Matter wave in the context of Wave–particle duality

The electron (e
, or β
in nuclear reactions) is a subatomic particle whose electric charge is negative one elementary charge. It is an elementary particle that comprises the ordinary matter that makes up the universe, along with up and down quarks.

Electrons are extremely lightweight particles. In atoms, an electron's matter wave occupies atomic orbitals around a positively charged atomic nucleus. The configuration and energy levels of an atom's electrons determine the atom's chemical properties. Electrons are bound to the nucleus to different degrees. The outermost or valence electrons are the least tightly bound and are responsible for the formation of chemical bonds between atoms to create molecules and crystals. These valence electrons also facilitate all types of chemical reactions by being transferred or shared between atoms. The inner electron shells make up the atomic core.

In physics and many other areas of science and engineering the intensity or flux of radiant energy is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy. In the SI system, it has units watts per square metre (W/m), or kg⋅s in base units. Intensity is used most frequently with waves such as acoustic waves (sound), matter waves such as electrons in electron microscopes, and electromagnetic waves such as light or radio waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.

The word "intensity" as used here is not synonymous with "strength", "amplitude", "magnitude", or "level", as it sometimes is in colloquial speech.

In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater amplitude (constructive interference) or lower amplitude (destructive interference) if the two waves are in phase or out of phase, respectively.Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves as well as in loudspeakers as electrical waves.

The Planck constant, or Planck's constant, denoted by ${\displaystyle h}$, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and a particle's momentum is equal to the wavenumber of the associated matter wave (the reciprocal of its wavelength) multiplied by the Planck constant.

The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation. Planck later referred to the constant as the "quantum of action". In 1905, Albert Einstein associated the "quantum" or minimal element of the energy to the electromagnetic wave itself. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

In quantum mechanics, an atomic orbital (/ˈɔːrbɪtəl/ ) is a function describing the location and wave-like behavior of an electron in an atom. This function describes an electron's charge distribution around the atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus.

Each orbital in an atom is characterized by a set of values of three quantum numbers n, ℓ, and m_ℓ, which respectively correspond to an electron's energy, its orbital angular momentum, and its orbital angular momentum projected along a chosen axis (magnetic quantum number). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of m_ℓ and −m_ℓ orbitals, and are often labeled using associated harmonic polynomials (e.g., xy, x − y) which describe their angular structure.

In physics, quantum tunnelling, barrier penetration, or simply tunnelling is a quantum mechanical phenomenon in which an object such as an electron or atom passes through a potential energy barrier that, according to classical mechanics, should not be passable due to the object not having sufficient energy to pass or surmount the barrier.

Tunnelling is a consequence of the wave nature of matter and quantum indeterminacy. The quantum wave function describes the states of a particle or other physical system and wave equations such as the Schrödinger equation describe their evolution. In a system with a short, narrow potential barrier, a small part of wavefunction can appear outside of the barrier representing a probability for tunnelling through the barrier.

In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The terms atomic orbital and molecular orbital were introduced by Robert S. Mulliken in 1932 to mean one-electron orbital wave functions. At an elementary level, they are used to describe the region of space in which a function has a significant amplitude.

In an isolated atom, the orbital electrons' location is determined by functions called atomic orbitals. When multiple atoms combine chemically into a molecule by forming a valence chemical bond, the electrons' locations are determined by the molecule as a whole, so the atomic orbitals combine to form molecular orbitals. The electrons from the constituent atoms occupy the molecular orbitals. Mathematically, molecular orbitals are an approximate solution to the Schrödinger equation for the electrons in the field of the molecule's atomic nuclei. They are usually constructed by combining atomic orbitals or hybrid orbitals from each atom of the molecule, or other molecular orbitals from groups of atoms. They can be quantitatively calculated using the Hartree–Fock or self-consistent field (SCF) methods.

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the frequency-dependent phase velocity and group velocity of each sinusoidal component of a wave in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency-dependence of wave propagation and attenuation.

Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation, even in the absence of geometric constraints and other media.

The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations.

Forward scattering is the deflection of waves by small angles so that they continue to move in close to the same direction as before the scattering. It can occur with all types of waves, for instance light, ultraviolet radiation, X-rays as well as matter waves such as electrons, neutrons and even water waves. It can be due to diffraction, refraction, and low angle reflection. It almost always occurs when the wavelength of the radiation used is small relative to the features which lead to the scattering. Forward scatter is essentially the reverse of backscatter.

Many different examples exist, and there are very large fields where forward scattering dominates, in particular for electron diffraction and electron microscopy, X-ray diffraction and neutron diffraction. In these the relevant waves are transmitted through the samples. One case where there is forward scattering in a reflection geometry is reflection high-energy electron diffraction.

Louis Victor Pierre Raymond, 7th Duc de Broglie (French: [də bʁɔj] ; 15 August 1892 – 19 March 1987) was a French theoretical physicist and aristocrat known for his contributions to quantum theory. In his 1924 Ph.D. thesis, he postulated the wave nature of electrons and suggested that all matter has wave properties. This concept is known as the de Broglie hypothesis, an example of wave–particle duality, and forms a central part of the theory of quantum mechanics. In 1929, de Broglie won the Nobel Prize in Physics, after the wave-like behaviour of matter was experimentally confirmed in 1927.

The wave-like behaviour of particles theorized by de Broglie was used by Erwin Schrödinger in his formulation of wave mechanics.De Broglie presented an alternative interpretation of these mechanics called the pilot-wave concept at the 1927 Solvay Conferences, but then abandoned it. In 1952, David Bohm developed a new form of the concept which became known as the de Broglie–Bohm theory. De Broglie revisited the idea in 1956, creating another version that incorporated ideas from Bohm and Jean-Pierre Vigier.