受迫谐振子的Cauchy问题
The Cauchy Problem for a Forced Harmonic Oscillator
摘要：
本文构造了含时哈密顿算符的一维薛定谔方程的Cauchy初值问题的显式解。首先利用广义Fourier变换和Heisenberg-Weyl群N（3）的表示关系，导出了相应的Green函数（传播子），然后将其推广到一般情况.一个三参数扩展的经典傅里叶积分讨论的副产品。作为一个应用，考虑了自旋粒子在均匀垂直磁场和电场中的运动，用Charlier多项式计算了朗道能级之间的跃迁振幅。此外，我们还解决了一个类似的扩散型方程的初值问题。
英文摘要：
We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schroedinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N(3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.
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