Matter wave

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https://en.wikipedia.org/wiki/Matter_wave

Matter waves are a central part of the theory of quantum mechanics, being half of wave–particle duality. At all scales where measurements have been practical, matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave.

The concept that matter behaves like a wave was proposed by French physicist Louis de Broglie (/dəˈbrɔɪ/) in 1924, and so matter waves are also known as de Broglie waves.

The de Broglie wavelength is the wavelength, λ, associated with a particle with momentum p through the Planck constant, h: $${\displaystyle \lambda =\frac{h}{p}.}$$

Wave-like behavior of matter has been experimentally demonstrated, first for electrons in 1927 (independently by Davisson and Germer and George Thomson) and later for other elementary particles, neutral atoms and molecules.

Matter waves have more complex velocity relations than solid objects and they also differ from electromagnetic waves (light). Collective matter waves are used to model phenomena in solid state physics; standing matter waves are used in molecular chemistry.

Matter wave concepts are widely used in the study of materials where different wavelength and interaction characteristics of electrons, neutrons, and atoms are leveraged for advanced microscopy and diffraction technologies.

History

Background

At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to Maxwell's equations, while matter was thought to consist of localized particles (see history of wave and particle duality). In 1900, this division was questioned when, investigating the theory of black-body radiation, Max Planck proposed that the thermal energy of oscillating atoms is divided into discrete portions, or quanta. Extending Planck's investigation in several ways, including its connection with the photoelectric effect, Albert Einstein proposed in 1905 that light is also propagated and absorbed in quanta, now called photons. These quanta would have an energy given by the Planck–Einstein relation: $${\displaystyle E=h\nu }$$ and a momentum vector ${\displaystyle \mathbf{p}}$ $${\displaystyle \left|\mathbf{p}\right|=p=\frac{E}{c}=\frac{h}{\lambda },}$$ where ν (lowercase Greek letter nu) and λ (lowercase Greek letter lambda) denote the frequency and wavelength of light respectively, c the speed of light, and h the Planck constant. In the modern convention, frequency is symbolized by f as is done in the rest of this article. Einstein's postulate was verified experimentally by K. T. Compton and O. W. Richardson and by A. L. Hughes in 1912 then more carefully including a measurement of the Planck constant in 1916 by Robert Millikan.

De Broglie hypothesis

Propagation of de Broglie waves in one dimension – real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the color opacity) of finding the particle at a given point x is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the slope decreases, so the amplitude diminishes again, and vice versa. The result is an alternating amplitude: a wave. Top: plane wave. Bottom: wave packet.

When I conceived the first basic ideas of wave mechanics in 1923–1924, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905.

— de Broglie

De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties, electrons also have wave-like properties. His thesis started from the hypothesis, "that to each portion of energy with a proper mass m₀ one may associate a periodic phenomenon of the frequency ν₀, such that one finds: hν₀ = m₀c². The frequency ν₀ is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory." (This frequency is also known as Compton frequency.)

To find the wavelength equivalent to a moving body, de Broglie set the total energy from special relativity for that body equal to hν: $${\displaystyle E=\frac{m{c}^2}{\sqrt{1-\frac{v^2}{c^2}}}=h\nu }$$

(Modern physics no longer uses this form of the total energy; the energy–momentum relation has proven more useful.) De Broglie identified the velocity of the particle, ${\displaystyle v}$, with the wave group velocity in free space: $${\displaystyle v_{\text{g}}\equiv \frac{\mathrm{\partial }\omega }{\mathrm{\partial }k}=\frac{d\nu }{d(1/\lambda )}}$$

(The modern definition of group velocity uses angular frequency ω and wave number k). By applying the differentials to the energy equation and identifying the relativistic momentum: $${\displaystyle p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}}$$

then integrating, de Broglie arrived at his formula for the relationship between the wavelength, λ, associated with an electron and the modulus of its momentum, p, through the Planck constant, h: $${\displaystyle \lambda =\frac{h}{p}.}$$

Schrödinger's (matter) wave equation

Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Erwin Schrödinger decided to find a proper three-dimensional wave equation for the electron. He was guided by William Rowan Hamilton's analogy between mechanics and optics (see Hamilton's optico-mechanical analogy), encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system – the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action.

In 1926, Schrödinger published the wave equation that now bears his name – the matter wave analogue of Maxwell's equations – and used it to derive the energy spectrum of hydrogen. Frequencies of solutions of the non-relativistic Schrödinger equation differ from de Broglie waves by the Compton frequency since the energy corresponding to the rest mass of a particle is not part of the non-relativistic Schrödinger equation. The Schrödinger equation describes the time evolution of a wavefunction, a function that assigns a complex number to each point in space. Schrödinger tried to interpret the modulus squared of the wavefunction as a charge density. This approach was, however, unsuccessful. Max Born proposed that the modulus squared of the wavefunction is instead a probability density, a successful proposal now known as the Born rule.

Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space

The following year, 1927, C. G. Darwin (grandson of the famous biologist Charles Darwin) explored Schrödinger's equation in several idealized scenarios. For an unbound electron in free space he worked out the propagation of the wave, assuming an initial Gaussian wave packet. Darwin showed that at time ${\displaystyle t}$ later the position ${\displaystyle x}$ of the packet traveling at velocity ${\displaystyle v}$ would be $${\displaystyle x_0+vt\pm \sqrt{{\sigma }^2+{\left(\frac{ht}{2\pi \sigma m}\right)}^2},}$$ where ${\displaystyle \sigma }$ is the uncertainty in the initial position. This position uncertainty creates uncertainty in velocity (the extra second term in the square root) consistent with Heisenberg's uncertainty relation. The wave packet spreads out as shown in the figure.

Experimental confirmation

In 1927, matter waves were first experimentally confirmed to occur in George Paget Thomson and Alexander Reid's diffraction experiment and the Davisson–Germer experiment, both for electrons.

Original electron diffraction camera made and used by Nobel laureate G P Thomson and his student Alexander Reid in 1925

 

Example original electron diffraction photograph from the laboratory of G. P. Thomson, recorded 1925–1927

The de Broglie hypothesis and the existence of matter waves has been confirmed for other elementary particles, neutral atoms and even molecules have been shown to be wave-like.

The first electron wave interference patterns directly demonstrating wave–particle duality used electron biprisms (essentially a wire placed in an electron microscope) and measured single electrons building up the diffraction pattern. A close copy of the famous double-slit experiment using electrons through physical apertures gave the movie shown.

Matter wave double slit diffraction pattern building up electron by electron. Each white dot represents a single electron hitting a detector; with a statistically large number of electrons interference fringes appear.

Electrons

Further information: Davisson–Germer experiment and Electron diffraction

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The diffracted electron intensity was measured, and was determined to have a similar angular dependence to diffraction patterns predicted by Bragg for x-rays. At the same time George Paget Thomson and Alexander Reid at the University of Aberdeen were independently firing electrons at thin celluloid foils and later metal films, observing rings which can be similarly interpreted. (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.) Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be exhibited only by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. The matter wave interpretation was placed onto a solid foundation in 1928 by Hans Bethe, who solved the Schrödinger equation, showing how this could explain the experimental results. His approach is similar to what is used in modern electron diffraction approaches.

This was a pivotal result in the development of quantum mechanics. Just as the photoelectric effect demonstrated the particle nature of light, these experiments showed the wave nature of matter.

Neutrons

See also: Neutron diffraction

Neutrons, produced in nuclear reactors with kinetic energy of around 1 MeV, thermalize to around 0.025 eV as they scatter from light atoms. The resulting de Broglie wavelength (around 180 pm) matches interatomic spacing and neutrons scatter strongly from hydrogen atoms. Consequently, neutron matter waves are used in crystallography, especially for biological materials. Neutrons were discovered in the early 1930s, and their diffraction was observed in 1936. In 1944, Ernest O. Wollan, with a background in X-ray scattering from his PhD work under Arthur Compton, recognized the potential for applying thermal neutrons from the newly operational X-10 nuclear reactor to crystallography. Joined by Clifford G. Shull, they developed neutron diffraction throughout the 1940s. In the 1970s, a neutron interferometer demonstrated the action of gravity in relation to wave–particle duality. The double-slit experiment was performed using neutrons in 1988.

Atoms

Interference of atom matter waves was first observed by Immanuel Estermann and Otto Stern in 1930, when a Na beam was diffracted off a surface of NaCl. The short de Broglie wavelength of atoms prevented progress for many years until two technological breakthroughs revived interest: microlithography allowing precise small devices and laser cooling allowing atoms to be slowed, increasing their de Broglie wavelength. The double-slit experiment on atoms was performed in 1991.

Advances in laser cooling allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the de Broglie wavelengths come into the micrometre range. Using Bragg diffraction of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold sodium atoms was explicitly measured and found to be consistent with the temperature measured by a different method.

Molecules

Recent experiments confirm the relations for molecules and even macromolecules that otherwise might be supposed too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes. The researchers calculated a de Broglie wavelength of the most probable C₆₀ velocity as 2.5 pm. More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of 10123 Da. As of 2019, this has been pushed to molecules of 25000 Da.

In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity. Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e., to certain decoherence mechanisms.

Others

Matter waves have been detected in van der Waals molecules, rho mesons, and Bose-Einstein condensate.

