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Kubo Formula
The Kubo formula, named for Ryogo Kubo who first presented the formula in 1957, is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Among numerous applications of the Kubo formula, one can calculate the charge and spin susceptibilities of systems of electrons in response to applied electric and magnetic fields. Responses to external mechanical forces and vibrations can be calculated as well. General Kubo formula Consider a quantum system described by the (time independent) Hamiltonian H_0. The expectation value of a physical quantity, described by the operator \hat, can be evaluated as: : \begin \left\langle\hat\right\rangle &= \operatorname\,\left hat\hat\right= \sum_n \left\langle n \left, \hat \ n \right\rangle e^ \\ \hat &= e^ = \sum_n , n \rangle\langle n , e^ \end where Z_0 = \operatorname\,\left hat\rho_0\right/math> is the partition function. Suppose now that just above some time t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ryogo Kubo
was a Japanese mathematical physicist, best known for his works in statistical physics and non-equilibrium statistical mechanics. Work In the early 1950s, Kubo transformed research into the linear response A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in informatio ... properties of near-equilibrium condensed-matter systems, in particular the understanding of electron transport and conductivity, through the Kubo formalism, a Green's function approach to linear response theory for quantum systems. In 1977 Ryogo Kubo was awarded the Boltzmann Medal ''for his contributions to the theory of non-equilibrium statistical mechanics, and to the theory of fluctuation phenomena''. He is cited particularly for his work in the establishment of the basic relations between transport coefficients and eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Response Function
A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance, see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related. Mathematical definition Denote the input of a system by h(t) (e.g. a force), and the response of the system by x(t) (e.g. a position). Generally, the value of x(t) will depend not only on the present value of h(t), but also on past values. Approximately x(t) is a weighted sum of the previous values of h(t'), with the weights given by the linear response function \chi(t-t'): x(t) = \int_^t dt'\, \chi(t-t') h(t') + \cdots\,. The explicit term on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory (quantum Mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system (e.g. its energy levels and eigenstates) can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system. Approximate Hamiltonians Perturbation theory is an important tool f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (statistical Mechanics)
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both hea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heaviside Function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as . The Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x > 0 \\ 0, & x \le 0 \end * using the Iverson bracket notation: H(x) := 0.html" ;"title=">0">>0/math> * an indicator function: H(x) := \mathbf_=\mathbf 1_(x) * t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent ''mixed states''. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, without describing their combined state. Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems, quantum decoherence, and quantum information. Definition an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the subject. The equation is named after Erwin Schrödinger, 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 a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may change if the potential V changes). This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures. In the Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. For time evolution from a state vector , \psi(t_0)\rangle at time 0 to a state vector , \psi(t)\rangle at time , the time-evolution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interaction Picture
In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables. The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. Most field-theoretical calculations use the interaction representation because they construct the solution to the many-body Schrödinger equation as the solution to the free-particle problem plus some unknown interaction parts. Equations that include operators acting at different times, which hold in the interaction picture, don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because time-dependent unitary transformations relate operators in one picture to the analogous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir Fock. In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory. Quantum many-bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Kubo Relations
The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions: :\gamma = \int_0^\infty \left\langle \dot(t) \dot(0) \right\rangle \;t. Thermal and mechanical transport processes Thermodynamic systems may be prevented from relaxing to equilibrium because of the application of a field (e.g. electric or magnetic field), or because the boundaries of the system are in relative motion (shear) or maintained at different temperatures, etc. This generates two classes of nonequilibrium system: mechanical nonequilibrium systems and thermal nonequilibrium systems. The standard example of an electrical transport process is Ohm's law, which states that, at least for sufficiently small applied voltages, the current ''I'' is linearly proportional to the applied voltage ''V'', : I = \sigma V.\, As the applied voltage increases one expects to see deviations from l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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