In
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
, a
matter wave or
de Broglie wave is the
waveA wave is a disturbance that propagates through space and time, usually with transference of energy. A mechanical wave is a wave that propagates or travels through a medium due to the restoring forces it produces upon deformation. There also exist waves capable of traveling through a vacuum,...
(wave-particle duality) of
matterThe term matter traditionally refers to the substance that all objects are made of. One common way to identify this "substance" is through its physical properties; a common definition of matter is anything that has mass and occupies a volume...
. The
de Broglie relations show that the
wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
is inversely proportional to the
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
of a particle and that the
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
is directly proportional to the particle's
kinetic energyThe kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its...
. The wavelength of matter is also called
de Broglie wavelength. The theory was advanced by Louis de Broglie in 1924 in his PhD thesis; he was awarded the Nobel Prize for Physics in 1929 for this work, which made him the first person to receive a Nobel Prize on a PhD thesis.
Historical context
After strides made by
Max PlanckMax Planck was a German physicist. He is considered to be the founder of the quantum theory, and thus one of the most important physicists of the twentieth century. Planck was awarded the Nobel Prize in Physics in 1918.-Biography:Planck came from a traditional, intellectual family...
(1858-1947) and
Albert EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
(1879-1955) in understanding the behavior of electrons and what would be known as quantum physics,
Niels BohrNiels Henrik David Bohr was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...
(1885-1962) began (among other things) trying to explain how electrons behave. He came up with new fundamental ideas about electrons and mathematically derived the Rydberg equation, an equation that was discovered only through trial and error. This equation explains the
energiesIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
of the
light emittedA spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
when hydrogen gas is compressed and electrified (similar to neon signs, but with hydrogen in this case). Unfortunately, his model only worked for the hydrogen-atom-configuration, but his ideas were so revolutionary that they broke up the classical view of electrons' behavior and paved the way for fresh new ideas in what would become quantum physics and quantum mechanics.
Louis de Broglie (1892-1987) tried to expand on Bohr's ideas, and he pushed for their application beyond
hydrogenHydrogen is the chemical element with atomic number 1. It is represented by the symbol H. At standard temperature and pressure, hydrogen is a colorless, odorless, nonmetallic, tasteless, highly flammable diatomic gas with the molecular formula H
2...
. In fact he looked for an equation which could explain the wavelength characteristics of all matter. His equation was experimentally confirmed in 1927 when physicists
Lester GermerLester Halbert Germer was an American physicist. With Clinton Davisson, he proved the wave-particle duality of matter in the Davisson-Germer experiment, which was important to the development of the electron microscope. These studies supported the theoretical work of DeBroglie...
and
Clinton DavissonClinton Joseph Davisson , was an American physicist who won the 1937 Nobel Prize in Physics for his discovery of electron diffraction. Davisson shared the Nobel Prize with George Paget Thomson, who independently discovered electron diffraction at about the same time as Davisson.-Early...
fired electrons at a crystalline
nickelNickel is a chemical element, with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. It is one of the four ferromagnetic elements at about room temperature, other three being iron, cobalt and gadolinium...
target and the resulting diffraction pattern was found to match the predicted values.. In de Broglie's equation an electron's wavelength is a function of Planck's constant (6.626×10
−34 jouleThe joule , named for James Prescott Joule, is the derived unit of energy in the International System of Units. It is the energy exerted by a force of one newton acting to move an object through a distance of one metre...
-seconds) divided by the object's momentum (nonrelativistically, its
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
multiplied by its
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
). When this momentum is very large (relative to Planck's constant), then an object's wavelength is very small. This is the case with every-day objects, such as a person; given the enormous momentum of a person compared with the very tiny Planck constant, the wavelength of a person would be so small (on the order of 10
−35 meter or smaller) as to be undetectable by any current measurement tools. On the other hand, many small particles (such as typical electrons in everyday materials) have a very low momentum compared to macroscopic objects. In this case, the de Broglie wavelength may be large enough that the particle's wave-like nature gives observable effects.
