Microscopic expression for heat in the adiabatic basis.

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Thermodynamics

Background:

• Understanding the dynamics of quantum systems under time-dependent perturbations is crucial.
• The behavior of the density matrix in non-equilibrium conditions requires a microscopic description.
• Thermodynamic quantities like heat generation need to be precisely defined in quantum systems.

Purpose of the Study:

• To derive a microscopic expression for the instantaneous diagonal elements of the density matrix in an adiabatic basis.
• To formulate a microscopic expression for heat generated by transitions between instantaneous energy levels.
• To investigate the non-negativity of heat for passive initial states.

Main Methods:

• Derivation of the density matrix elements using the evolution operator in the adiabatic basis.
• Formulation of the heat definition based on energy level transitions.
• Analysis of the derived expressions for stationary and passive initial density matrices.

Main Results:

• A microscopic expression for instantaneous diagonal density matrix elements rho(nn)(t) was obtained.
• The expression for heat generation was derived, showing it's non-negative for passive initial states.
• The results confirm basic thermodynamic expectations and offer a method for adiabatic expansion.

Conclusions:

• The study provides a microscopic framework for analyzing time-dependent quantum systems.
• The derived heat expression is consistent with thermodynamic principles, particularly for passive states.
• The findings facilitate systematic expansions of observables around the adiabatic limit.