that light was particles, but Christiaan Huygens took an opposing wave approach. Thomas Young's interference experiments in 1801, and François Arago's detection of the Poisson spot in 1819, validated Huygen's wave models. However the wave model was challenged in 1901 by Planck's law for black-body radiation. Max Planck heuristically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged oscillator in a cavity that contained black-body radiation could only change its energy in a minimal increment, E, that was proportional to the frequency of its associated electromagnetic wave. In 1905 Einstein interpreted the photoelectric effect also with discrete energies for photons. These both indicate particle behavior. Despite confirmation by various experimental observations, the photon theory (as it came to be called) remained controversial until Arthur Compton performed a series of experiments from 1922 to 1924 demonstrating the momentum of light. The experimental evidence of particle-like momentum and energy seemingly contradicted the earlier work demonstrating wave-like interference of light.

The contradictory evidence from electrons arrived in the opposite order. Many experiments by J. J. Thompson, Robert Millikan, and Charles Wilson among others had shown that free electrons had particle properties, for instance the measurement of their mass by Thompson in 1897. In 1924 Louis de Broglie in his PhD thesis Recherches sur la théorie des quanta introduced his theory of electron waves. He suggested that an electron around a nucleus could be thought of as being a standing wave, and that electrons and all matter could be considered as waves. He merged the idea of thinking about them as particles, and of thinking of them as waves. He proposed that particles are bundles of waves (wave packets) which move with a group velocity and have an effective mass. Both of these depend upon the energy, which in turn connects to the wavevector and the relativistic formulation of Albert Einstein a few years before.

This rapidly became part of what was called by Erwin Schrödinger undulatory mechanics, now called the Schrödinger equation or wave mechanics. Both the wave nature and the undulatory mechanics approach were experimentally confirmed for electron beams by experiments from two groups performed independently, the first the Davisson–Germer experiment, the other by George Paget Thomson and Alexander Reid. Alexander Reid, who was Thomson's graduate student, performed the first experiments, but he died soon after in a motorcycle accident and is rarely mentioned. These experiments were rapidly followed by the first non-relativistic diffraction model for electrons by Hans Bethe based upon the Schrödinger equation, which is very close to how electron diffraction is now described. Significantly, Davidsson and Germer noticed that their results could not be interpreted using a Bragg's law approach as the positions were systematically different; the approach of Bethe which includes the refraction due to the average potential yielded more accurate results.

Duality in terms of probability and measurement

To reconcile the conflicting behavior where some sometimes waves, sometimes particles is observed, the modern approach is to analyze a quantum system in terms of probabilities and measurements. Taking, as an example, a single electron, it would be described in terms of a wave function $\psi (r)$ as a function of position $r$. There will a corresponding probability of detecting the electron at a position $r$ of $|\psi (r)|^{2}$, for instance by the light it produces at a detector. At any position this value is a fraction of one, with the sum (integral) over all positions one. One electron can be detected anywhere; however if there is some very large number such as a million, approximately a million times $|\psi (r)|^{2}$ will be detected at each position. Because it is all probabilities, it won't be exact, but the more electrons are collected the closer it will become; the result will be a measurement of the probability distribution. One electron only produces light at a specific location, so when it is detected it is effectively a quantized particle. However, millions can either be detected at many positions, equivalent statistically to measuring a wave, or if the probability is only large in a small region, all the detections will be essentially the same so it is effectively being measured as a particle. Whether the electrons are behaving as particles or a waves depending on how they are being measured.

This can be illustrated by experimental data. Single-particle interferometry, specifically the double-slit experiment with electrons at low count rates has become a classic this purpose. Richard Feynman called this experiment "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]. Both particle and wave aspects are seen in different parts of this experiment. Note that exactly the same holds for photons and atoms, with the caveat of coherence which will be mentioned later.

Observing electrons as waves

Calculated pattern from a plane wave incident on a slit. The wave spreads out, characteristic of diffraction.

