具有“天花板”边界的线性势在位置和动量空间中的WKB传播子
WKB propagators in position and momentum space for a linear potential with a "ceiling" boundary
摘要：
作为具有势和边界条件的量子力学系统的半经典分析模型，我们构造了从不可穿透边界倾斜的线性势的WKB传播子。首先，我们找到了时间上从点$y$到点$x$的所有经典路径$t$，并计算了相应的作用函数和振幅函数。很大一部分时空是经典不可达的，这个区域的边界是一种不寻常的焦散线，在那里振幅消失而不是发散。我们表明，这条曲线是焦散线在通常意义上的反射边界时，由陡峭上升的光滑电位近似的极限。然后，为了改进WKB近似，我们在动量空间中构造初始数据的传播子;这需要对具有在时间$t$之后到达$x$的初始动量$p$的各种有趣的经典路径进行分类。这两个近似的传播子进行比较，通过将它们应用到高斯初始数据包的数值积分，结果显示物理预期的行为，与基于动量的传播子在经典禁止制度（大$t$）的优势。
英文摘要：
As a model for the semiclassical analysis of quantum-mechanical systems with both potentials and boundary conditions, we construct the WKB propagator for a linear potential sloping away from an impenetrable boundary. First, we find all classical paths from point $y$ to point $x$ in time $t$ and calculate the corresponding action and amplitude functions. A large part of space-time turns out to be classically inaccessible, and the boundary of this region is a caustic of an unusual type, where the amplitude vanishes instead of diverging. We show that this curve is the limit of caustics in the usual sense when the reflecting boundary is approximated by steeply rising smooth potentials. Then, to improve the WKB approximation we construct the propagator for initial data in momentum space; this requires classifying the interesting variety of classical paths with initial momentum $p$ arriving at $x$ after time $t$. The two approximate propagators are compared by applying them to Gaussian initial packets by numerical integration; the results show physically expected behavior, with advantages to the momentum-based propagator in the classically forbidden regime (large $t$).
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