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Spherium
The "spherium" model consists of two electrons trapped on the surface of a sphere of radius R. It has been used by Berry and collaborators to understand both weakly and strongly correlated systems and to suggest an "alternating" version of Hund's rule. Seidl studies this system in the context of density functional theory (DFT) to develop new correlation functionals within the adiabatic connection. Definition and solution The electronic Hamiltonian in atomic units is :\hat = - \frac - \frac + \frac where u is the interelectronic distance. For the singlet S states, it can be then shown that the wave function S(u) satisfies the Schrödinger equation :\left( \frac - 1 \right) \frac + \left(\frac - \frac \right) \frac + \frac S(u)= E S(u) By introducing the dimensionless variable x = u/2R, this becomes a Heun equation with singular points at x = -1, 0, +1. Based on the known solutions of the Heun equation, we seek wave functions of the form :S(u) = \sum_^\infty s_k\,u^k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Quantum-mechanical Systems With Analytical Solutions
Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation. It takes the form : \hat \psi\left(\mathbf, t\right) = \left - \frac \nabla^2 + V\left(\mathbf\right) \right\psi\left(\mathbf, t\right) = i\hbar \frac, where \psi is the wave function of the system, \hat is the Hamiltonian operator, and t is time. Stationary states of this equation are found by solving the time-independent Schrödinger equation, : \left - \frac \nabla^2 + V\left(\mathbf\right) \right\psi\left(\mathbf\right) = E \psi \left(\mathbf\right), which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies, or eigenvalues, can be found. These quantum-mechanical systems with analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron's mass is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum ( spin) of a half-integer value, expressed in units of the reduced Planck constant, . Being fermions, no two electrons can occupy the same quantum state, per the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: They can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heun Equation
In mathematics, the local Heun function H \ell (a,q;\alpha ,\beta, \gamma, \delta ; z) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point ''z'' = 0. The local Heun function is called a Heun function, denoted ''Hf'', if it is also regular at ''z'' = 1, and is called a Heun polynomial, denoted ''Hp'', if it is regular at all three finite singular points ''z'' = 0, 1, ''a''. Heun's equation Heun's equation is a second-order linear ordinary differential equation (ODE) of the form :\frac + \left frac+ \frac + \frac \right \frac + \frac w = 0. The condition \epsilon=\alpha+\beta-\gamma-\delta+1 is taken so that the characteristic exponents for the regular singularity at infinity are α and β (see below). The complex number ''q'' is called the accessory parameter. Heun's equation has four regular singular points: 0, 1, ''a'' and ∞ with exponents (0, 1 − ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review Letters
''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the '' Journal Citation Reports'' impact factor and the journal ''h''-index proposed by Google Scholar, many physicists and other scientists consider ''Physical Review Letters'' to be one of the most prestigious journals in the field of physics. ''According to Google Scholar, PRL is the journal with the 9th journal h-index among all scientific journals'' ''PRL'' is published as a print journal, and is in electronic format, online and CD-ROM. Its focus is rapid dissemination of significant, or notable, results of fundamental research on all topics related to all fields of physics. This is accomplished by rapid publication of short reports, called "Letters". Papers are published and available electronically one article at a time. When published ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an ''n''-sphere. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, 4-dimensional space () such that :\sum_^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted : :S^3 = \left\. It is often convenient to regard as the space with 2 complex dimensions () or the quaternions (). The unit 3-sphere is then given by :S^3 = \left\ or :S^3 = \left\. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Physics (journal)
''Molecular Physics'' is a peer-reviewed scientific journal covering research on the interface between chemistry and physics, in particular chemical physics and physical chemistry. It covers both theoretical and experimental molecular science, including electronic structure, molecular dynamics, spectroscopy, reaction kinetics, statistical mechanics, condensed matter and surface science. The journal was established in 1958 and is published by Taylor & Francis. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 1.962. The current editor-in-chief is Professor George Jackson (Imperial College London). A reprint of the first editorial and a full list of editors since its establishment can be found in the issue celebrating 50 years of the journal. Notable current and former editors * Christopher Longuet-Higgins (Founding Editor) * Joan van der Waals (Founding Editor) * John Shipley Rowlinson * A. David Buckingham * Lawrence D. Barron * Mart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kato Theorem
The Kato theorem, or Kato's cusp condition (after Japanese mathematician Tosio Kato), is used in computational quantum physics. It states that for generalized Coulomb potentials, the electron density has a cusp at the position of the nuclei, where it satisfies : Z_k = - \frac \frac , _ Here \mathbf denotes the positions of the nuclei, Z_k their atomic number and a_o is the Bohr radius. For a Coulombic system one can thus, in principle, read off all information necessary for completely specifying the Hamiltonian directly from examining the density distribution. This is also known as E. Bright Wilson's argument within the framework of density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-bo ... (DFT). The electron density of the ground state of a molecular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression ... [...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] |
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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