There are two different ways that electrons or any other type of quantum entity can manifest as a wave when millions of them are detected: diffraction and wave interference.

Single slit experiment

A classic example of the first is a single slit experiment. As illustrated in the Figure, if we have any form of wave incident on a small aperture or slit, it spreads out after the aperture, what is also called Fresnel diffraction. Electrons, atoms and also light exhibit this type of wave phenomena. There is extensive experimental and theoretical data demonstrating this for electron diffraction, X-ray diffraction, neutron diffraction as well as helium, sodium and even larger molecules. In all cases a detector placed (in terms of the figure) vertically some distance from the slit will show oscillations in the number of detected species, consistent with the Fresnel diffraction.

Double slit experiment

Adding to diffraction, interference is an additional wave behavior. An electron double slit experiment is shown schematically in figure (a) below, together with results for each slit individually in (b) and the two slit combination in (c), all at high electron intensity so many millions of electrons are being detected.

Left half: schematic setup for electron double-slit experiment with masking; inset micrographs of slits and mask; Right half: results for slit 1, slit 2 and both slits open.

(The schematic distorts the geometry to illustrate the concept; in practice the slits are very small and far from the source).

As shown in (b) with one or the other slit open a smooth peak shows up, only diffraction is present. When both slits are open as in (c) the intensity in the pattern oscillates, characteristic of wave interference.

Observing electrons as particles

Having observed wave behavior for large numbers, now change the experiment, lowering the intensity of the electron source until only one or two are detected per second, appearing as individual dots, single quantized particles. As shown in the movie clip below, the dots on the detector seem at first to be random. After some time a pattern emerges. Eventually we see an alternating pattern of light and dark bands shown in the image at the top. It is the probability when we have millions of electrons that gives us a result which is comparable to what one has in classic wave interference. Little can be said about a single electron except that, when detected, it is a particle (due to wave function collapse). Averaged over large numbers we have a wave.

Experimental electron double slit diffraction pattern. Across the middle of the image the intensity alternates from high to low showing interference in the signal from the two slits. Bottom: movie of the pattern build up dot by dot. Click on the thumbnail to enlarge the movie.

Probability, coherence and density matrix

The electron single and double-slit experiments demonstrate particle behavior when viewing individual events, and wave behavior when events are summed up. Unlike water waves, the modulus square of the quantum wave amplitudes is a probability distribution for the observed electron counts. The electrons observed act like samples from a continuous pattern predicted by the quantum wave equation. One caveat: coherence, which requires that we go beyond a simple explanation in terms of a single wavefunction. Consider the two slits at $r_{1}$and $r_{2}$. If we write

$\psi (r_{1})=|\psi (r_{1})|\exp(i\phi _{1})$

$\psi (r_{2})=|\psi (r_{2})|\exp(i\phi _{2})$

with phase terms $\psi _{1}$ and $\psi _{2}$, the mutual coherence is defined as

$\Gamma (r_{1},r_{2})=\psi (r_{1})\psi ^{*}(r_{2})=|\psi (r_{1})\psi (r_{2})|\exp(i(\phi _{1}-\phi _{2}))=|\psi (r_{1})\psi (r_{2})|\exp(i\Delta \phi )$

Image (a-c) and line trace (d-f) for high coherence to low coherence for a double slit experiment.

The question now is whether the phase difference $\Delta \phi$ is a constant for every single electron (or photon, atom) or not when we consider millions being detected. If it is constant, ideally $\Delta \phi =0$, then the system is coherent and the oscillating fringes detailed above are observed. If, instead, the phase difference is statistically random then the system is incoherent and the interference fringes are not observed, just a smoothly varying intensity due to diffrsction. The same type of approach can also be used within quantum mechanics, and is called the density matrix formalism.

