# A Novel Class of Phase Space Representations for the Exact Population Dynamics of Two-State Quantum Systems and the Relation to Triangle Window Functions

**Authors:** Xiangsong Cheng, Xin He, Jian Liu  
**arXiv:** [2404.04868](https://arxiv.org/abs/2404.04868)  
**Date:** 2024-04-07  
**Summary page:** https://www.summarizepaper.com/fr/arxiv-id/2404.04868v1/

## Abstract

Isomorphism of the two-state system is heuristic in understanding the dynamical or statistical behavior of the simplest yet most quantum system that has no classical counterpart. We use constraint phase space [developed in J. Chem. Phys. 2016, 145, 204105; 2019, 151, 024105 and J. Phys. Chem. Lett. 2021, 12, 2496-2501], non-covariant phase space functions, time-dependent weight functions, and time-dependent normalization factors to construct a novel class of phase space representations of the exact population dynamics of the two-state quantum system. The equations of motion of the trajectory on constraint phase space are isomorphic to the time-dependent Schr\"odinger equation. The contribution of each trajectory to the integral expression for the population dynamics is always positive semi-definite. We also prove that the triangle window function approach, albeit empirically proposed in J. Chem. Phys. 2016, 145, 144108, is related to a special case of the novel class and leads to an isomorphic representation of the exact population dynamics of the two-state quantum system.