De Broglie relation
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Matter waves are a central part of the theory of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, being an example of
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...
. All
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
exhibits
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
-like behavior. For example, a beam of
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s can be
diffracted Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
just like a beam of light or a water wave. In most cases, however, the wavelength is too small to have a practical impact on day-to-day activities. The concept that matter behaves like a wave was proposed by French physicist
Louis de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to Old quantum theory, quantum theory. In his 1924 PhD thesis, he pos ...
() in 1924. It is also referred to as the ''de Broglie hypothesis''. Matter waves are referred to as ''de Broglie waves''. The ''de Broglie wavelength'' is the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, , associated with a massive particle (i.e., a particle with mass, as opposed to a massless particle) and is related to its
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, , through the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, : : \lambda = \frac=\frac. Wave-like behavior of matter was first experimentally demonstrated by
George Paget Thomson Sir George Paget Thomson, FRS (; 3 May 189210 September 1975) was a British physicist and Nobel laureate in physics recognized for his discovery of the wave properties of the electron by electron diffraction. Education and early life Thomson ...
's thin metal diffraction experiment, and independently in the
Davisson–Germer experiment The Davisson–Germer experiment was a 1923-27 experiment by Clinton Davisson and Lester Germer at Western Electric (later Bell Labs), in which electrons, scattered by the surface of a crystal of nickel metal, displayed a diffraction pattern. ...
, both using electrons; and it has also been confirmed for other
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
s, neutral
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
s and even
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s. For v = \frac its value is the same as the
Compton wavelength The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...
.


Historical context

At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
, while matter was thought to consist of localized particles (see history of wave and particle duality). In 1900, this division was exposed to doubt, when, investigating the theory of
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spe ...
,
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
proposed that light is emitted in discrete quanta of energy. It was thoroughly challenged in 1905. Extending Planck's investigation in several ways, including its connection with the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...
,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
proposed that light is also propagated and absorbed in quanta; now called
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s. These quanta would have an energy given by the
Planck–Einstein relation The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck relation, Planck equation, and Planck formula, ...
: :E=h\nu and a momentum :p=\frac=\frac where (lowercase Greek letter nu) and (lowercase Greek letter lambda) denote the frequency and wavelength of the light, the speed of light, and the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. In the modern convention, frequency is symbolized by ''f'' as is done in the rest of this article. Einstein's postulate was confirmed experimentally by Robert Millikan and Arthur Compton over the next two decades.


De Broglie hypothesis

De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties,
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s also have wave-like properties. De Broglie did not simplify his equation into the one that bears his name. He did conclude that ., translated in 2004 by A. F. Kracklauer as He also referred to Einstein’s famous relativity equation. Thus, it was a simple step to get to the equation that bears his name. Also, by rearranging the momentum equation stated in the above section, we find a relationship between the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, , associated with an electron and its
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, , through the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, : : \lambda = \frac. The relationship has since been shown to hold for all types of matter: all matter exhibits properties of both particles and waves. In 1926,
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
published an equation describing how a matter wave should evolve – the matter wave analogue of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
— and used it to derive the
energy spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
. 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 The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
of a particle is not part of the non-relativistic Schrödinger equation.


Experimental confirmation

Matter waves were first experimentally confirmed to occur in
George Paget Thomson Sir George Paget Thomson, FRS (; 3 May 189210 September 1975) was a British physicist and Nobel laureate in physics recognized for his discovery of the wave properties of the electron by electron diffraction. Education and early life Thomson ...
's cathode ray diffraction experiment and the Davisson-Germer experiment for electrons, and the de Broglie hypothesis has been confirmed for other elementary particles. Furthermore, neutral atoms and even molecules have been shown to be wave-like.


