|
Tensor Operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator. The general notion of scalar, vector, and tensor operators In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pure Mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Completeness Condition
In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if ''T'' is an operator, applying the squaring function ''s'' → ''s''2 to ''T'' yields the operator ''T''2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator or the exponential e^. The 'scope' here means the kind of ''function of an operator'' which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its focus is different than the holomorphic functional calculus. More precisely, the Borel functional calculus allows for applying an arbitrary Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. Motivation If ''T' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 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 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 indices. When any two indices are interchanged, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Baker–Campbell–Hausdorff Formula
In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation e^X e^Y = e^Z for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in X and Y and iterated commutators thereof. The first few terms of this series are: Z = X + Y + \frac ,Y+ \frac ,[X,Y + \frac [Y,[Y,X + \cdots\,, where "\cdots" indicates terms involving higher Commutator#Identities_(ring_theory)">commutators of X and Y. If X and Y are sufficiently small elements of the Lie algebra \mathfrak g of a Lie group G, the series is convergent. Meanwhile, every element g sufficiently close to the identity in G can be expressed as g = e^X for a small X in \mathfrak g. Thus, we can say that ''near the identity'' the group multiplication in G—written as e^X e^Y = e^Z—can be expressed in purely Lie alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ''discontinuous'' rigid transformations such as reflections. For example, the angular velocity of a rotating object is a pseudovector because, when the object is reflected in a mirror, the reflected image rotates in such a way so that ''its'' angular velocity "vector" is ''not'' the mirror image of the angular velocity "vector" of the ''original'' object; for true vectors (also known as ''polar vectors''), the reflection "vector" and the original "vector" ''must'' be mirror images. One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Kinetic Energy
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Robert and Halliday, David (1960) ''Physics'', Section 7-5, Wiley International Edition The kinetic energy of an object is equal to the work, or force ( F) in the direction of motion times its displacement ( s), needed to accelerate the object from rest to its given speed. The same amount of work is done by the object when decelerating from its current speed to a state of rest. The SI unit of energy is the joule, while the English unit of energy is the foot-pound. In relativistic mechanics, \fracmv^2 is a good approximation of kinetic energy only when ''v'' is much less than the speed of light. History and etymology The adjective ''kinetic'' has its roots in the Greek word κίνησις ''kinesis'', meaning "motion". The dichoto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Potential Energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity or those in a spring. The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of Potentiality and Actuality, ''potentiality''. Common types of potential energy include gravitational potential energy, the elastic potential energy of a deformed spring, and the electric potential energy of an electric charge and an electric field. The unit for energy in the International System of Units (SI) is the joule (symbol J). Potential energy is associated with forces that act on a body in a way that the total Work (physics), work done by these forces on the body depends only on the initial and final positions of the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Energy Operator
In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry. Definition It is given by: \hat = i\hbar\frac It acts on the wave function (the probability amplitude for different configurations of the system) \Psi\left(\mathbf, t\right) Application The energy operator corresponds to the full energy of a system. The Schrödinger equation describes the space- and time-dependence of the slow changing (non- relativistic) wave function of a quantum system. The solution of the Schrödinger equation for a bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta. Schrödinger equation Using the energy operator in the Schrödinger equation: i\hbar\frac \Psi(\mathbf,\,t) = \hat H \Psi(\mathbf,t) one obtains: \hat\Psi(\mathbf, t) = \hat \Psi(\mathbf, t) where ''i'' is the imaginary unit, ''ħ'' is the reduce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quantum Number
In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantum numbers includes the principal, azimuthal, magnetic, and spin quantum numbers. To describe other systems, different quantum numbers are required. For subatomic particles, one needs to introduce new quantum numbers, such as the flavour of quarks, which have no classical correspondence. Quantum numbers are closely related to eigenvalues of observables. When the corresponding observable commutes with the Hamiltonian of the system, the quantum number is said to be " good", and acts as a constant of motion in the quantum dynamics. History Electronic quantum numbers In the era of the old quantum theory, starting from Max Planck's proposal of quanta in his model of blackbody radiation (1900) and Albert Einstein's adaptation o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Associated Legendre Polynomial
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left[ \ell (\ell + 1) - \frac \right] P_\ell^m(x) = 0, or equivalently \frac \left[ \left(1 - x^2\right) \frac P_\ell^m(x) \right] + \left[ \ell (\ell + 1) - \frac \right] P_\ell^m(x) = 0, where the indices ''ℓ'' and ''m'' (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if ''ℓ'' and ''m'' are integers with 0 ≤ ''m'' ≤ ''ℓ'', or with trivially equivalent negative values. When in addition ''m'' is even, the function is a polynomial. When ''m'' is zero and ''ℓ'' integer, these functions are identical to the Legendre polynomials. In general, when ''ℓ'' and ''m'' are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Spherical Harmonic
In mathematics and Outline of physical science, physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, every function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of Trigonometric functions, circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of Rotation group SO(3), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