Traveling matter waves

Waves have more complicated concepts for velocity than solid objects. The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by $${\displaystyle \psi (\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}-i\omega t},}$$ where ${\displaystyle \mathbf{r}}$ is a position in real space, ${\displaystyle \mathbf{k}}$ is the wave vector in units of inverse meters, ω is the angular frequency with units of inverse time and ${\displaystyle t}$ is time. (Here the physics definition for the wave vector is used, which is ${\displaystyle 2\pi }$ times the wave vector used in crystallography, see wavevector.) The de Broglie equations relate the wavelength λ to the modulus of the momentum ${\displaystyle |\mathbf{p}|=p}$, and frequency f to the total energy E of a free particle as written above:$${\displaystyle \begin{array}{rl}& \lambda =\frac{2\pi }{|\mathbf{k}|}=\frac{h}{p}\\ & f=\frac{\omega }{2\pi }=\frac{E}{h}\end{array}}$$ where h is the Planck constant. The equations can also be written as $${\displaystyle \begin{array}{rl}& \mathbf{p}=\hslash \mathbf{k}\\ & E=\hslash \omega .\end{array}}$$ Here, ħ = h/2π is the reduced Planck constant. The second equation is also referred to as the Planck–Einstein relation.

Group velocity

In the de Broglie hypothesis, the velocity of a particle equals the group velocity of the matter wave. In isotropic media or a vacuum the group velocity of a wave is defined by: $${\displaystyle {\mathbf{v}}_{\mathbf{g}}=\frac{\mathrm{\partial }\omega (\mathbf{k})}{\mathrm{\partial }\mathbf{k}}}$$ The relationship between the angular frequency and wavevector is called the dispersion relationship. For the non-relativistic case this is: $${\displaystyle \omega (\mathbf{k})\approx \frac{m_0{c}^2}{\hslash }+\frac{\hslash k^2}{2m_0},}$$ where ${\displaystyle m_0}$ is the rest mass. Applying the derivative gives the (non-relativistic) matter wave group velocity: $${\displaystyle {\mathbf{v}}_{\mathbf{g}}=\frac{\hslash \mathbf{k}}{m_0}.}$$ For comparison, the group velocity of light, with a dispersion ${\displaystyle \omega (k)=ck}$, is the speed of light ${\displaystyle c}$.

As an alternative, using the relativistic dispersion relationship for matter waves $${\displaystyle \omega (\mathbf{k})=\sqrt{k^2{c}^2+{\left(\frac{m_0{c}^2}{\hslash }\right)}^2},}$$ then $${\displaystyle {\mathbf{v}}_{\mathbf{g}}=\frac{\mathbf{k}{c}^2}{\omega }.}$$ This relativistic form relates to the phase velocity as discussed below.

For non-isotropic media we use the Energy–momentum form instead: $${\displaystyle \begin{array}{rl}{\mathbf{v}}_{\mathrm{g}}& =\frac{\mathrm{\partial }\omega }{\mathrm{\partial }\mathbf{k}}=\frac{\mathrm{\partial }(E/\hslash )}{\mathrm{\partial }(\mathbf{p}/\hslash )}=\frac{\mathrm{\partial }E}{\mathrm{\partial }\mathbf{p}}=\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{p}}\left(\sqrt{p^2{c}^2+m_0^2{c}^4}\right)\\ & =\frac{\mathbf{p}{c}^2}{\sqrt{p^2{c}^2+m_0^2{c}^4}}\\ & =\frac{\mathbf{p}{c}^2}{E}.\end{array}}$$

But (see below), since the phase velocity is ${\displaystyle {\mathbf{v}}_{\mathrm{p}}=E/\mathbf{p}=c^2/\mathbf{v}}$, then $${\displaystyle \begin{array}{rl}{\mathbf{v}}_{\mathrm{g}}& =\frac{\mathbf{p}{c}^2}{E}\\ & =\frac{c^2}{{\mathbf{v}}_{\mathrm{p}}}\\ & =\mathbf{v},\end{array}}$$ where ${\displaystyle \mathbf{v}}$ is the velocity of the center of mass of the particle, identical to the group velocity.

Phase velocity

The phase velocity in isotropic media is defined as: $${\displaystyle {\mathbf{v}}_{\mathbf{p}}=\frac{\omega }{\mathbf{k}}}$$ Using the relativistic group velocity above: $${\displaystyle {\mathbf{v}}_{\mathbf{p}}=\frac{c^2}{{\mathbf{v}}_{\mathbf{g}}}}$$ This shows that ${\displaystyle {\mathbf{v}}_{\mathbf{p}}\cdot {\mathbf{v}}_{\mathbf{g}}=c^2}$ as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey ${\displaystyle {\mathbf{v}}_{\mathbf{p}}\cdot {\mathbf{v}}_{\mathbf{g}}=c^2}$, as both ${\displaystyle |{\mathbf{v}}_{\mathbf{p}}|=c}$ and ${\displaystyle |{\mathbf{v}}_{\mathbf{g}}|=c}$. Since for matter waves, ${\displaystyle |{\mathbf{v}}_{\mathbf{g}}|<c}$, it follows that ${\displaystyle |{\mathbf{v}}_{\mathbf{p}}|>c}$, but only the group velocity carries information. The superluminal phase velocity therefore does not violate special relativity, as it does not carry information.

