在Mathematica 6.0中用吸收边界条件和源项求解含时薛定谔方程
Solving the time-dependent Schrödinger equation with absorbing boundary conditions and source terms in Mathematica 6.0
摘要：
近几十年来，人们对含时薛定谔方程的数值解进行了大量的研究。一方面，一些数值方法不需要任何形式的矩阵求逆，但源项不容易实现到这些计划;另一方面，一些方法涉及矩阵求逆可以实现源项以自然的方式，但不容易实现到一些计算软件程序广泛使用的非专家在编程（如Mathematica）。利用标准的Crank-Nicholson方法和演化算子的凯莱形式的有限差分表示，提出了一种求解含时薛定谔方程的简单方法。在这里，这种标准的数值方案已被简化，通过反演解析的位置表示的发展算子的矩阵。N × N矩阵的解析求逆让我们可以轻松地完全实现数值方法，无论是否有源项，都可以在Mathematica中实现，甚至可以在任何用于科学计算的数值计算语言或计算软件中实现。
英文摘要：
In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrödinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrödinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.
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