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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 this 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 to the Schrödinger equation: i\hbar\frac \Psi(\mathbf,\,t) = \hat H \Psi(\mathbf,t) can be obtained: \hat\Psi(\mathbf, t) = \hat \Psi(\mathbf, t) where ''i'' is the imaginary unit, ''ħ'' is the reduced Planck ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for z=f(x, y, \ldots), the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: :f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Broglie Relation
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons 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, , associated with a massive particle (i.e., a particle with mass, as opposed to a massless particle) and is related to its momentum, , through the Planck constant, : : \lambda = \frac=\frac. Wave-like behavior of matter was first experimentally demonstrated by George Paget Thomson's thin metal diffraction experiment, and independently in the Davisson–Germer experiment, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, the value of such a field can be written as :F(\vec x,t) = G(\vec x \cdot \vec n, t), where \vec n is a unit-length vector, and G(d,t) is a function that gives the field's value as dependent on only two real parameters: the time t, and the scalar-valued displacement d = \vec x \cdot \vec n of the point \vec x along the direction \vec n. The displacement is constant over each plane perpendicular to \vec n. The values of the field F may be scalars, vectors, or any other physical or mathematical quantity. They can be complex numbers, as in a complex exponential plane wave. When the values of F are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector \vec n, and a transverse wave if they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. In quantum mechanics, it means the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space. Classical free particle The classical free particle is characterized by a fixed velocity v. The momentum is given by \mathbf=m\mathbf and the kinetic energy (equal to total energy) by E=\fracmv^2=\frac where ''m'' is the mass of the particle and v is the vector velocity of the particle. Quantum free particle Mathematical description A free particle with mass m in non-relativistic quantum mechanics is described by the free Schrödinger equation: - \frac \nabla^2 \ \psi(\mathbf, t) = i\hbar\frac \psi (\mathbf, t) where ''ψ'' is the wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum Operator
In quantum mechanics, the momentum operator is the operator (physics), operator associated with the momentum (physics), linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is: \hat = - i \hbar \frac where is Planck's reduced constant, the imaginary unit, is the spatial coordinate, and a partial derivative (denoted by \partial/\partial x) is used instead of a total derivative () since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows: \hat\psi = - i \hbar \frac In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by , i.e. it is a multiplication operator, just as the position operator is a multiplication operator in the position represen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein–Gordon Equation
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation E^2 = (pc)^2 + \left(m_0c^2\right)^2\,. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian,) the practical utility is limited. The equation can be put into the form of a Schrödinger equation. In this form it is expressed as two coupled differential equations, each of first order in time. The so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for the speed at which conventional matter or energy (and thus any signal carrying information) can travel through space. All forms of electromagnetic radiation, including visible light, travel at the speed of light. For many practical purposes, light and other electromagnetic waves will appear to propagate instantaneously, but for long distances and very sensitive measurements, their finite speed has noticeable effects. Starlight viewed on Earth left the stars many years ago, allowing humans to study the history of the universe by viewing distant objects. When communicating with distant space probes, it can take minutes to hours for signals to travel from Earth to the spacecraft and vice versa. In computing, the speed of light fixes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