For non-isotropic media, then $${\displaystyle {\mathbf{v}}_{\mathrm{p}}=\frac{\omega }{\mathbf{k}}=\frac{E/\hslash }{\mathbf{p}/\hslash }=\frac{E}{\mathbf{p}}.}$$

Using the relativistic relations for energy and momentum yields $${\displaystyle {\mathbf{v}}_{\mathrm{p}}=\frac{E}{\mathbf{p}}=\frac{m{c}^2}{m\mathbf{v}}=\frac{\gamma m_0{c}^2}{\gamma m_0\mathbf{v}}=\frac{c^2}{\mathbf{v}}.}$$ The variable ${\displaystyle \mathbf{v}}$ can either be interpreted as the speed of the particle or the group velocity of the corresponding matter wave—the two are the same. Since the particle speed ${\displaystyle |\mathbf{v}|<c}$ for any particle that has nonzero mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e., $${\displaystyle |{\mathbf{v}}_{\mathrm{p}}|>c,}$$ which approaches c when the particle speed is relativistic. The superluminal phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article on Dispersion (optics) for further details.

Special relativity

Using two formulas from special relativity, one for the relativistic mass energy and one for the relativistic momentum $${\displaystyle \begin{array}{rl}E& =m{c}^2=\gamma m_0{c}^2\\ \mathbf{p}& =m\mathbf{v}=\gamma m_0\mathbf{v}\end{array}}$$ allows the equations for de Broglie wavelength and frequency to be written as $${\displaystyle \begin{array}{rl}& \lambda =\frac{h}{\gamma m_{0}v}=\frac{h}{m_{0}v}\sqrt{1-\frac{v^2}{c^2}}\\ & f=\frac{\gamma m_0{c}^2}{h}=\frac{m_0{c}^2}{h\sqrt{1-\frac{v^2}{c^2}}},\end{array}}$$ where ${\displaystyle v=|\mathbf{v}|}$ is the velocity, ${\displaystyle \gamma }$ the Lorentz factor, and ${\displaystyle c}$ the speed of light in vacuum. This shows that as the velocity of a particle approaches zero (rest) the de Broglie wavelength approaches infinity.

Four-vectors

Main article: Four-vector

Using four-vectors, the de Broglie relations form a single equation: $${\displaystyle \mathbf{P}=\hslash \mathbf{K},}$$ which is frame-independent. Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by: $${\displaystyle \mathbf{K}=\left(\frac{{\omega }_0}{c^2}\right)\mathbf{U},}$$ where

• Four-momentum ${\displaystyle \mathbf{P}=\left(\frac{E}{c},\mathbf{p}\right)}$
• Four-wavevector ${\displaystyle \mathbf{K}=\left(\frac{\omega }{c},\mathbf{k}\right)}$
• Four-velocity ${\displaystyle \mathbf{U}=\gamma (c,\mathbf{u})=\gamma (c,v_{\mathrm{g}}\hat{\mathbf{u}})}$

General matter waves

The preceding sections refer specifically to free particles for which the wavefunctions are plane waves. There are significant numbers of other matter waves, which can be broadly split into three classes: single-particle matter waves, collective matter waves and standing waves.

Single-particle matter waves

The more general description of matter waves corresponding to a single particle type (e.g. a single electron or neutron only) would have a form similar to $${\displaystyle \psi (\mathbf{r})=u(\mathbf{r},\mathbf{k})\exp (i\mathbf{k}\cdot \mathbf{r}-iE(\mathbf{k})t/\hslash )}$$ where now there is an additional spatial term ${\displaystyle u(\mathbf{r},\mathbf{k})}$ in the front, and the energy has been written more generally as a function of the wave vector. The various terms given before still apply, although the energy is no longer always proportional to the wave vector squared. A common approach is to define an effective mass which in general is a tensor ${\displaystyle m_{ij}^{\ast }}$ given by $${\displaystyle {m_{ij}^{\ast }}^{-1}=\frac{1}{{\hslash }^2}\frac{{\mathrm{\partial }}^{2}E}{\mathrm{\partial }{k}_i\mathrm{\partial }{k}_j}}$$ so that in the simple case where all directions are the same the form is similar to that of a free wave above.$${\displaystyle E(\mathbf{k})=\frac{{\hslash }^2{\mathbf{k}}^2}{2m^{\ast }}}$$In general the group velocity would be replaced by the probability current $${\displaystyle \mathbf{j}(\mathbf{r})=\frac{\hslash }{2mi}\left({\psi }^{\ast }(\mathbf{r})\mathrm{\nabla }\psi (\mathbf{r})-\psi (\mathbf{r})\mathrm{\nabla }{\psi }^{\ast }(\mathbf{r})\right)}$$ where ${\displaystyle \mathrm{\nabla }}$ is the del or gradient operator. The momentum would then be described using the kinetic momentum operator,^[59] $${\displaystyle \mathbf{p}=-i\hslash \mathrm{\nabla }}$$ The wavelength is still described as the inverse of the modulus of the wavevector, although measurement is more complex. There are many cases where this approach is used to describe single-particle matter waves:

• Bloch wave, which form the basis of much of band structure as described in Ashcroft and Mermin, and are also used to describe the diffraction of high-energy electrons by solids.
• Waves with angular momentum such as electron vortex beams.
• Evanescent waves, where the component of the wavevector in one direction is complex. These are common when matter waves are being reflected, particularly for grazing-incidence diffraction.

