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Quantum Mechanical Potentials
This is a list of potential energy functions that are frequently used in quantum mechanics and have any meaning. One-dimensional potentials * Rectangular potential barrier * Delta potential (aka "contact potential") * Double delta potential * Step potential * Periodic potential * Barrier potential * Gaussian potential * Eckart potential Wells * Quantum well * Potential well * Finite potential well * Infinite potential well * Double-well potential * Semicircular potential well * Circular potential well * Spherical potential well * Triangular potential well Interatomic potentials * Interatomic potential * Bond order potential * EAM potential * Coulomb potential * Buckingham potential * Lennard-Jones potential * Morse potential * Morse/Long-range potential * Rosen–Morse potential * Trigonometric Rosen–Morse potential * Stockmayer potential * Pöschl–Teller potential * Axilrod–Teller potential * Mie potential Oscillators * Harmonic potential (harmonic os ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object, the elastic potential energy of an extended spring, and the electric potential energy of an electric charge in an electric field. The unit for energy in the International System of Units (SI) is the joule, which has the symbol J. The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, that are called ''conservative forces'', can be represented at every point in space by vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Potential Well
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of Euclid's Elements. The names used for modern classification are eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axilrod–Teller Potential
The Axilrod–Teller potential in molecular physics, is a three-body potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ... that results from a third-order perturbation correction to the attractive London dispersion interactions (instantaneous induced dipole-induced dipole) : V_= E_ \left \frac \right where r_ is the distance between atoms i and j, and \gamma_ is the angle between the vectors \mathbf_ and \mathbf_. The coefficient E_ is positive and of the order V\alpha^, where V is the ionization energy and \alpha is the mean atomic polarizability; the exact value of E_ depends on the magnitudes of the dipole matrix elements and on the energies of the p orbitals. References * Chemical bonding Quantum mechanical potentials {{electromagnetism-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pöschl–Teller Potential
In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions. Definition In its symmetric form is explicitly given by : V(x) =-\frac\mathrm^2(x) and the solutions of the time-independent Schrödinger equation : -\frac\psi''(x)+ V(x)\psi(x)=E\psi(x) with this potential can be found by virtue of the substitution u=\mathrm, which yields : \left 1-u^2)\psi'(u)\right+\lambda(\lambda+1)\psi(u)+\frac\psi(u)=0 . Thus the solutions \psi(u) are just the Legendre functions P_\lambda^\mu(\tanh(x)) with E=\frac, and \lambda=1, 2, 3\cdots, \mu=1, 2, \cdots, \lambda-1, \lambda. Moreover, eigenvalues and scattering data can be explicitly computed. In the special case of integer \lambda, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stockmayer Potential
The Stockmayer potential is a mathematical model for representing the interactions between pairs of atoms or molecules. It is defined as a Lennard-Jones potential with a point electric dipole moment The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The .... A Stockmayer liquid consists of a collection of spheres with point dipoles embedded at the centre of each. These spheres interact both by Lennard-Jones and dipolar interactions. In the absence of the point dipoles, the spheres face no rotational friction and the translational dynamics of such LJ spheres have been studied in detail. This system, therefore, provides a simple model where the only source of rotational friction is dipolar interactions References M. E. Van Leeuwe "Deviation from corresponding-states behaviour for polar fluids" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Rosen–Morse Potential
The trigonometric Rosen–Morse potential, named after the physicists Nathan Rosen and Philip M. Morse, is among the exactly solvable quantum mechanical potentials. Definition In dimensionless units and modulo additive constants, it is defined as where r is a relative distance, \lambda is an angle rescaling parameter, and R is so far a matching length parameter. Another parametrization of same potential is which is the trigonometric version of a one-dimensional hyperbolic potential introduced in molecular physics by Nathan Rosen and Philip M. Morse and given by, a parallelism that explains the potential's name. The most prominent application concerns the V_^(\chi) parametrization, with \ell non-negative integer, and is due to Schrödinger who intended to formulate the hydrogen atom problem on Albert Einstein's closed universe, R^1\otimes S^3, the direct product of a time line with a three-dimensional closed space of positive constant curvature, the hypersphere S^, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pöschl–Teller Potential
