共振态的概率解释
Probabilistic Interpretation of Resonant States
摘要：
我们提供共振态的概率解释。这是通过证明共振波函数的模平方的积分（即，常规范数）与时间无关，因此导致概率守恒。这与排除概率解释的双正交基的常规使用相反，因为共振状态的波函数在空间中呈指数发散。另一方面，共振态随时间呈指数衰减，因为动量从中心散射区域泄漏出来。这种动量泄漏也是共振态空间指数发散的原因。正是通过结合共振态的相反的时间和空间行为，我们得出了这些状态的概率解释。在扩展的空间域上规范化共振波函数的物理需要出现，因为粒子泄漏出包含势范围的区域并逃逸到无穷大，并且必须将它们包括在粒子数的总计数中。
英文摘要：
We provide probabilistic interpretation of resonant states. This we do by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviors of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particle number.
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