Collective matter waves

See also: List of quasiparticles

Other classes of matter waves involve more than one particle, so are called collective waves and are often quasiparticles. Many of these occur in solids – see Ashcroft and Mermin. Examples include:

• In solids, an electron quasiparticle is an electron where interactions with other electrons in the solid have been included. An electron quasiparticle has the same charge and spin as a "normal" (elementary particle) electron and, like a normal electron, it is a fermion. However, its effective mass can differ substantially from that of a normal electron. Its electric field is also modified, as a result of electric field screening.
• A hole is a quasiparticle which can be thought of as a vacancy of an electron in a state; it is most commonly used in the context of empty states in the valence band of a semiconductor. A hole has the opposite charge of an electron.
• A polaron is a quasiparticle where an electron interacts with the polarization of nearby atoms.
• An exciton is an electron and hole pair which are bound together.
• A Cooper pair is two electrons bound together so they behave as a single matter wave.

Standing matter waves

See also: Standing wave

Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and matter waves (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.

The third class are matter waves which have a wavevector, a wavelength and vary with time, but have a zero group velocity or probability flux. The simplest of these, similar to the notation above would be $${\displaystyle \cos (\mathbf{k}\cdot \mathbf{r}-\omega t)}$$ These occur as part of the particle in a box, and other cases such as in a ring. This can, and arguably should be, extended to many other cases. For instance, in early work de Broglie used the concept that an electron matter wave must be continuous in a ring to connect to the Bohr–Sommerfeld condition in the early approaches to quantum mechanics. In that sense atomic orbitals around atoms, and also molecular orbitals are electron matter waves.

Matter waves vs. electromagnetic waves (light)

Schrödinger applied Hamilton's optico-mechanical analogy to develop his wave mechanics for subatomic particles. Consequently, wave solutions to the Schrödinger equation share many properties with results of light wave optics. In particular, Kirchhoff's diffraction formula works well for electron optics and for atomic optics. The approximation works well as long as the electric fields change more slowly than the de Broglie wavelength. Macroscopic apparatus fulfill this condition; slow electrons moving in solids do not.

Beyond the equations of motion, other aspects of matter wave optics differ from the corresponding light optics cases.

Sensitivity of matter waves to environmental condition. Many examples of electromagnetic (light) diffraction occur in air under many environmental conditions. Obviously visible light interacts weakly with air molecules. By contrast, strongly interacting particles like slow electrons and molecules require vacuum: the matter wave properties rapidly fade when they are exposed to even low pressures of gas. With special apparatus, high velocity electrons can be used to study liquids and gases. Neutrons, an important exception, interact primarily by collisions with nuclei, and thus travel several hundred feet in air.

Dispersion. Light waves of all frequencies travel at the same speed of light while matter wave velocity varies strongly with frequency. The relationship between frequency (proportional to energy) and wavenumber or velocity (proportional to momentum) is called a dispersion relation. Light waves in a vacuum have linear dispersion relation between frequency: ${\displaystyle \omega =ck}$. For matter waves the relation is non-linear: $${\displaystyle \omega (k)\approx \frac{m_0{c}^2}{\hslash }+\frac{\hslash k^2}{2m_0}.}$$ This non-relativistic matter wave dispersion relation says the frequency in vacuum varies with wavenumber (${\displaystyle k=1/\lambda }$) in two parts: a constant part due to the de Broglie frequency of the rest mass (${\displaystyle \hslash {\omega }_0=m_0{c}^2}$) and a quadratic part due to kinetic energy. The quadratic term causes rapid spreading of wave packets of matter waves.

Coherence The visibility of diffraction features using an optical theory approach depends on the beam coherence, which at the quantum level is equivalent to a density matrix approach. As with light, transverse coherence (across the direction of propagation) can be increased by collimation. Electron optical systems use stabilized high voltage to give a narrow energy spread in combination with collimating (parallelizing) lenses and pointed filament sources to achieve good coherence. Because light at all frequencies travels the same velocity, longitudinal and temporal coherence are linked; in matter waves these are independent. For example, for atoms, velocity (energy) selection controls longitudinal coherence and pulsing or chopping controls temporal coherence.

Optically shaped matter waves Optical manipulation of matter plays a critical role in matter wave optics: "Light waves can act as refractive, reflective, and absorptive structures for matter waves, just as glass interacts with light waves." Laser light momentum transfer can cool matter particles and alter the internal excitation state of atoms.

Multi-particle experiments While single-particle free-space optical and matter wave equations are identical, multiparticle systems like coincidence experiments are not.