The distance over which there is coherence is called the coherence length in classical electromagnetic theory, and in quantum mechanics the quantum coherence. The light from a laser is highly coherent, with a coherence length of centimeters or more; that from a normal light bulb is almost completely incoherent. When the coherence length is very small, as show to the right of the figure, then there is no interference behavior, but the diffraction continues; when it is large (left) there is both interference and diffraction. The coherence length of the detector also matters, since in most cases it is very small, for instance the "dots" in the video above. Whether the behavior is that of particles or a waves, manifests diffraction or interference depends on experimental details, the wave-particle duality.

Observing photons as particles

Photoelectric effect in a solid

Main articles: Photoelectric effect and Compton scattering

In 1887, Heinrich Hertz observed that when light with sufficient frequency hits a metallic surface, the surface emits cathode rays, what are now called electrons. In 1902, Philipp Lenard discovered that the maximum possible energy of an ejected electron is unrelated to its intensity. This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the incident radiation. In 1905, Albert Einstein suggested that the energy of the light must occur a finite number of energy quanta. He postulated that electrons can receive energy from an electromagnetic field only in discrete units (quanta or photons): an amount of energy E that was related to the frequency f of the light by

$E=hf$

A photon of wavelength $\lambda$ comes in from the left, collides with a target at rest, and a new photon of wavelength $\lambda '$ emerges at an angle $\theta$. The target recoils, and the photons have provided momentum to the target.

where h is the Planck constant (6.626×10⁻³⁴ J⋅s). Only photons of a high enough frequency (above a certain threshold value which is the work function) could knock an electron free. For example, photons of blue light had sufficient energy to free an electron from the metal he used, but photons of red light did not. One photon of light above the threshold frequency could release only one electron; the higher the frequency of a photon, the higher the kinetic energy of the emitted electron, but no amount of light below the threshold frequency could release an electron. Despite confirmation by various experimental observations, the photon theory (as it came to be called later) remained controversial until Arthur Compton performed a series of experiments from 1922 to 1924 demonstrating the momentum of light.

Both discrete (quantized) energies and also momentum are, classically, particle attributes. There many other examples where photons display particle-type properties, for instance in solar sails, where sunlight could propel a space vehicle and laser cooling where the momentum is used to slow down (cool) atoms.

Interpretation

The observation of wave-particle duality, that some experiments produce wave-like results and some produce particle-like results seem contradictory. As Albert Einstein wrote:

It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.

Different interpretations of quantum mechanics analyze this contradiction in different ways.

Conventional interpretation

The interpretation adopted by Nobel laureates in quantum physics, Bohr, Heisenberg, Schrodinger,, Feynman, and Zeilinger among others interprets the wave as "a probability list for all possible measurement outcomes and not a real physical object". "Each property becomes 'real' only when the electron interacts with an instrument specifically designed to reveal that property." These properties are revealed only have many individual measurements: "...the simplest interpretation one might give of the wave-corpuscular duality is a statistical interpretation: namely that the intensity of the wave at each point of the screen gives the probability of occurrence of an impact at that point".

Complimentarity interpretation

Complimentarity derives from Niels Bohr's 1926 presentation in Como, Italy, at a scientific celebration of the work of Alessandro Volta 100 years previous. Bohr's subject was complementarity, the idea that measurements of quantum events provide complementary information through seemingly contradictory results. In his Como lecture he says: "our interpretation of the experimental material rests essentially upon the classical concepts." Direct observation being impossible, observations of quantum effects are necessarily classical. Whatever the nature of quantum events, our only information will arrive via classical results. If experiments sometimes produce wave results and sometimes particle results, that is the nature of light and of the ultimate constituents of matter. While Bohr's presentation was not well received at that time, it did crystallize the issues ultimately leading to the modern wave-particle duality concept.

