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.^[1] 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,^[2]: 87 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 the light, c the speed of light, and h the Planck constant.^[3] In the modern convention, frequency is symbolized by f as is done in the rest of this article. Einstein's postulate was verified experimentally^[2]: 89 by K. T. Compton and O. W. Richardson^[4] and by A. L. Hughes^[5] in 1912 then more carefully including a measurement of the Planck constant in 1916 by Robert Millikan^[6]

De Broglie hypothesis

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^[7]

De Broglie, in his 1924 PhD thesis,^[8] 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."^{[9][8]: 8 [10][11][12][13]} (This frequency is also known as Compton frequency.)

To find the wavelength equivalent to a moving body, de Broglie^[2]: 214 set the total energy from special relativity for that body equal to hν: $${\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=h\nu }$$

(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, v, with the wave group velocity in free space: $${\displaystyle v_{\text{g}}\equiv {\frac {\partial \omega }{\partial k}}={\frac {d\nu }{d(1/\lambda )}}}$$

(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 as 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:^[14] $${\displaystyle \lambda ={\frac {h}{p}}.}$$

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.^[15]

In 1926, Schrödinger published the wave equation that now bears his name^[16] – 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.^[17][18][19] Max Born proposed that the modulus squared of the wavefunction is instead a probability density, a successful proposal now known as the Born rule.^[17]

The following year, 1927, C. G. Darwin (grandson of the famous biologist) explored Schrödinger's equation in several idealized scenarios.^[20] 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 $${\displaystyle x_{0}+vt\pm {\sqrt {\sigma ^{2}+(ht/2\pi \sigma m)^{2}}}}$$ 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 show 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^[21] and the Davisson–Germer experiment,^[22][23] 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.^[24]

The first electron wave interference patterns directly demonstrating wave–particle duality used electron biprisms^[25][26] (essentially a wire placed in an electron microscope) and measured single electrons building up the diffraction pattern. Recently, a close copy of the famous double-slit experiment^[27]: 260 using electrons through physical apertures gave the movie shown.^[28]

Group velocity

In the de Broglie hypothesis, the velocity of a particle equals the group velocity of the matter wave.^[2]: 214 In isotropic media or a vacuum the group velocity of a wave is defined by: $${\displaystyle \mathbf {v_{g}} ={\frac {\partial \omega (\mathbf {k} )}{\partial \mathbf {k} }}}$$ The relationship between the angular frequency and wavevector is called the dispersion relationship. For the non-relativistic case this is: $${\displaystyle \omega (\mathbf {k} )\approx {\frac {m_{0}c^{2}}{\hbar }}+{\frac {\hbar k^{2}}{2m_{0}}}\,.}$$ where ${\displaystyle m_{0}}$ is the rest mass. Applying the derivative gives the (non-relativistic) matter wave group velocity: $${\displaystyle \mathbf {v_{g}} ={\frac {\hbar \mathbf {k} }{m_{0}}}}$$ 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 $${\displaystyle \omega (\mathbf {k} )={\sqrt {k^{2}c^{2}+\left({\frac {m_{0}c^{2}}{\hbar }}\right)^{2}}}\,.}$$ then $${\displaystyle \mathbf {v_{g}} ={\frac {\mathbf {k} c^{2}}{\omega }}}$$ This relativistic form relates to the phase velocity as discussed below.

For non-isotropic media we use the Energy–momentum form instead: $${\displaystyle {\begin{aligned}\mathbf {v} _{\mathrm {g} }&={\frac {\partial \omega }{\partial \mathbf {k} }}={\frac {\partial (E/\hbar )}{\partial (\mathbf {p} /\hbar )}}={\frac {\partial E}{\partial \mathbf {p} }}={\frac {\partial }{\partial \mathbf {p} }}\left({\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}\right)\\&={\frac {\mathbf {p} c^{2}}{\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}}\\&={\frac {\mathbf {p} c^{2}}{E}}.\end{aligned}}}$$

