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Berry Connection And Curvature
In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. Berry phase and cyclic adiabatic evolution In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian H(\mathbf R) depends on a (vector) parameter \mathbf R that varies with time t. If the n'th eigenvalue \varepsilon_n(\mathbf R) remains non-degenerate everywhere along the path and the variation with time ''t'' is sufficiently slow, then a system initially in the normalized eigenstate , n(\mathbf R(0))\rangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berry Phase
In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by S. Pancharatnam (1956), in classical optics and by H. C. Longuet-Higgins (1958)See page 12 in molecular physics; it was generalized by Sir Michael Berry in (1984). It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase. It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi-Civita Symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the natural numbers , for some positive integer . It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric tensor, antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon or , or less commonly the Latin lower case . Index notation allows one to display permutations in a way compatible with tensor analysis: \varepsilon_ where ''each'' index takes values . There are indexed values of , which can be arranged into an -dimensional array. The key defining property of the symbol is ''total antisymmetry'' in the ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall Effect
The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879. A Hall effect can also occur across a void or hole in a semiconductor or metal plate, when current is injected via contacts that lie on the boundary or edge of the void or hole, and the charge flows outside the void or hole, in the metal or semiconductor. This Hall effect becomes observable in a perpendicular applied magnetic field across voltage contacts that lie on the boundary of the void on either side of a line connecting the current contacts. It exhibits apparent sign reversal in comparison to the standard "ordinary Hall effect" in the simply connected specimen, and depends only on the current injected from within the void. Superposition may also be realized in the Hall effect: first imagine the stand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Diametric. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field. Paramagnetic materials have a weak induced magnetization in a magnetic field, which disappears when the magnetic field is removed. Ferromagnetic and ferrimagnetic materials have strong magnetization in a magnetic field, and can be ''magnetized'' to have magnetization in the absence of an external field, becoming a permanent magnet. Magnetization is not necessarily uniform within a material, but may vary between different points. Magneti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Polarization
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.''Introduction to Electrodynamics'' (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ''McGraw Hill Encyclopaedia of Physics'' (2nd Edition), C.B. Parker, 1994, Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brillouin Zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in ''k''-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Space
In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the ''direct lattice''. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice (e.g., a lattice of a crystal), the reciprocal lattice exists in reciprocal space (also known as '' momentum space'' or less commonly as ''K-space'', due to the relationship between the Pontryagin duals momentum and position). The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively. The reciprocal lattice is the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch Theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where \mathbf is position, \psi is the wave function, u is a periodic function with the same periodicity as the crystal, the wave vector \mathbf is the crystal momentum vector, e is Euler's number, and i is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids. Named after Swiss physicist Felix Bloch, the description of electrons in terms of Bloch functions, termed Bloch electrons (or less often ''Bloch Waves''), underlies the concept of electronic band structures. These eigenstates are written with subscripts as \psi_, where n is a discr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Hall Effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhibits steps that take on the quantized values : R_ = \frac = \frac , where is the Hall voltage, is the channel current, is the elementary charge and is Planck's constant. The divisor can take on either integer () or fractional () values. Here, is roughly but not exactly equal to the filling factor of Landau levels. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether is an integer or fraction, respectively. The striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. Since the electron density remains constant when the Fermi level is in a clean spectral gap, this situation corres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets), permanent magnets, elementary particles (such as electrons), various molecules, and many astronomical objects (such as many planets, some moons, stars, etc). More precisely, the term ''magnetic moment'' normally refers to a system's magnetic dipole moment, the component of the magnetic moment that can be represented by an equivalent magnetic dipole: a magnetic north and south pole separated by a very small distance. The magnetic dipole component is sufficient for small enough magnets or for large enough distances. Higher-order terms (such as the magnetic quadrupole moment) may be needed in addition to the dipole moment for extended objects. The magnetic dipole moment of an object is readily defined in terms of the torque that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_\mathrm &= \begin 0&1\\ 1&0 \end \\ \sigma_2 = \sigma_\mathrm &= \begin 0& -i \\ i&0 \end \\ \sigma_3 = \sigma_\mathrm &= \begin 1&0\\ 0&-1 \end \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