Electrons

In 1927 at Bell Labs,
Clinton Davisson Clinton Joseph Davisson (October 22, 1881 – February 1, 1958) was an American physicist who won the 1937 Nobel Prize in Physics for his discovery of electron diffraction in the famous Davisson–Germer experiment. Davisson shared the Nobel Priz ...
and Lester Germer fired slow-moving
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s at a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
line
nickel Nickel is a chemical element with symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive but large pieces are slow ...
target. The angular dependence of the diffracted electron intensity was measured, and was determined to have the same
diffraction pattern Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a ...
as those predicted by
Bragg Bragg may refer to: Places * Bragg City, Missouri, United States * Bragg, Texas, a ghost town, United States * Bragg, West Virginia, an unincorporated community, United States *Electoral district of Bragg, a state electoral district in South Austra ...
for
x-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10  picometers to 10  nanometers, corresponding to frequencies in the range 30&nb ...
s. At the same time George Paget Thomson at the University of Aberdeen was independently firing electrons at very thin metal foils to demonstrate the same effect. 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. When the de Broglie wavelength was inserted into the Bragg condition, the predicted diffraction pattern was observed, thereby experimentally confirming the de Broglie hypothesis for electrons. This was a pivotal result in the development of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Just as the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...
demonstrated the particle nature of light, the
Davisson–Germer experiment The Davisson–Germer experiment was a 1923-27 experiment by Clinton Davisson and Lester Germer at Western Electric (later Bell Labs), in which electrons, scattered by the surface of a crystal of nickel metal, displayed a diffraction pattern. ...
showed the wave-nature of matter, and completed the theory of
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...
. For
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s this idea was important because it meant that not only could any particle exhibit wave characteristics, but that one could use
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
s to describe phenomena in matter if one used the de Broglie wavelength.


Neutral atoms

Experiments with Fresnel diffraction and an
atomic mirror In physics, an atomic mirror is a device which reflects neutral atoms in the similar way as a conventional mirror reflects visible light. Atomic mirrors can be made of electric fields or magnetic fields, electromagnetic waves or just silicon waf ...
for
specular reflection Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction,
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
and allow
quantum reflection Quantum reflection is a uniquely quantum phenomenon in which a compact object, such as a neutron or a small molecule, reflects smoothly and in a wavelike fashion from a much larger surface, such as a pool of mercury. In contrast, a classically beha ...
by the tails of the attractive potential. Advances in
laser cooling Laser cooling includes a number of techniques in which atoms, molecules, and small mechanical systems are cooled, often approaching temperatures near absolute zero. Laser cooling techniques rely on the fact that when an object (usually an atom) a ...
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 diffraction In physics and chemistry , Bragg's law, Georg Wulff, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposit ...
of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold
sodium Sodium is a chemical element with the symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable ...
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
holography Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other applications. In principle, i ...
, 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. The effect has also been used to explain the spatial version of the
quantum Zeno effect The quantum Zeno effect (also known as the Turing paradox) is a feature of quantum-mechanical systems allowing a particle's time evolution to be slowed down by measuring it frequently enough with respect to some chosen measurement setting. Somet ...
, in which an otherwise unstable object may be stabilised by rapidly repeated observations.


Molecules

Recent experiments even 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 en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...
demonstrated diffraction for molecules as large as
fullerene A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...
s. 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 10,123 u. As of 2019, this has been pushed to molecules of 25,000 u. Still one step further than Louis de Broglie go theories which in quantum mechanics eliminate the concept of a pointlike classical particle and explain the observed facts by means of wavepackets of matter waves alone.


De Broglie relations

The de Broglie equations relate the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
to the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, and
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
to the total energy of a
free particle In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
: \begin & \lambda = \frac\\ & f = \frac \end where ''h'' is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. The equations can also be written as \begin & \mathbf p = \hbar \mathbf k\\ & E = \hbar \omega\\ \end or \begin & \mathbf p = \hbar \boldsymbol \beta\\ & E = \hbar \omega\\ \end where is the reduced Planck constant, is the
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
, is the
phase constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
, and is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
. In each pair, the second equation is also referred to as the
Planck–Einstein relation The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck relation, Planck equation, and Planck formula, ...
, since it was also proposed by
Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
and Einstein.


Special relativity

Using two formulas from
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
, one for the relativistic mass energy and one for the
relativistic momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
:E = m c^2 = \gamma m_0 c^2 :\mathbf = m\mathbf = \gamma m_0 \mathbf allows the equations to be written as :\begin&\lambda =\,\, \frac \, =\, \frac \,\,\,\, \sqrt\\ & f = \frac = \frac \end where m_0 denotes the particle's
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
, v its
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
, \gamma the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
, and c the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in a vacuum. See below for details of the derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with phase velocity (equal to the product of the particle's frequency and its wavelength). In the case of a non-
dispersive medium In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to o ...
, they happen to be equal, but otherwise they are not.


