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Uncertainty Relations in the Madelung Picture
Moise Bonilla-Licea1, Dieter Schuch2
1Departamento de Física y Matemáticas, Universidad Iberoamericana, Prolongación Paseo de la Reforma 880, Mexico City 01219, Mexico.
Entropy (Basel, Switzerland)
|January 21, 2022
Summary
This study reveals how quantum uncertainties in position and momentum are linked to the wave function's phase and amplitude. It highlights the crucial role of phase-dependent position-momentum correlations in quantum mechanics.
Area of Science:
- Quantum Mechanics
- Mathematical Physics
Background:
- The complex Schrödinger equation governs quantum systems.
- Madelung's transformation separates it into coupled real equations for wave function amplitude and phase.
- These equations resemble classical hydrodynamics but retain quantum information.
Purpose of the Study:
- To elucidate the connection between quantum mechanical uncertainties and the wave function's phase and amplitude.
- To demonstrate how position-momentum correlations influence these uncertainties.
- To illustrate these relationships using generalized coherent states.
Main Methods:
- Reformulation of the complex Schrödinger equation into coupled real equations for amplitude and phase.
- Analysis of quantum mechanical uncertainties (position and momentum) in terms of these real functions.
- Application of generalized coherent states for explicit illustration.
Main Results:
- Quantum uncertainties are directly related to the wave function's phase and amplitude.
- The phase component significantly contributes to uncertainties, particularly through position-momentum correlations.
- Generalized coherent states provide a clear framework for observing these effects.
Conclusions:
- The phase of the wave function is critical for understanding quantum uncertainties.
- Position-momentum correlations, mediated by the phase, are fundamental to quantum mechanical uncertainty relations.
- This hydrodynamic-like formulation offers insights into the nature of quantum uncertainty.
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