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: Abstract: The wavefunctions and energy eigenvalues for the ring-shaped oscillator are obtained using the ideas of supersymmetric quantum mechanics and shape invaricace.
中文: 文摘:采用量子力学中的超对称性和形不变性，讨论环形振子的波函数和能量本征值.
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英文: Applications to physics: the oscillator and planetary motion.
中文: 物理上的应用：谐振和行星运动。
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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: Each machine cycle takes 12 oscillator or clock cycles.
中文: 每个机器周期为12个振荡器或时钟周期。
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