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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク List of quantum-mechanical systems with analytical solutions ： ウィキペディア英語版 List of quantum-mechanical systems with analytical solutions Much insight in quantum mechanics can be gained from understanding the solutions to the time-dependent non-relativistic Schrödinger equation in an appropriate configuration space. In vector Cartesian coordinates $\backslash mathbf$, the equation takes the form :$$ H \psi\left(\mathbf, t\right) = \left(T + V\right) \, \psi\left(\mathbf, t\right) = \left(- \frac \nabla^2 + V\left(\mathbf\right) \right ) \psi\left(\mathbf, t\right) = i\hbar \frac in which $\backslash psi$ is the wavefunction of the system, H is the Hamiltonian operator, and T and V are the operators for the kinetic energy and potential energy, respectively. (Common forms of these operators appear in the square brackets.) The quantity ''t'' is the time. Stationary states of this equation are found by solving the eigenvalue-eigenfunction (time-independent) form of the Schrödinger equation, :$$ \left(- \frac \nabla^2 + V\left(\mathbf\right) \right ) \psi\left(\mathbf\right) = E \psi \left(\mathbf\right) or any equivalent formulation of this equation in a different coordinate system other than Cartesian coordinates. For example, systems with spherical symmetry are simplified when expressed with spherical coordinates. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. Fortunately, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies can be found. These quantum-mechanical systems with analytical solutions are listed below, and are quite useful for teaching and gaining intuition about quantum mechanics. == Solvable systems == *The free particle *The delta potential *The Double well Dirac delta potential *The particle in a box / infinite potential well *The finite potential well *The One-dimensional triangular potential *The particle in a ring or ring wave guide *The particle in a spherically symmetric potential *The quantum harmonic oscillator *The hydrogen atom or hydrogen-like atom e.g. positronium *The hydrogen atom in a spherical cavity with Dirichlet boundary conditions〔T.C. Scott and Wenxing Zhang, ''Efficient hybrid-symbolic methods for quantum mechanical calculations'', Comput. Phys. Commun. 191, pp. 221-234, 2015 ().〕 *The Hydrogen Molecular ion (Solutions in terms of generalized Lambert W function) *The particle in a one-dimensional lattice (periodic potential) *The Morse potential *The step potential *The linear rigid rotor *The symmetric top *The Hooke's atom *The Spherium atom *Zero range interaction in a harmonic trap〔http://www.springerlink.com/content/k86t52r653522rk6/〕 *The Quantum pendulum *The Rectangular potential barrier *The Inverse square root potential〔http://arxiv.org/abs/1509.00019〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of quantum-mechanical systems with analytical solutions」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.