The wave-like behavior of small-momentum particles is analogous to that of light. As an example, electron microscopes use electrons, instead of light, to see very small objects. Since electrons typically have more momentum than photons, their de Broglie wavelength will be smaller, resulting in better spatial resolution.
The de Broglie relations
The de Broglie equations relate the wavelength and frequency to the momentum and energy , respectively, as
and
where is Planck's constant. The two equations are often written as
where is the reduced Planck's constant (also known as
Dirac's constant, pronounced "h-bar"), is the angular wavenumber, and is the
angular frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
.
Using results from
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
, the equations can be written as
and
where is the particle's rest mass, is the particle's
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
, is the
Lorentz factorThe Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ. It gets its name from its earlier...
, and is the
speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
in a vacuum.
See the article on
group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
for detail on the argument and derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with
phase velocityThe phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
(equal to the product of the particle's frequency and its wavelength).
Elementary particles
In 1927 at Bell Labs,
Clinton DavissonClinton Joseph Davisson , was an American physicist who won the 1937 Nobel Prize in Physics for his discovery of electron diffraction. Davisson shared the Nobel Prize with George Paget Thomson, who independently discovered electron diffraction at about the same time as Davisson.-Early...
and
Lester GermerLester Halbert Germer was an American physicist. With Clinton Davisson, he proved the wave-particle duality of matter in the Davisson-Germer experiment, which was important to the development of the electron microscope. These studies supported the theoretical work of DeBroglie...
firedThe Davisson–Germer experiment was a physics experiment conducted in 1927 which confirmed the de Broglie hypothesis, which says that particles of matter have wave properties...
slow-moving electrons at a crystalline
nickelNickel is a chemical element, with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. It is one of the four ferromagnetic elements at about room temperature, other three being iron, cobalt and gadolinium...
target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same
diffraction patternDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
as those predicted by
BraggSir William Lawrence Bragg, CH, OBE, MC, FRS was an English physicist who shared the Nobel Prize in Physics in 1915 with his father Sir William Henry Bragg. He was the director of the Cavendish Laboratory, Cambridge when the epochal discovery of the structure of DNA was made by James D...
for x-rays. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. Therefore, the presence of any
diffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the
Bragg conditionIn physics, Bragg's law states that when X-rays hit an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency ; this phenomenon is known as the Rayleigh scattering...
, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.
This was a pivotal result in the development of
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
. Just as
Arthur ComptonArthur Holly Compton was an American physicist and Nobel laureate in physics for his discovery of the Compton effect. He served as Chancellor of Washington University in St. Louis from 1945 to 1953.-Early years:...
demonstrated the particle nature of light, the
Davisson-Germer experimentThe Davisson–Germer experiment was a physics experiment conducted in 1927 which confirmed the de Broglie hypothesis, which says that particles of matter have wave properties...
showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
s to describe phenomena in matter if one uses the de Broglie wavelength.
Since the original Davisson-Germer experiment for electrons, the de Broglie hypothesis has been confirmed for other elementary particles.
The wavelength of a thermalized electron in a non-metal at room temperature (300 K) is about 8 nm.
Neutral atoms
Experiments with
Fresnel diffractionIn optics, Fresnel diffraction or near-field diffraction is a process of diffraction that occurs when a wave passes through an aperture and diffracts in the near field, causing any diffraction pattern observed to differ in size and shape, depending on the distance between the aperture and the...
and
specular reflectionSpecular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction is reflected into a single outgoing direction...
of neutral atoms
confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo
diffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
,
interferenceIn physics, interference is the addition of two or more waves that results in a new wave pattern.Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same...
and allow
quantum reflectionQuantum reflection is a physical phenomenon involving the reflection of a matter wave from an attractive potential. In classical physics, such a phenomenon is not possible; for instance when one magnet is pulled toward another, you do not expect one of the magnets to suddenly Quantum reflection is...
by the tails of the attractive potential
.