Group velocity

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
first explained the
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...
of light in 1905.
Louis de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to Old quantum theory, quantum theory. In his 1924 PhD thesis, he pos ...
hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded, should always equal the group velocity of the corresponding wave. The magnitude of the group velocity is equal to the particle's speed. Both in relativistic and non-relativistic quantum physics, we can identify the group velocity of a particle's wave function with the particle velocity.
Quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that :v_g = \frac = \frac = \frac where is the total
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
of the particle, is its
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, is the reduced Planck constant. For a free non-relativistic particle it follows that :\begin v_g &= \frac = \frac \left( \frac\frac \right)\\ &= \frac\\ &= v \end where is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
of the particle and its velocity. Also in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
we find that :\begin v_g &= \frac = \frac \left( \sqrt \right)\\ &= \frac\\ &= \frac \end where is the rest mass of the particle and is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in a vacuum. But (see below), using that the phase velocity is , therefore :\begin v_g &= \frac\\ &= \frac\\ &= v \end where is the velocity of the particle regardless of wave behavior.


Phase velocity

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, particles also behave as waves with
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
phases. The phase velocity is equal to the product of the frequency multiplied by the wavelength. By the de Broglie hypothesis, we see that :v_\mathrm = \frac = \frac = \frac. Using relativistic relations for energy and momentum, we have :v_\mathrm = \frac = \frac = \frac = \frac = \frac where ''E'' is the total energy of the particle (i.e.
rest energy The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
plus
kinetic energy In physics, the kinetic energy of an object is the energy that 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 stated velocity. Having gained this energy during its acc ...
in the kinematic sense), ''p'' the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, \gamma the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
, ''c'' the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, and β the speed as a fraction of ''c''. The variable ''v'' can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed v < c for any particle that has mass (according to
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
), the phase velocity of matter waves always exceeds ''c'', i.e. :v_\mathrm > c, \, and as we can see, it approaches ''c'' when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, because phase propagation carries no energy. See the article on '' Dispersion (optics)'' for details.


Four-vectors

Using four-vectors, the De Broglie relations form a single equation: \mathbf= \hbar\mathbf which is
frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (con ...
-independent. Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by: \mathbf = \left(\frac\right)\mathbf where *
Four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
\mathbf = \left(\frac, \vec \right) * Four-wavevector \mathbf = \left(\frac, \vec \right) = \left(\frac, \frac\mathbf \right) *
Four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
\mathbf = \gamma(c,\vec) = \gamma(c,v_g \hat)


Interpretations

The purpose of de Broglie’s 81 page thesis was to create an improved version of the Bohr atom through
pilot wave theory In theoretical physics, the pilot wave theory, also known as Bohmian mechanics, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets quan ...
. De Broglie presented his thesis on pilot wave theory at the 1927
Solvay Conference The Solvay Conferences (french: Conseils Solvay) have been devoted to outstanding preeminent open problems in both physics and chemistry. They began with the historic invitation-only 1911 Solvay Conference on Physics, considered a turning point i ...
. The thesis of de Broglie involved the hypothesis that a standing wave guided the electrons in the
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
of the atom. The thesis had an unusual analysis that higher energy photons obey the Wien Law and are particle-like while lower energy photons obey the
Rayleigh–Jeans law In physics, the Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength λ, it is: B_ (T) = \ ...
and are wave-like. Particle physics tends to treat all forces by particle-particle interaction causing Richard Feynman to say that there are no waves just particles. And recently, there have been some theories that try to explain the
Interpretations of quantum mechanics An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraord ...
which try to resolve whether either the particle or the wave aspect is fundamental in nature, seeking to explain the other as an emergent property. Some interpretations, such as hidden variable theory, treat the wave and the particle as distinct entities. Yet others propose some intermediate entity that is neither quite wave nor quite particle but only appears as such when we measure one or the other property. The
Copenhagen interpretation The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as feat ...
states that the nature of the underlying reality is unknowable and beyond the bounds of scientific inquiry. Schrödinger's acknowledges that his quantum mechanical equation is based in part on the thesis of de Broglie. Schrödinger emphasized that his equation was different in that it was in multi-dimensional space. In his lecture as both wave mechanics and matrix mechanics were both new concepts, he tries to imply his formula is superior as does Heisenberg in his speech. At the Fifth Solvay Conference in 1927,
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
reported: In 1955, Heisenberg showed that the waves of the quantum mechanical equations were reinterpreted as probabilities rather than classical waves stating: It is mentioned above that the "displaced quantity" of the Schrödinger wave has values that are dimensionless complex numbers. According to Heisenberg, rather than being of some ordinary physical quantity such as, for example, Maxwell's electric field intensity, or mass density, the Schrödinger-wave packet's "displaced quantity" is a probability amplitude. He wrote that instead of using the term 'wave packet', it is preferable to speak of a probability packet. The Schrödinger equation probability amplitude is interpreted as the calculation of the probability of the location or momentum of discrete particles. Heisenberg recites Duane's account of particle diffraction by probabilistic quantal translation momentum transfer, which allows, for example in Young's two-slit experiment, each diffracted particle probabilistically to pass discretely through a particular slit. Schrödinger originally proposed that his matter wave was 'composed of smeared matter,’ but the
Born rule The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findi ...
changed the psi function to be understood as a description of probability rather than a description of the actual electron charge density. These ideas may be expressed in ordinary language as follows. In the account of ordinary physical waves, a 'point' refers to a position in ordinary physical space at an instant of time, at which there is specified a 'displacement' of some physical quantity. But in the account of quantum mechanics, a 'point' refers to a configuration of the system at an instant of time, every particle of the system being in a sense present in every 'point' of configuration space, each particle at such a 'point' being located possibly at a different position in ordinary physical space. There is no explicit definite indication that, at an instant, this particle is 'here' and that particle is 'there' in some separate 'location' in configuration space. This conceptual difference entails that, in contrast to de Broglie's pre-quantum mechanical wave description, the quantum mechanical probability packet description does not directly and explicitly express the Aristotelian idea, referred to by Newton, that causal efficacy propagates through ordinary space by contact, nor the Einsteinian idea that such propagation is no faster than light. In contrast, these ideas are so expressed in the classical wave account, through the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
, though it is inadequate for the observed quantal phenomena. The physical reasoning for this was first recognized by Einstein.


