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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:

Wave-like behavior of matter has been experimentally demonstrated, first for electrons in 1927 and for other elementary particles, neutral atoms and molecules in the years since.

Introduction

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:

and a momentum vector ${\displaystyle \mathbf {p} }$ 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:

(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:

(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:

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]

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

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.

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]

Electrons

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target.^[22][23] 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.^[21] (Alexander Reid, who was Thomson's graduate student, performed the first experiments but he died soon after in a motorcycle accident^[29] 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.^[30] The matter wave interpretation was placed onto a solid foundation in 1928 by Hans Bethe,^[31] who solved the Schrödinger equation,^[16] showing how this could explain the experimental results. His approach is similar to what is used in modern electron diffraction approaches.^[32][33]

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

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. In 1944, Ernest O. Wollan, with a background in X-ray scattering from his PhD work^[34] 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^[35] neutron diffraction throughout the 1940s. In the 1970s a neutron interferometer demonstrated the action of gravity in relation to wave–particle duality in a neutron interferometer.^[36]

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.^[37] 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.^[38]

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.^[39]

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.^[40] 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.^[41] As of 2019, this has been pushed to molecules of 25000 Da.^[42]

In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity.^[43] 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.^[44][45]

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

where ${\displaystyle \mathbf {r} }$ is a position in real space, ${\displaystyle \mathbf{k}}$Zdroj:https://en.wikipedia.org?pojem=Matter_waves
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