Ehrenfest framework

Within the Ehrenfest framework [12], valid for exact as well as variational approximate methods, either the time-variational principle of Frenkel [11],... 

We describe the linearly damped harmonic quantum oscillator in Heisenberg s interpretation by Onsager s thermodynamic equations. Ehrenfest s theorem is also discussed in this framework. We have also shown that the quantum mechanics of the dissipative processes exponentially decay to classical statistical theory. 

Different pathways of vibrational relaxation of diatoms in thermal collisions with atoms are discussed in the framework of the Ehrenfest adiabatic principle and generalized Landau-Teller model. Since the efficiency of different energy-transfer channels depend very strongly on the value of the Ehrenfest exponent, it is possible to assign, for given collision partners and the heat-bath temperature, the vibrational energy transfer events to VT, VRT or VR processes. 

To conclude, we have seen that for a given wave function and Hamiltonian, the Ehrenfest theorem can be instrumentalized to derive explicit expressions for the density and current-density distributions by rewriting it in such a way that the continuity equation results. We will rely on this option in the relativistic framework in chapters 5, 8, and 12 to define these distributions for relativistic Hamiltonian operators and various approximations of N-particle wave functions. From the derivation, it is obvious that the definition of the current density is determined by the commutator of the Hamiltonian operator with the position operator of a particle. All terms of the Hamiltonian which depend on the momentum operator of the same particle will produce contributions to the current density. In section 5.4.3 we will encounter a case in which the momentum operator is associated with an external vector potential so that an additional term will show up in the commutator. Then, the definition of the current density has to be extended and the additional term can be attributed to an (external-field) induced current density. 

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