De Broglie's phase wave and periodic phenomenon

De Broglie's thesis started from the hypothesis, "that to each portion of energy with a proper mass one may associate a periodic phenomenon of the frequency , such that one finds: . The frequency is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory."MacKinnon, E. (1976). De Broglie's thesis: a critical retrospective, ''Am. J. Phys.'' 44: 1047–1055
(This frequency is also known as Compton frequency.) De Broglie followed his initial hypothesis of a periodic phenomenon, with frequency , associated with the energy packet. He used the special theory of relativity to find, in the frame of the observer of the electron energy packet that is moving with velocity v, that its frequency was apparently reduced to :\nu_1 = \nu_0 \sqrt\,. De Broglie reasoned that to a stationary observer this hypothetical intrinsic particle periodic phenomenon appears to be in phase with a wave of wavelength \lambda and frequency f that is propagating with phase velocity v_\mathrm p. De Broglie called this wave the "phase wave" («onde de phase» in French). This was his basic matter wave conception. He noted, as above, that v_\mathrm p > c, and the phase wave does not transfer energy.Bacciagaluppi, G., Valentini, A. (2009). ''Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference'', Cambridge University Press, Cambridge UK, , pp. 30–88. While the concept of waves being associated with matter is correct, de Broglie did not leap directly to the final understanding of quantum mechanics with no missteps. There are conceptual problems with the approach that de Broglie took in his thesis that he was not able to resolve, despite trying a number of different fundamental hypotheses in different papers published while working on, and shortly after publishing, his thesis. These difficulties were resolved by
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
, who developed the wave mechanics approach, starting from a somewhat different basic hypothesis.


See also

*
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
*
Compton wavelength The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...
*
Faraday wave Faraday waves, also known as Faraday ripples, named after Michael Faraday (1791–1867), are nonlinear standing waves that appear on liquids enclosed by a vibrating receptacle. When the vibration frequency exceeds a critical value, the flat hydrost ...
* Kapitsa–Dirac effect *
Matter wave clock A matter wave clock is a type of clock whose principle of operation makes use of the apparent wavelike properties of matter. Matter waves were first proposed by Louis de Broglie and are sometimes called de Broglie waves. They form a key aspect of ...
*
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
* Theoretical and experimental justification for the Schrödinger equation *
Thermal de Broglie wavelength In physics, the thermal de Broglie wavelength (\lambda_, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in ...
* De Broglie–Bohm theory


References


Further reading

* L. de Broglie, ''Recherches sur la théorie des quanta'' (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, ''Ann. Phys.'' (Paris) 3, 22 (1925)
English translation by A.F. Kracklauer.

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. . pp. 203–4, 222–3, 236. * * An extensive review article "Optics and interferometry with atoms and molecules" appeared in July 2009: https://web.archive.org/web/20110719220930/http://www.atomwave.org/rmparticle/RMPLAO.pdf.
"Scientific Papers Presented to Max Born on his retirement from the Tait Chair of Natural Philosophy in the University of Edinburgh"
1953 (Oliver and Boyd)


External links

* {{DEFAULTSORT:Matter Wave Waves Matter Foundational quantum physics