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Path Integral Monte Carlo
Path integral Monte Carlo (PIMC) is a quantum Monte Carlo method used to solve quantum statistical mechanics problems numerically within the path integral formulation. The application of Monte Carlo methods to path integral simulations of condensed matter systems was first pursued in a key paper by John A. Barker. The method is typically (but not necessarily) applied under the assumption that symmetry or antisymmetry under exchange can be neglected, i.e., identical particles are assumed to be quantum Boltzmann particles, as opposed to fermion and boson particles. The method is often applied to calculate thermodynamic properties such as the internal energy, heat capacity, or free energy. As with all Monte Carlo method based approaches, a large number of points must be calculated. In principle, as more path descriptors are used (these can be "replicas", "beads," or "Fourier coefficients," depending on what strategy is used to represent the paths), the more quantum (and the less cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Monte Carlo
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the quantum many-body problem. The diverse flavors of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem. Quantum Monte Carlo methods allow for a direct treatment and description of complex many-body effects encoded in the wave function, going beyond mean-field theory. In particular, there exist numerically exact and polynomially-scaling algorithms to exactly study static properties of boson systems without geometrical frustration. For fermions, there exist very good approximations to their static properties and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exchange Interaction
In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles. While sometimes called an exchange force, or, in the case of fermions, Pauli repulsion, its consequences cannot always be predicted based on classical ideas of force. Both bosons and fermions can experience the exchange interaction. The wave function of identical particles, indistinguishable particles is subject to exchange symmetry: the wave function either changes sign (for fermions) or remains unchanged (for bosons) when two particles are exchanged. The exchange symmetry alters the Expectation value (quantum mechanics), expectation value of the distance between two indistinguishable particles when their wave functions overlap. For fermions the expectation value of the distance increases, and for bosons it decreases (compared to distinguishable particles). The exchange interaction arises from the combination of exchange symmetry and the Coulomb's l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Monte Carlo
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the quantum many-body problem. The diverse flavors of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem. Quantum Monte Carlo methods allow for a direct treatment and description of complex many-body effects encoded in the wave function, going beyond mean-field theory. In particular, there exist numerically exact and polynomially-scaling algorithms to exactly study static properties of boson systems without geometrical frustration. For fermions, there exist very good approximations to their static properties and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of Molecule, molecules, Material, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed Wave function, wave functions as well as to observable properties such as structures, spectra, and Thermodynamics, thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics. Chemists rely heavily on spectroscopy through which information regarding the Quantization (physics), quantization of energy on a molecular scale can be obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Algorithm
In quantum computing, a quantum algorithm is an algorithm that runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is generally reserved for algorithms that seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement. Problems that are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Integral Molecular Dynamics
Path integral molecular dynamics (PIMD) is a method of incorporating quantum mechanics into molecular dynamics simulations using Feynman path integrals. In PIMD, one uses the Born–Oppenheimer approximation to separate the wavefunction into a nuclear part and an electronic part. The nuclei are treated quantum mechanically by mapping each quantum nucleus onto a classical system of several fictitious particles connected by springs (harmonic potentials) governed by an effective Hamiltonian, which is derived from Feynman's path integral. The resulting classical system, although complex, can be solved relatively quickly. There are now a number of commonly used condensed matter computer simulation techniques that make use of the path integral formulation including centroid molecular dynamics (CMD), ring polymer molecular dynamics (RPMD), and the Feynman–Kleinert quasi-classical Wigner (FK–QCW) method. The same techniques are also used in path integral Monte Carlo (PIMC). There are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Option Pricing
In finance, a price (premium) is paid or received for purchasing or selling options. The calculation of this premium will require sophisticated mathematics. Premium components This price can be split into two components: intrinsic value, and time value (also called "extrinsic value"). Intrinsic value The ''intrinsic value'' is the difference between the underlying spot price and the strike price, to the extent that this is in favor of the option holder. For a call option, the option is in-the-money if the underlying spot price is higher than the strike price; then the intrinsic value is the underlying price minus the strike price. For a put option, the option is in-the-money if the ''strike'' price is higher than the underlying spot price; then the intrinsic value is the strike price minus the underlying spot price. Otherwise the intrinsic value is zero. For example, when a DJI call (bullish/long) option is 18,000 and the underlying DJI Index is priced at $18,050 then there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Modeling
Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio of a business, project, or any other investment. Typically, then, financial modeling is understood to mean an exercise in either asset pricing or corporate finance, of a quantitative nature. It is about translating a set of hypotheses about the behavior of markets or agents into numerical predictions. At the same time, "financial modeling" is a general term that means different things to different users; the reference usually relates either to accounting and corporate finance applications or to quantitative finance applications. Accounting In corporate finance and the accounting profession, ''financial modeling'' typically entails financial statement forecasting; usually the preparation of detailed company-specific models used for deci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microcanonical Ensemble
In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it cannot exchange energy or particles with its environment, so that (by conservation of energy) the energy of the system does not change with time. The primary macroscopic variables of the microcanonical ensemble are the total number of particles in the system (symbol: ), the system's volume (symbol: ), as well as the total energy in the system (symbol: ). Each of these is assumed to be constant in the ensemble. For this reason, the microcanonical ensemble is sometimes called the ensemble. In simple terms, the microcanonical ensemble is defined by assigning an equal probability to every microstate whose energy falls within a range centered at . All other microstates are given a probability of zero. Since the probabilities must add up to 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand Canonical Ensemble
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system. The thermodynamic variables of the grand canonical ensemble are chemical potential (symbol: ) and absolute temperature (symbol: . The ensemble is also dependent on mechanical variables such as volume (symbol: , which influence the nature of the system's internal states. This ensemble is therefore sometimes called the ensemble, as each of these three quantitie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-point Energy
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly Quantum fluctuation, fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle. Therefore, even at absolute zero, atoms and molecules retain some vibrational motion. Apart from atoms and molecules, the empty space of Vacuum state, the vacuum also has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating Field (physics), fields: matter fields, whose Quantum, quanta are fermions (i.e., leptons and quarks), and Force field (physics), force fields, whose quanta are bosons (e.g., photons and gluons). All these fields have zero-point energy. These fluctuating zero-point fields lead to a kind of reintroduction of an Luminiferous aether, aether in physics since some systems can detect the existence of this energy. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Statistical Mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. It relies on constructing density matrices that describe quantum systems in thermal equilibrium. Its applications include the study of collections of identical particles, which provides a theory that explains phenomena including superconductivity and superfluidity. Density matrices, expectation values, and entropy In quantum mechanics, probabilities for the outcomes of experiments made upon a system are calculated from the quantum state describing that system. Each physical system is associated with a vector space, or more specifically a Hilbert space. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. A density operator, the mathematical r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
