Time-dependent momentum expectation values from different quantum probability and flux densities

Area of Science:

• Quantum mechanics
• Theoretical physics
• Computational chemistry

Background:

• The Ehrenfest theorem provides a link between classical and quantum mechanics.
• Time-dependent expectation values are crucial for understanding quantum dynamics.
• Quantum mechanical flux density (j̲) is related to probability density (ρ) via the continuity equation.

Purpose of the Study:

• To explore two equivalent methods for evaluating the time-dependent expectation value of a momentum operator based on the Ehrenfest theorem.
• To compare the properties and coordinate/time-dependence of the quantum mechanical flux density (j̲) and a related quantity (j̲̃).
• To investigate the physical interpretations and information content of these different flux densities and their associated densities (ρ̃).

Main Methods:

• Application of the Ehrenfest theorem to derive expressions for the time-dependent expectation value of the momentum operator.
• Utilizing a model for the coupled dynamics of an electron and a proton.
• Analysis of the coordinate and time-dependence of the quantum mechanical flux density (j̲) and the alternative flux density (j̲̃).
• Examination of the properties of the associated densities (ρ̃) and their relation to wave packet dispersion.

Main Results:

• Two distinct but equivalent methods for calculating the mean momentum were identified, involving different integrand functions (j̲ and j̲̃).
• Despite yielding identical mean momentum, the functions j̲ and j̲̃ exhibit significantly different behaviors in terms of coordinate and time-dependence.
• The alternative flux density (j̲̃) was found to directly reflect temporal changes in probability density (ρ).
• The derived density (ρ̃) was shown to contain information about wave packet dispersion in different spatial directions.

Conclusions:

• The study highlights that while the mean momentum is invariant, the underlying flux densities and densities provide complementary insights into quantum system dynamics.
• The flux density j̲̃ and density ρ̃ offer unique perspectives on temporal evolution and spatial dispersion, respectively.
• These findings contribute to a deeper understanding of quantum mechanical descriptions of particle dynamics.