Time evolution, cyclic solutions and geometric phases for the generalized time-dependent harmonic oscillator
Time evolution, cyclic solutions and geometric phases for the generalized time-dependent harmonic oscillator
复制标题
广义瞬态谐振子的时间演化、循环解和几何相位
DOI：
10.1088/0305-4470/37/4/020
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发表时间：
2004-01
期刊：
影响因子：
--
通讯作者：
林琼桂
中科院分区：
文献类型：
--
作者：
林琼桂
研究了广义时变谐振子。虽然已有几种求解该模型的方法，但本文提出了一种新的方法，它非常适合于研究循环解和几何相。在这种方法中，寻找薛定谔方程的时间演化算子被简化为求解三维空间中在双曲面上运动的c数向量的常微分方程。循环解并不存在于所有时间间隔。给出了循环解存在的充分必要条件。在一定的时间区间内，可能存在所有的解都有确定宇称，甚至所有的解都是循环的。给出了出现这种情况的标准。重新建立了已知的循环解的非绝热几何相位与经典汉内角成正比的关系。然而，这只对特殊的循环解有效。对于更一般的方程，非绝热几何相可能包含一个额外的项。对几个具有相对简单哈密顿量的情况进行了详细的讨论和求解。大多数情况下存在循环解。运动的模式，比方说，有限或无限，不能简单地由哈密顿量的性质（椭圆或双曲，等等）来决定。对于具有确定性质的哈密顿量，运动可以从一种模式转变为另一种模式，也就是说，如果哈密顿量中的某个参数经过某个临界值，则可能发生某种相变。
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrodinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
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影响因子：
8.6
作者：
Lewis;Lawrence;Harris
通讯作者：
Lewis;Lawrence;Harris
影响因子：
8.6
作者：
Y. Ge;M. Child
通讯作者：
Y. Ge;M. Child
DOI：
10.1103/physreva.42.5107
发表时间：
1990-11
期刊：
Physical review. A, Atomic, molecular, and optical physics
影响因子：
--
作者：
Shun-Jin Wang
通讯作者：
Shun-Jin Wang
DOI：
10.1103/physrevb.38.11907
发表时间：
1988-12
期刊：
Physical review. B, Condensed matter
影响因子：
--
作者：
Wu;Li
通讯作者：
Wu;Li
DOI：
10.1103/physreva.52.3352
发表时间：
1995-10
期刊：
Physical review. A, Atomic, molecular, and optical physics
影响因子：
--
作者：
Ji;Kim;Soh
通讯作者：
Ji;Kim;Soh