But (see below), since the phase velocity is ${\displaystyle \mathbf {v} _{\mathrm {p} }=E/\mathbf {p} =c^{2}/\mathbf {v} }$, then $${\displaystyle {\begin{aligned}\mathbf {v} _{\mathrm {g} }&={\frac {\mathbf {p} c^{2}}{E}}\\&={\frac {c^{2}}{\mathbf {v} _{\mathrm {p} }}}\\&=\mathbf {v} ,\end{aligned}}}$$ 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: $${\displaystyle \mathbf {v_{p}} ={\frac {\omega }{\mathbf {k} }}}$$ Using the relativistic group velocity above:^[2]: 215 $${\displaystyle \mathbf {v_{p}} ={\frac {c^{2}}{\mathbf {v_{g}} }}}$$ This shows that ${\displaystyle \mathbf {v_{p}} \cdot \mathbf {v_{g}} =c^{2}}$ as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey ${\displaystyle \mathbf {v_{p}} \cdot \mathbf {v_{g}} =c^{2}}$, as both ${\displaystyle |\mathbf {v_{p}} |=c}$ and ${\displaystyle |\mathbf {v_{g}} |=c}$. Since for matter waves, ${\displaystyle |\mathbf {v_{g}} |<c}$, it follows that ${\displaystyle |\mathbf {v_{p}} |>c}$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 $${\displaystyle \mathbf {v} _{\mathrm {p} }={\frac {\omega }{\mathbf {k} }}={\frac {E/\hbar }{\mathbf {p} /\hbar }}={\frac {E}{\mathbf {p} }}.}$$

Using the relativistic relations for energy and momentum yields $${\displaystyle \mathbf {v} _{\mathrm {p} }={\frac {E}{\mathbf {p} }}={\frac {mc^{2}}{m\mathbf {v} }}={\frac {\gamma m_{0}c^{2}}{\gamma m_{0}\mathbf {v} }}={\frac {c^{2}}{\mathbf {v} }}.}$$ 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., $${\displaystyle |\mathbf {v} _{\mathrm {p} }|>c,}$$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 $${\displaystyle {\begin{aligned}E&=mc^{2}=\gamma m_{0}c^{2}\\[1ex]\mathbf {p} &=m\mathbf {v} =\gamma m_{0}\mathbf {v} \end{aligned}}}$$ allows the equations for de Broglie wavelength and frequency to be written as $${\displaystyle {\begin{aligned}&\lambda =\,\,{\frac {h}{\gamma m_{0}v}}\,=\,{\frac {h}{m_{0}v}}\,\,\,{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\[2.38ex]&f={\frac {\gamma \,m_{0}c^{2}}{h}}={\frac {m_{0}c^{2}}{h{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}},\end{aligned}}}$$ where ${\displaystyle v=|\mathbf {v} |}$ is the velocity, ${\displaystyle \gamma }$ the Lorentz factor, and ${\displaystyle c}$ the speed of light in vacuum.^[55][56] This shows that as the velocity of a particle approaches zero (rest) the de Broglie wavelength approaches infinity.

Four-vectors

Using four-vectors, the de Broglie relations form a single equation: $${\displaystyle \mathbf {P} =\hbar \mathbf {K} ,}$$ 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 ${\displaystyle \mathbf {U} =\gamma (c,{\mathbf {u} })=\gamma (c,v_{\mathrm {g} }{\hat {\mathbf {u} }})}$

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 $${\displaystyle \psi (\mathbf {r} )=u(\mathbf {r} ,\mathbf {k} )\exp(i\mathbf {k} \cdot \mathbf {r} -iE(\mathbf {k} )t/\hbar )}$$ 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 ${\displaystyle m_{ij}^{*}}$ given by $${\displaystyle {m_{ij}^{*}}^{-1}={\frac {1}{\hbar ^{2}}}{\frac {\partial ^{2}E}{\partial k_{i}\partial k_{j}}}}$$ so that in the simple case where all directions are the same the form is similar to that of a free wave above.$${\displaystyle E(\mathbf {k} )={\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m^{*}}}}$$In general the group velocity would be replaced by the probability current^[57] $${\displaystyle \mathbf {j} (\mathbf {r} )={\frac {\hbar }{2mi}}\left(\psi ^{*}(\mathbf {r} )\mathbf {\nabla } \psi (\mathbf {r} )-\psi (\mathbf {r} )\mathbf {\nabla } \psi ^{*}(\mathbf {r} )\right)}$$ where ${\displaystyle \nabla }$ is the del or gradient operator. The momentum would then be described using the kinetic momentum operator,^[57] $${\displaystyle \mathbf {p} =-i\hbar \nabla }$$ 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.^[58][33]
• Waves with angular momentum such as electron vortex beams.^[59]
• 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

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.^[60] 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.^[60] 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

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.^[61] In that sense atomic orbitals around atoms, and also molecular orbitals are electron matter waves.^[62][63][64]

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.^[83][84]

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 or properties of individualized complex molecules.^[85]