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: 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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