In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions. Definition In its symmetric form is explicitly given by : V(x) =-\frac\mathrm^2(x) and the solutions of the time-independent Schrödinger equation : -\frac\psi''(x)+ V(x)\psi(x)=E\psi(x) with this potential can be found by virtue of the substitution u=\mathrm, which yields : \left 1-u^2)\psi'(u)\right+\lambda(\lambda+1)\psi(u)+\frac\psi(u)=0 . Thus the solutions \psi(u) are just the Legendre functions P_\lambda^\mu(\tanh(x)) with E=\frac, and \lambda=1, 2, 3\cdots, \mu=1, 2, \cdots, \lambda-1, \lambda. Moreover, eigenvalues and scattering data can be explicitly computed. In the special case of integer \lambda, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse/Long-range Potential
The Morse/Long-range potential (MLR potential) is an interatomic interaction model for the potential energy of a diatomic molecule. Due to the simplicity of the regular Morse potential (it only has three adjustable parameters), it is very limited in its applicability in modern spectroscopy. The MLR potential is a modern version of the Morse potential which has the correct theoretical long-range form of the potential naturally built into it. It has been an important tool for spectroscopists to represent experimental data, verify measurements, and make predictions. It is useful for its extrapolation capability when data for certain regions of the potential are missing, its ability to predict energies with accuracy often better than the most sophisticated ''ab initio'' techniques, and its ability to determine precise empirical values for physical parameters such as the dissociation energy, equilibrium bond length, and long-range constants. Cases of particular note include: # the c- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Potential
The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Due to its simplicity (only three fitting parameters), it is not used in modern spectroscopy. However, its mathematical form inspired the MLR ( Morse/Long-range) potential, which is the most popular potential energy function used for fitting spectroscopic data. Potential energy function The Morse potential energy function is of the form :V(r) = D_e ( 1-e^ )^2 Here r is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lennard-Jones Potential
The Lennard-Jones potential (also termed the LJ potential or 12-6 potential) is an intermolecular pair potential. Out of all the intermolecular potentials, the Lennard-Jones potential is probably the one that has been the most extensively studied. It is considered an archetype model for simple yet realistic intermolecular interactions. The Lennard-Jones potential models soft repulsive and attractive ( van der Waals) interactions. Hence, the Lennard-Jones potential describes electronically neutral atoms or molecules. It is named after John Lennard-Jones. The commonly used expression for the Lennard-Jones potential is V_\text(r) = 4\varepsilon \left \left(\frac\right)^ - \left(\frac\right)^6 \right, where r is the distance between two interacting particles, \varepsilon is the depth of the potential well (usually referred to as 'dispersion energy'), and \sigma is the distance at which the particle-particle potential energy V is zero (often referred to as 'size of the particle'). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buckingham Potential
In theoretical chemistry, the Buckingham potential is a formula proposed by Richard Buckingham which describes the Pauli exclusion principle and van der Waals energy \Phi_(r) for the interaction of two atoms that are not directly bonded as a function of the interatomic distance r. It is a variety of interatomic potentials. :\Phi_(r) = A \exp \left(-Br\right) - \frac Here, A, B and C are constants. The two terms on the right-hand side constitute a repulsion and an attraction, because their first derivatives with respect to r are negative and positive, respectively. Buckingham proposed this as a simplification of the Lennard-Jones potential, in a theoretical study of the equation of state for gaseous helium, neon and argon. As explained in Buckingham's original paper and, e.g., in section 2.2.5 of Jensen's text,F. Jensen, ''Introduction to Computational Chemistry'', 2nd ed., Wiley, 2007, the repulsion is due to the interpenetration of the closed electron shells. "There is ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb Potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used. In classical electrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar quantity denoted by or occasionally , equal to the electric potential energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
