实时间和虚时间的含时矩形势薛定谔方程的精确解
An exact solution of the time-dependent Schrödinger equation with a rectangular potential for real and imaginary time
摘要：
利用多重散射理论，得到了具有非对称矩形势的一维含时薛定谔方程的传播子。它允许考虑的反射和透射过程的粒子散射在潜在的跳跃（与传统的类波图片），并占非经典的反直觉的贡献的向后移动的组成部分的波包归因于粒子。该传播子完全解决了相应的时间相关的薛定谔方程（定义波函数$ψ(x,t)$），并允许考虑从潜在的向下台阶/阱的粒子反射和粒子隧穿势垒作为时间的函数的量子力学效应。这些结果与一些基本问题有关，如在量子力学中测量时间（隧穿时间、到达时间、停留时间）。对于虚时间，它表示逆温度$(t\rightarrow-i{\hbar}/k_{B}T)$，所获得的传播子是等价的密度矩阵的粒子是在热浴中，并受到矩形势。如果转移到虚时间$(t\rightarrow-it)$，则在空间基中计算的传播子的矩阵元素提供了具有矩形势的类扩散方程的解。所得到的精确结果作为从初等函数的积分，从而允许的概率密度$|ψ(x,t)|^2$，密度矩阵和扩散方程的解决方案的数值可视化。所获得的结果也可能是有用的自旋电子学应用，由于事实上，非对称（自旋相关）的矩形电位可以模拟分层磁性纳米结构的电位分布。
英文摘要：
A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function $ψ(x,t)$) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature $(t\rightarrow-i{\hbar}/k_{B}T)$, the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time $(t\rightarrow-it)$, the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density $|ψ(x,t)|^2$, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
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