Underlying non‐Hermitian character of the Born rule in open quantum systems

Abstract

The absolute value squared of the probability amplitude corresponding to the overlap of an initial state with a continuum wave solution to the Schrödinger equation of the problem, has the physical interpretation provided by the Born rule. Here, it is shown that for an open quantum system, the above probability may be written in an exact analytical fashion as an expansion in terms of the non‐Hermitian resonance (quasinormal) states and complex poles to the problem which provides an underlying non‐Hermitian character of the Born rule.

Heisenberg uncertainty relations for the non-Hermitian resonance-state solutions to the Schrödinger equation

“…Since resonance states correspond to non-Hermitian solutions to the Schrödinger equation and therefore lie outside the standard formalism of quantum mechanics, the above results have prompt us to explore some fundamental issues concerning these states. One of these refers to the Born rule [27] and the one discussed here concerns the Heisenberg uncertainty relations. This work, therefore, explores the Heisenberg uncertainty relations using resonance states.…”

“…Since the 1990's up to the present time, one may find in the literature an increasing number of works dealing with distinct aspects of these states, as discussed, for example in Refs. [20][21][22][23][24][25][26][27] and references therein. It is worth mentioning the generalization of the phenomenon of diffraction in time, first discussed by Moshinsky [28], to potentials of finite range by García-Calderón and Rubio [29].…”