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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 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 () 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, 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.


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 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,
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 theor ...
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 alwa ...
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 Arthur Holly Compton (September 10, 1892 – March 15, 1962) was an American physicist who won the Nobel Prize in Physics in 1927 for his 1923 discovery of the Compton effect, which demonstrated the particle nature of electromagnetic radia ...
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 published an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
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 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-to ...
. 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.


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 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 t ...
target. The angular dependence of the diffracted electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg 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 In physics and chemistry , Bragg's law, 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 superposition of wave ...
, 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 demonstrated the particle nature of light, the Davisson–Germer experiment 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 ...
s to describe phenomena in matter if one used the de Broglie wavelength.


Neutral atoms

Experiments with Fresnel diffraction and an atomic mirror 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 su ...
of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection 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) ...
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 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, 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, 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 fullerenes. The researchers calculated a De Broglie wavelength of the most probable C60 velocity as 2.5 pm. More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of 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: \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, is the phase constant, 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 ti ...
. 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 and
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
.


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 law ...
, one for the relativistic mass energy and one for the relativistic momentum :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, 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, 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 fo ...
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, 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 theor ...
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 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 chemistry, ...
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 ...
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 ele ...
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 law ...
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 fo ...
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 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 acce ...
in the
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
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, ''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 fo ...
, 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 law ...
), 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 Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (). The special theory of relativity implies that only particles with z ...
phase velocity does not violate special relativity, because phase propagation carries no energy. See the article on ''
Dispersion (optics) 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 ...
'' for details.


Four-vectors

Using four-vectors, the De Broglie relations form a single equation: \mathbf= \hbar\mathbf which is frame-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 In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
\mathbf = \left(\frac, \vec \right) = \left(\frac, \frac\mathbf \right) * Four-velocity \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. 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 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 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 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 In physics, hidden-variable theories are proposals to provide explanations of quantum mechanical phenomena through the introduction of (possibly unobservable) hypothetical entities. The existence of fundamental indeterminacy for some measure ...
, 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 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 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 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, 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 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, ...
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, who developed the wave mechanics approach, starting from a somewhat different basic hypothesis.


See also

* Bohr model * Compton wavelength * Faraday wave * Kapitsa–Dirac effect * Matter wave clock *
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 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. The motivation uses photons, which are relat ...
* Thermal de Broglie wavelength * 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