含时线性势的薛定谔方程的解
Solutions of the Schrödinger equation for the time-dependent linear potential
摘要：
利用Lewis-Riesenfeld不变量理论，得到了含时线性势的薛定谔方程对应于二次型Lewis-Riesenfeld不变量$I_{\rm q}(t)$的解.本文强调指出，要得到含时薛定谔方程的通解，首先要找到完备的Lewis-Riesenfeld不变量集.对于具有含时线性势的本量子系统，线性$I_{\rm l}(t)$和二次$I_{\rm q}(t)$（其中后一个$I_{\rm q}(t)$不能写为前一个$I_{\rm l}(t)$的平方，{ \it即}，关系式$I_{\rm q}(t)= cI_{\rm l}^{2}(t)$并不总是成立）将形成Lewis-Riesenfeld不变量的完整集合。本文还证明了Bekkar { \it et al.}最近的是对应于线性$I_{\rm l}(t)$的一个，它是形成完整集合的不变量之一。此外，我们还讨论了与Bekkar { \it et al.}的评论[Phys. Rev. A { \bf 68}，016101（2003）]相关的一些主题。关于Guedes的工作[Phys.Rev.A { \bf 63}，034102（2001）]和Guedes的相应答复[Phys.Rev.A { \bf 68}，016102（2003）]。
英文摘要：
By making use of the Lewis-Riesenfeld invariant theory, the solution of the Schrödinger equation for the time-dependent linear potential corresponding to the quadratic-form Lewis-Riesenfeld invariant $I_{\rm q}(t)$ is obtained in the present paper. It is emphasized that in order to obtain the general solutions of the time-dependent Schrödinger equation, one should first find the complete set of Lewis-Riesenfeld invariants. For the present quantum system with a time-dependent linear potential, the linear $I_{\rm l}(t)$ and quadratic $I_{\rm q}(t)$ (where the latter $I_{\rm q}(t)$ cannot be written as the squared of the former $I_{\rm l}(t)$, {\it i.e.}, the relation $I_{\rm q}(t)= cI_{\rm l}^{2}(t)$ does not hold true always) will form a complete set of Lewis-Riesenfeld invariants. It is also shown that the solution obtained by Bekkar {\it et al.} more recently is the one corresponding to the linear $I_{\rm l}(t)$, one of the invariants that form the complete set. In addition, we discuss some related topics regarding the comment [Phys. Rev. A {\bf 68}, 016101 (2003)] of Bekkar {\it et al.} on Guedes's work [Phys. Rev. A {\bf 63}, 034102 (2001)] and Guedes's corresponding reply [Phys. Rev. A {\bf 68}, 016102 (2003)].
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