以下为句子列表：
英文: A damped harmonic oscillator without a driving force was studied using canonical tramsformation; An exact wavefunction and energy level for the damped harmonic oscillator has Been obtained.
中文: 摘要对于无外界驱动力且阻力与速度成正比的阻尼谐振子，通过正则变换，得出了阻尼谐振子的严格波函数及其相应能级。
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英文: By using some formulae of confluent hyper-geometric function, the results of radial operators on the normalizing radial wave function of a 3-D isotropic harmonic oscillator are derived, and the ladder operators are given to calculate the matrix elements.
中文: 摘要利用合流超几何函数的递推关系式得到径向算符对归一化径向函数的作用结果，得出其升降算符通项，便于求解矩阵元。
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英文: Concepts covered include: Schrodinger's equation applied to the free particle, tunneling, the harmonic oscillator, and hydrogen atom, variational methods, Fermi-Dirac, Bose-Einstein, and Boltzmann distribution functions, and simple models for metals, semi
中文: 这些概念包括：应用于自由粒子的薛丁格方程式、穿遂效应、谐振子、氢原子、变分法、费米─狄拉克（译注一）、波司─爱因斯坦（译注二）、波兹曼分布函数（译注三）、简单金属模型、半导体以及包括电子显微镜、穿透式扫瞄电子显微镜、热射极、原子力显微镜等的元件。
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英文: Coordinate Transformation, Matrix, Vectors, Newton's Law, Conservation Theorems, Simple Harmonic Oscillator, Non-linear Oscillations, Gravitation, Euler's Equation when Auxiliany Condition are Imposed, The Delta Notation, Lagrangian and Hamilitonian Dynam
中文: 座标变换、矩阵、向量、牛顿定律、守恒定律、简谐振动、非线性振动、引力、尤拉式方程式及附加条件、δ符号、拉格兰及汉米尔顿力学、中心运动、多粒子系统动力学、非惯性参考座标运动、刚体动力学、耦会振动、本微向量正交性、连续系统。
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英文: We solve the master equation by system-reservoir method with Born-Markov approximation for the composite system with one simple harmonic oscillator and the reservoir composed with many other harmonic oscillators surrounded.
中文: 摘要:我们使用系统与热库的方法，并使用波恩-马可夫近似来求解一模型：单一简谐振子与一群简谐振子的热库。
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