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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 the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...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 : \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 eigenvalues E are quantized because of the periodic boundary conditions, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle In A Spherically Symmetric Potential
In quantum mechanics, a spherically symmetric potential is a system of which the potential only depends on the radial distance from the spherical center and a location in space. A particle in a spherically symmetric potential will behave accordingly to said potential and can therefore be used as an approximation, for example, of the electron in a hydrogen atom or of the formation of chemical bonds. In the general time-independent case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:\hat = \frac + V() Here, m_0 is the mass of the particle, \hat is the momentum operator, and the potential V(r) depends only on the vector magnitude of the position vector, that is, the radial distance from the origin (hence the spherical symmetry of the problem). To describe a particle in a spherically symmetric system, it is convenient to use spherical coordinates; denoted by r, \theta and \phi. The time-independent Schrödin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gas In A Box
Gas is a state of matter that has neither a fixed volume nor a fixed shape and is a compressible fluid. A ''pure gas'' is made up of individual atoms (e.g. a noble gas like neon) or molecules of either a single type of atom ( elements such as oxygen) or from different atoms ( compounds such as carbon dioxide). A ''gas mixture'', such as air, contains a variety of pure gases. What distinguishes gases from liquids and solids is the vast separation of the individual gas particles. This separation can make some gases invisible to the human observer. The gaseous state of matter occurs between the liquid and plasma states, the latter of which provides the upper-temperature boundary for gases. Bounding the lower end of the temperature scale lie degenerative quantum gases which are gaining increasing attention. High-density atomic gases super-cooled to very low temperatures are classified by their statistical behavior as either Bose gases or Fermi gases. For a comprehensive listing of ... [...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 one-dimensional potential well For the one-dimensional case on the ''x''-axis, the time-independent Schrödinger equation can be written as: where * \hbar is the reduced Planck constant, * m ... [...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 the movement of a free particle 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 ... [...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 : \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 eigenvalues E are quantized because of the periodic boundary conditions, a ... [...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 and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such 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 o ... [...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 the movement of a free particle 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane 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. More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides. Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle. The term ''angle'' is also used for the size, magnitude (mathematics), magnitude or Physical quantity, quantity of these types of geometric figures and in this context an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter), psi, respectively). Wave functions are complex number, complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density function, probability density of measurement in quantum mechanics, measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
