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Semicircle Potential Well
In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The particle follows the path of a semicircle from 0 to \pi where it cannot escape, because the potential from \pi to 2 \pi is infinite. Instead there is total reflection, meaning the particle bounces back and forth between 0 to \pi . The Schrödinger equation for a free particle which is restricted to a semicircle (technically, whose configuration space is the circle S^1) is Wave function Using cylindrical coordinates on the 1-dimensional semicircle, the wave function depends only on the angular coordinate, and so Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as The moment of inertia for a semicircle, best expressed in cylindrical coordinates, is I \ \stackrel\ \iiint_V r^2 \,\rho(r,\phi,z)\,r dr\,d\phi\,dz \!. Solving the integral, one finds that the moment of inertia of a semicircle is I=m s^2 , exact ... [...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] |
Particle In A Ring
In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S^1) is : -\frac\nabla^2 \psi = E\psi Wave function Using polar coordinates on the 1-dimensional ring of radius R, the wave function depends only on the angular coordinate, and so Problems and Solutions to accompany Physical Chemistry: a Molecular Approach : \nabla^2 = \frac \frac Requiring that the wave function be periodic in \ \theta with a period 2 \pi (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions : \int_^ \left, \psi ( \theta ) \^2 \, d\theta = 1\ , and : \ \psi (\theta) = \ \psi ( \theta + 2\pi) Under these conditions, the solution to the Schrödinger equation is given by : \psi_(\theta) = \frac\, e^ Energy eigenvalues The energy e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle In A Spherically Symmetric Potential
In the quantum mechanics description of a particle in spherical coordinates, a spherically symmetric potential, is a potential that depends only on the distance between the particle and a defined centre point. One example of a spherical potential is the electron within a hydrogen atom. The electron's potential depends only on its distance from the proton in the atom's nucleus. This spherical potential can be derived from Coulomb's law. In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form: \hat = \frac + V(r) Where m_0 is the mass of the particle, \hat is the momentum operator, and the potential V(r) depends only on r, the modulus of the radius vector. The possible quantum states of the particle are found by using the above Hamiltonian to solve the Schrödinger equation for its eigenvalues, which are wave functions. To describe these spherically symmetric systems, it is natural to use spheri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gas In A Box
In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions. This simple model can be used to describe the classical ideal gas as well as the various quantum ideal gases such as the ideal massive Fermi gas, the ideal massive Bose gas as well as black body radiation (photon gas) which may be treated as a massless Bose gas, in which thermalization is usually assumed to be facilitated by the interaction of the photons with an equilibrated mass. Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics, and considering the limit of a very large box, the Thomas–Fermi approximation (named after Enrico Fermi and Llewellyn Thomas) is used to express the degeneracy of the energy states as a differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delta Function Potential
In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential. The delta potential well is a limiting case of the finite potential well, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential. This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Potential Well
The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than the potential energy barrier of the walls it cannot be found outside the box. In the quantum interpretation, there is a non-zero probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls (cf quantum tunnelling). Particle in a 1-dimensional box For the 1-dimensional case on the ''x''-axis, the time-independent Schrödinger equation can be written as: where *\hbar = \frac is the reduced Planck's constant, *h is Plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle In A Box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes. The particle in a box mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle In A Ring
In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S^1) is : -\frac\nabla^2 \psi = E\psi Wave function Using polar coordinates on the 1-dimensional ring of radius R, the wave function depends only on the angular coordinate, and so Problems and Solutions to accompany Physical Chemistry: a Molecular Approach : \nabla^2 = \frac \frac Requiring that the wave function be periodic in \ \theta with a period 2 \pi (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions : \int_^ \left, \psi ( \theta ) \^2 \, d\theta = 1\ , and : \ \psi (\theta) = \ \psi ( \theta + 2\pi) Under these conditions, the solution to the Schrödinger equation is given by : \psi_(\theta) = \frac\, e^ Energy eigenvalues The energy e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the ''number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate of a po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle In A Box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes. The particle in a box mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two rays lie in the plane (geometry), plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection. ''Angle'' is also used to designate the measurement, measure of an angle or of a Rotation (mathematics), rotation. This measure is the ratio of the length of a arc (geometry), circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation ... [...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 tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
