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Uncertainty Relations in the Madelung Picture Including a Dissipative Environment
Dieter Schuch1, Moise Bonilla-Licea2
1Institute of Theoretical Physics, Goethe-University Frankfurt, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany.
Entropy (Basel, Switzerland)
|February 25, 2023
Summary
This study explores quantum mechanics with a dissipative environment, revealing how nonlinearities impact uncertainty dynamics. Generalized coherent states illustrate changes in quantum energy and uncertainty products, linking to thermodynamics.
Area of Science:
- Quantum Mechanics
- Quantum Thermodynamics
Background:
- Madelung's hydrodynamic formulation relates quantum uncertainties to wave function phase and amplitude.
- Previous work established this relationship in a non-dissipative context.
Purpose of the Study:
- To investigate the impact of a dissipative environment on quantum uncertainty dynamics.
- To analyze changes in quantum mechanical energy and uncertainty products due to environmental interactions.
- To explore connections between quantum properties and environmental thermodynamics.
Main Methods:
- Incorporation of a dissipative environment using a nonlinear modified Schrödinger equation.
- Description of environmental effects via a complex logarithmic nonlinearity.
- Illustration using generalized coherent states.
Main Results:
- The nonlinear term introduces significant changes to uncertainty dynamics.
- Quantum mechanical energy and uncertainty products are affected by the dissipative environment.
- Connections are established between quantum mechanical contributions and thermodynamic properties.
Conclusions:
- Dissipative environments alter quantum uncertainty dynamics in Madelung's formulation.
- The nonlinear modified Schrödinger equation provides a framework for studying these effects.
- This research bridges quantum mechanics and thermodynamics through environmental interactions.
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