论文标题
二维电位散射中的传播波近似
Propagating-wave approximation in two-dimensional potential scattering
论文作者
论文摘要
我们引入了一种非扰动近似方案，用于在两个维度上执行散射计算，涉及忽略逃生波对散射幅度的贡献。这对应于用相关的依赖能量的非本地电位$ {\ Mathscr {v}} _ k $替换交互潜力$ v $，该潜在$ {\ mathscr {v}} _ k $不会与evaneScent Wave搭配。 Schrödinger方程的散射解决方案$ψ（\ Mathbf {r}）$，$（ - \ nabla^2+{\ MathScr {v}} _ k）ψ（\ Mathbf {r}）= k^2ψ（\ Mathbf {r}） $ \tildeψ（\ mathbf {p}）$消失了，除非$ \ mathbf {p} $对应于幅度等于$ k $的经典粒子的动量。我们为这类非本地电位构建了传输矩阵，并根据进化算子的进化量子来探索其表示有效的非整体量子系统。我们表明，上面的近似值减少到弱电位的第一个诞生近似值，并且与半经典近似相似，在高能下有效。此外，我们确定了无限的复杂电势，该近似方案是准确的。我们还讨论了该方案的实用和数学方面。
We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $ψ(\mathbf{r})$ of the Schrödinger equation, $(-\nabla^2+{\mathscr{V}}_k)ψ(\mathbf{r})=k^2ψ(\mathbf{r})$, has the remarkable property that their Fourier transform $\tildeψ(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.
