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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 analyt ... [...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] |
Quantum Harmonic Oscillator
量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最も量子力学における重要なモデル系。さらに、これは正確な 解析解法が知られている数少ない量子力学系の1つである。 author=Griffiths, David J. , title=量子力学入門 , エディション=2nd , 出版社=プレンティス・ホール , 年=2004 , isbn=978-0-13-805326-0 , author-link=David Griffiths (物理学者) , URL アクセス = 登録 , url=https://archive.org/details/introductiontoel00grif_0 One-dimensional harmonic oscillator Hamiltonian and energy eigenstates 粒子の ハミルトニアン は次のとおりです。 \hat H = \frac + \frac k ^2 = \frac + \frac m \omega^2 ^2 \, , ここで、 は粒子の質量、 は力定数、\omega = \sqrt は 動子の [角周波数 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangular Potential Barrier
In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left. Although classically a particle behaving as a point mass would be reflected if its energy is less than a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Pendulum
The quantum pendulum is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be solved relatively easily. Schrödinger equation Using Lagrangian mechanics from classical mechanics, one can develop a Hamiltonian for the system. A simple pendulum has one generalized coordinate (the angular displacement \phi) and two constraints (the length of the string and the plane of motion). The kinetic and potential energies of the system can be found to be :T = \frac m l^2 \dot^2, :U = mgl (1 - \cos\phi). This results in the Hamiltonian :\hat = \frac + mgl (1 - \cos\phi). The time-dependent Schrödinger equation for the system is :i \hbar \frac = -\frac \frac + mgl (1 - \cos\phi) \Psi. One must solve the time-independent Schrödinger equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hooke's Atom
Hooke's atom, also known as harmonium or hookium, refers to an artificial helium-like atom where the Coulombic electron-nucleus interaction potential is replaced by a harmonic potential. This system is of significance as it is, for certain values of the force constant defining the harmonic containment, an exactly solvable ground-state many-electron problem that explicitly includes electron correlation. As such it can provide insight into quantum correlation (albeit in the presence of a non-physical nuclear potential) and can act as a test system for judging the accuracy of approximate quantum chemical methods for solving the Schrödinger equation. The name "Hooke's atom" arises because the harmonic potential used to describe the electron-nucleus interaction is a consequence of Hooke's law. Definition Employing atomic units, the Hamiltonian defining the Hooke's atom is :\hat = -\frac\nabla^_ -\frac\nabla^_ + \frack(r^_+r^_) + \frac. As written, the first two terms are the kin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Rotor
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the ''linear rotor'' requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water (asymmetric rotor), ammonia (symmetric rotor), or methane (spherical rotor). Linear rotor The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses and the values of the masses are the only characteristics of the rigid model. However, for many actual diatomics this model is too restrictive since distances are usually not completely fixed. Corrections on the rigid model can be made to compensate for small variations in the distance. Even in such a case the rigid rotor model is a u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Step Potential
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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] |
Particle In A One-dimensional Lattice Of Finite Length ''L = N A'' (''N'' Is A Positive Integer, ''a'' Is The Potential Period)
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic particles like atoms and molecules, to macroscopic particles like powders and other granular materials. Particles can also be used to create scientific models of even larger objects depending on their density, such as humans moving in a crowd or celestial bodies in motion. The term ''particle'' is rather general in meaning, and is refined as needed by various scientific fields. Anything that is composed of particles may be referred to as being particulate. However, the noun ''particulate'' is most frequently used to refer to pollutants in the Earth's atmosphere, which are a suspension of unconnected particles, rather than a connected particle aggregation. Conceptual properties The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle In A One-dimensional Lattice (periodic Potential)
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice. It is a generalization of the free electron model, which assumes zero potential inside the lattice. Problem definition When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is , the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period . According to Bloch's theorem, the wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as: \psi (x) = e^ u(x), where is a periodic function which satisfies . It is the Bloch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Boundary Condition
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, ''essential'' or Dirichlet boundary condition is defined by weighted-integral form of a differential equation. The dependent unknown ''u in the same form as the weight function w'' appearing in the boundary expression is termed a ''primary variable'', and its specification constitutes the ''essential'' or Dirichlet boundary condition. The question of finding solutions to such equations is known as the Dirichlet problem. In applied sciences, a Dirichlet boundary condition may also be referred to as a fixed boundary condition. Examples ODE For an ordinary differential equation, for instance, y'' + y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
