Probabilistic Interpretation of Resonant States

TL;DR

This paper offers a probabilistic interpretation of resonant states by demonstrating that their wave function norms over expanding regions remain constant in time, reconciling divergence issues with probability conservation.

Contribution

It introduces a method to interpret resonant states probabilistically by integrating wave functions over expanding domains, addressing divergence problems in conventional approaches.

Findings

Integral of resonance wave functions over expanding domains is time-independent.

Resonant states decay exponentially in time due to particle leakage.

The approach reconciles divergence with probability conservation in resonant states.

Abstract

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.