Advances in
laser coolingLaser cooling refers to the number of techniques in which atomic and molecular samples are cooled through the interaction with one or more laser light fields...
have allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the thermal de Broglie wavelengths come into the micrometre range. Using
Bragg diffractionBragg diffraction was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays...
of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold
sodiumSodium is a metallic element with a symbol Na and atomic number 11. It is a soft, silvery-white, highly reactive metal and is a member of the alkali metals within "group 1"...
atoms was explicitly measured and found to be consistent with the temperature measured by a different method.
.
This effect has been used to demonstrate atomic
holographyHolography is a technique that allows the light scattered from an object to be recorded and later reconstructed so that it appears as if the object is in the same position relative to the recording medium as it was when recorded...
, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.
Waves of molecules
Recent experiments even confirm the relations for molecules and even macromolecules, which are normally considered too large to undergo quantum mechanical effects. In 1999, a research team in
ViennaVienna is the capital of the Republic of Austria and also one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.7 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre. It is the 10th largest city by...
demonstrated diffraction for molecules as large as fullerenes.
In general, the De Broglie hypothesis is expected to apply to any well isolated object.
Spatial Zeno effect
The matter wave leads to the spatial version of the Zeno effect. If an object (particle) is observed with frequency
in a half-space (say, ), then this observation prevents the particle, which stays in the half-space from entry into this half-space
. Such an "observation" can be realized with a set of rapidly moving absorbing ridges, filling one half-space. In the system of coordinates related to
the ridges, this phenomenon appears as a
specular reflectionSpecular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction is reflected into a single outgoing direction...
of a particle from a
ridged mirrorIn atomic physics, a ridged mirror is a kind of atomic mirror, designed for the specular reflection of neutral particles coming at the grazing incidence angle, characterised in the following: in order to reduce the mean attraction of particles to the surface and increase the reflectivity, this...
, assuming the grazing incidence (small values of the grazing angle).
Such a ridged mirror is universal; while we consider the idealised "absorption" of the de Broglie wave at the ridges, the reflectivity is determined by wavenumber and does not depend on other properties of a particle.
See also
- Atomic de Broglie microscope
- Atomic mirror
In physics, an atomic mirror is a device which reflects neutral atoms in the similar way as the conventional mirror reflects visible light.Atomic mirrors can be made of electric fields or magnetic fields, electromagnetic waves...
- Quantum reflection
Quantum reflection is a physical phenomenon involving the reflection of a matter wave from an attractive potential. In classical physics, such a phenomenon is not possible; for instance when one magnet is pulled toward another, you do not expect one of the magnets to suddenly Quantum reflection is...
- Bohr model
In atomic physics, the Bohr model, devised by Niels Bohr, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing...
- Theoretical and experimental justification for the Schrödinger equation
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles...
- Thermal de Broglie wavelength
In physics, the Thermal de Broglie wavelength is defined for a freeideal gas of massive particles in equilibrium as:where* h is Planck's constant* m is the mass of a gas particle* k is Boltzmann's constant...
- Atom optics
Atom optics is the area of physics which deals with beams of cold, slowly moving neutral atoms, as a special case of a particle beam.Like an optical beam, the atomic beam may exhibit diffraction and interference, and can be focused with...
- Ridged mirror
In atomic physics, a ridged mirror is a kind of atomic mirror, designed for the specular reflection of neutral particles coming at the grazing incidence angle, characterised in the following: in order to reduce the mean attraction of particles to the surface and increase the reflectivity, this...
- Zeno effect
Further reading
- Broglie, Louis de, The wave nature of the electron Nobel Lecture, 12, 1929
- Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Co. ISBN 0-7167-4345-0. pp. 203–4, 222-3, 236.
- Web version of Thesis, translated (English)
- Steven S. Zumdahl, Chemical Principles 5th Edition, (2005) Houghton Mifflin Company.
- René-Louis Vallée: L'énergie électromagnétique matérielle et gravitationnelle, Paris, 1971. Vallée based lot of his research on De Broglie and worked together with him. This manuscript can be found here