Applications of matter waves

The following subsections provide links to pages describing applications of matter waves as probes of materials or of fundamental quantum properties. In most cases these involve some method of producing travelling matter waves which initially have the simple form ${\displaystyle \exp (i\mathbf{k}\cdot \mathbf{r}-i\omega t)}$, then using these to probe materials.

As shown in the table below, matter wave mass ranges over 6 orders of magnitude and energy over 9 orders but the wavelengths are all in the picometre range, comparable to atomic spacings. (Atomic diameters range from 62 to 520 pm, and the typical length of a carbon–carbon single bond is 154 pm.) Reaching longer wavelengths requires special techniques like laser cooling to reach lower energies; shorter wavelengths make diffraction effects more difficult to discern. Therefore, many applications focus on material structures, in parallel with applications of electromagnetic waves, especially X-rays. Unlike light, matter wave particles may have mass, electric charge, magnetic moments, and internal structure, presenting new challenges and opportunities.

Various matter wave wavelengths

matter	mass	kinetic energy	wavelength	reference
Electron	1/1823 Da	54 eV	167 pm	Davisson–Germer experiment
Electron	1/1823 Da	5×104 eV	5 pm	Tonomura et al.
He atom, H2 molecule	4 Da		50 pm	Estermann and Stern
Neutron	1 Da	0.025 eV	181 pm	Wollan and Shull
Sodium atom	23 Da		20 pm	Moskowitz et al.
Helium	4 Da	0.065 eV	56 pm	Grisenti et al.
Na2	23 Da	0.00017 eV	459 pm	Chapman et al.
C60 fullerene	720 Da	0.2 eV	5 pm	Arndt et al.
C70 fullerene	841 Da	0.2 eV	2 pm	Brezger et al.
polypeptide, Gramicidin A	1860 Da		360 fm	Shayeghi et al.
functionalized oligoporphyrins	25000 Da	17 eV	53 fm	Fein et al.

Electrons

Electron diffraction patterns emerge when energetic electrons reflect or penetrate ordered solids; analysis of the patterns leads to models of the atomic arrangement in the solids.

They are used for imaging from the micron to atomic scale using electron microscopes, in transmission, using scanning, and for surfaces at low energies.

The measurements of the energy they lose in electron energy loss spectroscopy provides information about the chemistry and electronic structure of materials. Beams of electrons also lead to characteristic X-rays in energy dispersive spectroscopy which can produce information about chemical content at the nanoscale.

Quantum tunneling explains how electrons escape from metals in an electrostatic field at energies less than classical predictions allow: the matter wave penetrates of the work function barrier in the metal.

Scanning tunneling microscope leverages quantum tunneling to image the top atomic layer of solid surfaces.

Electron holography, the electron matter wave analog of optical holography, probes the electric and magnetic fields in thin films.

Neutrons

Neutron diffraction complements x-ray diffraction through the different scattering cross sections and sensitivity to magnetism.

Small-angle neutron scattering provides way to obtain structure of disordered systems that is sensitivity to light elements, isotopes and magnetic moments.

Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films.

Neutral atoms

Atom interferometers, similar to optical interferometers, measure the difference in phase between atomic matter waves along different paths.

Atom optics mimic many light optic devices, including mirrors, atom focusing zone plates.

Scanning helium microscopy uses He atom waves to image solid structures non-destructively.

Quantum reflection uses matter wave behavior to explain grazing angle atomic reflection, the basis of some atomic mirrors.

Quantum decoherence measurements rely on Rb atom wave interference.

Molecules

Quantum superposition revealed by interference of matter waves from large molecules probes the limits of wave–particle duality and quantum macroscopicity.

Matter-wave interfererometers generate nanostructures on molecular beams that can be read with nanometer accuracy and therefore be used for highly sensitive force measurements, from which one can deduce a plethora of properties of individualized complex molecules.

Social determinism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Social_determinism

Social determinism is the theory that social interactions alone determine individual behavior (as opposed to biological or objective factors).

A social determinist would only consider social dynamics like customs, cultural expectations, education, and interpersonal interactions as the contributing factors to shape human behavior. Non-social influences, like biology, would be ignored in their contribution towards behavior. Thus, in line with the nature-nurture debate, social determinism is analogous to the 'nurture' side of the argument.

Overview

Social determinism was studied by the French philosopher Émile Durkheim (1858–1917), who was considered the father of social science. Social determinism is most commonly understood in opposition to biological determinism.

However, within the media studies discipline, social determinism is understood as the counterpart of technological determinism. Technological determinism is the notion that technological change and development are inevitable and that the characteristics of any given technology determine how it is used by the society in which it is developed. The concept of technological determinism is dependent on the premise that social changes come about as a result of the new capabilities that new technologies enable.

Technological determinism

Social determinism perceives technology as a result of the society in which it is developed. A number of contemporary media theorists have provided persuasive accounts of social determinism, including Lelia Green (2001).

