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Mathematical Models for Unstable Quantum Systems and Gamow States
Manuel Gadella1, Sebastián Fortín2, Juan Pablo Jorge3,4
1Departamento de Física Teórica, Atómica y Optica, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Spain.
Entropy (Basel, Switzerland)
|June 24, 2022
Summary
This study explores quantum resonances in unstable systems using Gamow states within Rigged Hilbert Spaces. It reveals that Gamow states, while describing exponential decay, are not standard pure or mixed states and can explain phenomena like the Loschmidt echo.
Area of Science:
- Quantum Mechanics
- Theoretical Physics
Background:
- Unstable quantum systems are characterized by resonances.
- Gamow states, defined in Rigged Hilbert Spaces, model the exponential decay of resonances.
Purpose of the Study:
- To review definitions and properties of quantum resonances and Gamow states.
- To analyze the nature of Gamow states within algebraic formalisms.
- To explore applications of Gamow states, including the Loschmidt echo.
Main Methods:
- Review of theoretical results on non-relativistic quantum unstable systems.
- Construction and analysis of Gamow states in Rigged Hilbert Spaces.
- Application of algebraic formalism to study states and observables.
Main Results:
- Gamow states represent the purely exponential decaying part of resonances.
- Gamow states are neither pure states nor mixtures in a standard viewpoint.
- Modified time evolution shows non-commuting observables can commute for Gamow states over time.
Conclusions:
- Gamow states offer a framework for understanding quantum resonances and their decay.
- The properties of Gamow states challenge standard quantum state definitions.
- Gamow states provide a potential explanation for the Loschmidt echo phenomenon.
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