Wave packet interpretation

Wave packet without dispersion, that remains narrow

Wave packet with dispersion, which changes shape

In 1926 Schrödinger introduced his wave equation and in the next paper he proposed that the electron around atoms existed in the form of a wave packet, an idea de Broglie also proposed. a combination of waves bundled together such that their outline looks particle-like. His analysis turn out to be correct only for the particular (unrealistic) example he used; both Lorentz and Heisenberg immediately pointed out that in many other cases the electron bundle would rapidly spread-out, eliminating any resemblance to a localized particle. While this concept was not so useful to describe atomic features, there are many other examples of electron wavepackets. One of the most common is in scanning transmission electron microscopy where wavepackets which can be as small as 0.1nm across two dimensions but rather long (a micron) in the other are used to image materials.

Pilot wave interpretation

Main article: de Broglie–Bohm theory

The pilot wave model was originally developed by de Broglie and further developed by David Bohm into the hidden variable theory. The phrase “hidden variable” is misleading since the variable in question is the positions of the particles. Instead of duality, the pilot wave model proposes that both wave and particle are present with the wave guiding the particle in a deterministic fashion. The wave in question is the wavefunction obeying Schrödinger's equation. Bohm's formulation is intended to be classical, but it has to incorporate a distinctly non-classical feature: a nonlocal force ("quantum potential") acting on the particles.

Bohm's original purpose (1952) “was to show that an alternative to the Copenhagen interpretation is at least logically possible. Soon after he set the project aside and did not revive it until he met Basil Hiley in 1961 when both were at Birbeck College (University of London). Bohm and Hiley then wrote extensively on the theory and it gained a wider audience. 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 annihiliation 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.

See also

• Afshar experiment – Controversial physics experimentPages displaying wikidata descriptions as a fallback
• Basic concepts of quantum mechanics – Non-technical introduction to quantum physicsPages displaying short descriptions of redirect targets
• Complementarity (physics) – Quantum physics concept
• Compton scattering – Scattering of photons off charged particles
• Einstein's thought experiments – Albert Einstein's hypothetical situations to argue scientific points
• Electron diffraction – Bending of electron beams due to electrostatic interactions with matter
• Englert–Greenberger–Yasin duality relation – A relation of quantum opticsPages displaying short descriptions of redirect targets
• Einstein–Podolsky–Rosen paradox – Historical critique of quantum mechanicsPages displaying short descriptions of redirect targets
• Hanbury Brown and Twiss effect – Quantum correlations related to wave-particle duality
• Kapitsa–Dirac effect – Diffraction of matter by a standing wave of lightPages displaying wikidata descriptions as a fallback
• Many-worlds interpretation – Interpretation of quantum mechanics
• Measurement problem – Theoretical problem in quantum physics
• Neutron diffraction – Technique to investigate atomic structures using neutron scattering
• Photoelectron effect – Emission of electrons when light hits a materialPages displaying short descriptions of redirect targets
• Photon – Elementary particle or quantum of light
• Photon polarization – Quantum explanation of electromagnetic polarization
• Quantum mysticism – Pseudo-science purporting to build on the principles of quantum mechanics
• Scattering theory – Range of physical processesPages displaying short descriptions of redirect targets
• Wheeler's delayed choice experiment – Number of quantum physics thought experimentsPages displaying short descriptions of redirect targets
• X-ray crystallography – Technique used for determining crystal structures and identifying mineral compounds

Further reading

• Schrodinger, Edwin (1998). "What is an Elementary Particle?". In Castellani, Elena (ed.). Interpreting bodies: classical and quantum objects in modern physics. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01725-9.

External links

• R. Nave. "Wave–Particle Duality" (Web page). HyperPhysics. Georgia State University, Department of Physics and Astronomy. Retrieved December 12, 2005.
• Ash, Arvin. "Pilot Wave theory (Bohmian mechanics), Penrose & Transactional Interpretation explained simply". Retrieved 2023-08-15.
• Ash, Arvin. "Copenhagen vs Many Worlds Interpretation of Quantum Mechanics - Explained simply". Retrieved 2023-08-15.
• Hossenfelder, Sabine. "Understanding Quantum Mechanics #5: Decoherence". Retrieved 2023-08-15.