In her book Technoculture, Green examines in detail the workings of a social determinist perspective, and argues "social processes determine technology for social purposes." She claims that every technological development throughout history was born of a social need, be this need economical, political or military.

According to Green (2001), technology is always developed with a particular purpose or objective in mind. As the development of technology is necessarily facilitated by financial funding, a social determinist perspective recognizes that technology is always developed to benefit those who are capable of funding its development.

Thus, social determinists perceive that technological development is not only determined by the society in which it occurs, but that it is inevitably shaped by the power structures that exist in that society.

Background

Social determinism branches off the overarching concept of determinism, which is the notion that pre-existing causes determine all events, circumstances, or behaviors.

Determinism proposes that all behavior has either an external or internal cause.

Theories and concepts

Listed below are some theories and concepts that relate to the perspective of social determinism. As psychology is a constantly developing field, this is not a definitive list but includes significant studies and ideas.

Behaviorism

Social determinism aligns with the concept of behaviorism, which is the study of observable human behavior. Behaviorists believe that an individual's behavior can be explained by the response to the environment around them. Classical conditioning and operant conditioning provide an example of socially deterministic factors on behavior. These processes of conditioning provide evidence to suggest that behavior is learned and associated with consequences from the environment. Conditioning has been argued to be deterministic, as there is a lack of free will in the response of learning.

Social determinism aligns with the theory of behaviorism and conditioning, due to the social influences and environmental factors that determine a person's behavior.

Obedience

The concept of compliance and following orders relates to social determinism, as it follows the idea that individuals follow orders based on environmental and social variables.

This relates to the concept of social influence determining behavior, as demonstrated from the Milgram Experiment conducted in 1963. This study looked at environmental stimuli and social pressure on the responses of participants, namely whether they would inflict harm on another person.

The results from this study showed that participants were more or less likely to follow orders based on the perceived authority and prestige of the experimenter.

Milgram's study was replicated in the 21st century, with similar findings developed, suggesting the conclusions withstand the test of time. These studies into destructive obedience are prime example of how individuals are predetermined by their social environment, causing them to behave in certain ways which they may not have under different circumstances.

Sociocultural theory of cognitive development

The sociocultural theory emphasises how social contexts and interactions can determine children's behavior. Lev Vygotsky developed this theory prior to his death in 1934; his manuscripts and essays were translated and published posthumously, allowing the theory to be developed.

Vygotsky explores how inputs from an individual's society, culture and interactions contribute to development, demonstrating the socially deterministic aspects in advancing mental abilities. His essays are compiled in his book, Mind in Society, which focuses on the effect of cultural and parental interaction on child development.

Examples of the theory in practice can be seen cross-culturally, looking at the life-cycle of human development, with changes in development dependent on their context. Children of war, poverty and famine are likely to have lower cognitive abilities and mental illnesses like PTSD and anxiety. This can be owed to the trauma from their environments and distress from exposure to stressful social contexts. This reinforces the concept that their mental health, behavioural responses and abilities are determined by the societal interactions and environment.

Social learning theory

The social learning theory provides a model which accounts for the range of learning experiences that occur during child development and interaction. This theory was proposed by Albert Bandura in 1977. The theory is behavioral and cognitive in nature, suggesting that learning is a cognitive process that occurs based on the social context, with reinforcement and modelling playing a key role.

Bandura provided evidence to suggest that a child's development and behavior is determined by the social interactions they have in their lives. Bandura illustrated this concept through observational learning, demonstrated in his 1961 Bobo doll experiment. This experiment looked at imitation, namely looking at whether children copied the behaviour of an adult. In this contex, it was aggressive physical and verbal behaviour. Through the experiment and further research, Bandura outlined the contribution of modelling and social behavior in determining children's behavior, providing evidence for the concept of social determinism.

Cross-cultural studies

Cross-cultural studies demonstrate how cultural variation can significantly impact an individual's inherent beliefs and behavior. Considering the results of cross-cultural psychological studies, the social context has a significant impact on a person's behavior and responses. This is notable when considering socialization and attitudes towards mental health and crises.

This section looks at a limited number of human behavioral responses and interactions, and how the context plays a key part in the individuality of response.

Helping behaviour

Levine et al.'s 2001 study was conducted in 23 large, global cities. The study looked at the likelihood of individuals from different cultures on helping in non-emergency situations. The results indicated that altruistic behaviour varied depending on the society that an individual was a part of. This variation was owed to the factors such as the normality of amiable social behavior (simpatia), economic productivity, socialization and cultural traditions.

Levine's study supports the concept of social determinism, as it suggests that helping behavior is primarily influenced by socialization and cultural determinants.

Well-being

Across different cultures, the population have varied opinions on the 'ideal' level of subjective well-being. Evidence has shown that it is important to consider individual perspectives when rating happiness and well-being. Attempts to identify a universal indicator for subjective well-being has been unsuccessful, due to the significant variation in cultural contexts.

Social interactions and context play heavily into individual desires to express certain emotions; for example, studies have shown that East Asians tend to have lower levels of well-being, and emotions that are considered 'positive' and desired differ. For example, European Americans enjoy feeling high-energy emotions, like excitement, whilst individuals from Hong Kong prefer calmer states. Similarly, the nature of a society being individualist or collectivist can play a part in ideals of well-being. Studies have suggested that individuals within collectivist societies have lower life satisfaction due to stringent cultural norms and amplified societal pressure.

Well-being is a good example of social determination. It demonstrates that an individual's perspectives on what constitutes as being satisfied and well is dependent on socialization and cultural context.

Conflict resolution

Cultural and social practices play a critical role in the way individuals handle conflict. It appears that societal aspects, like collectivism, can explain differences in approaches to dealing with conflicts.  A quasi-experimental study found that Mexicans use negotiating techniques far more than their US counterparts when dealing with conflict, a finding owed to the collectivist nature of society and social cues.

Ideology

The creation of an ideology within the society of the individual can cause an individual's actions and reactions to stimuli to become predetermined to adhere to the social rules imposed on them.

Ideologies can be created using social institutions such as schooling, which "have become the terrain upon which contending forces express their social and political interest," or the mass media, which has "significant power in shaping the social agenda and framing of public opinion to support that agenda."

Social determinism can favor a political party's agenda by setting social rules so that the individual considers the party's agenda to be morally correct, an example being the 2010 G20 summit riots in Toronto. An individual's view on the subject was influenced by the media and their reactions are predetermined by that social form of control. "We have been taught to think that censorship is the main mechanism of how the media uses information as a form of social control, but in fact what is said, and how it is selectively presented, is a far more powerful form of information control."

Arguments against social determinism

Biology

The arguments that are against social determinism largely fall under biological determinism, which aligns closely with the 'nature' side of the nature vs nurture debate.

Social pre-wiring hypothesis

Scientific studies have shown that social behavior is partly inherited and can influence infants and also even influence foetuses. "Wired to be social" means that infants are not taught that they are social beings, but they are born with inherited social skills.

Social pre-wiring refers to the ontogeny of social interaction, which is informally referred to as, "wired to be social." This concept deals with the study of fetal social behavior and social interactions in a multi-fetal environment. Specifically, the theory questions whether there is a propensity to socially oriented action already present before birth. Research in the theory concludes that newborns are born into the world with a unique genetic wiring to be social.

Circumstantial evidence supporting the social pre-wiring hypothesis can be revealed when examining newborns' behavior. Newborns, not even hours after birth, have been found to display a preparedness for social interaction. This preparedness is expressed in ways such as their imitation of facial gestures. This observed behavior cannot be contributed to any current form of socialization. Rather, newborns most likely inherit to some extent social behavior and identity through genetics.

Principal evidence of this theory is uncovered by examining twin pregnancies. The main argument is, if there are social behaviors that are inherited and developed before birth, then one should expect twin foetuses to engage in some form of social interaction before they are born. Thus, ten foetuses were analyzed over a period of time using ultrasound techniques. Using kinematic analysis, the results of the experiment were that the twin foetuses would interact with each other for longer periods and more often as the pregnancies went on. Researchers were able to conclude that the performance of movements between the co-twins were not accidental but specifically aimed.

The social pre-wiring hypothesis was proved correct, "The central advance of this study is the demonstration that 'social actions' are already performed in the second trimester of gestation. Starting from the 14th week of gestation twin foetuses plan and execute movements specifically aimed at the co-twin. These findings force us to predate the emergence of social behavior: when the context enables it, as in the case of twin foetuses, other-directed actions are not only possible but predominant over self-directed actions." This suggests that there are inherent, biological factors which are responsible for factors like social behaviour, which disputes the argument of social determinism.

Traumatic brain injuries

Findings from head-injury studies suggest that some aspects of behavior can change after a traumatic brain injury. Significant brain damage is associated with poorer decision making, reduced regulation ability and changes in personality.

The 1848 case of Phineas Gage is the first recorded case study into the localisation of brain function, providing evidence to show that personality and behaviour is determined by brain structure. After a large rod was driven through his head, destroying most of his left frontal lobe, his personality shifted to become significantly more hostile and aggressive. Accounts from his doctor, family and friends claimed after the accident his personality and behaviors changed so radically that he was "no longer Gage".

Researchers have argued this provides evidence for the nature side of the debate on behaviour, as evidence has shown that it was the physical trauma that caused the shift in Gage's social interactions and perspectives. This is reinforced by research into brain tumors and contemporary studies into brain injuries. The location of a tumor can have a significant impact on personality and cognitive abilities, suggesting that behavior and socialization is not solely owed to social aspects.

Neuroscientific evidence into brain localisation and function suggests that once the integrity of the brain is disturbed, there are far-reaching consequences with changes in personality, emotions and behaviour usually experienced.