Ehrenfest s equations

Mean field theories of mixed quantum-classical systems are based on approximations that neglect correlations in Ehrenfest s equations of motion for the evolution of the position and momentum operators of the heavy-mass nuclear degrees of freedom. The approximate evolution equations take the form of Newton s equations of motion where the forces that the nuclear degrees of freedom experience involve mean forces determined from the time-evolving wave function of the system. 

This section considers the theoretical background for calculating the molecular properties of a quantum mechanical subsystem exposed to a structured environment and interacting with an externally applied electromagnetic field. The time evolution of the expectation value of any operator A is determined using Ehrenfest s equation ... 

Our next step is to determine the modifications of the response functions due to the interactions between the molecular subsystem and the structured environment, such as an aerosol particle. In order to do so, we consider the time-evolution of expectation values of the time-transformed operators from Eq. (65) and we do this using Ehrenfest s equation... 

In order to determine time-dependent molecular properties utilizing the MCSCF/MM approach it is necessary to consider the time evolution of the appropriate operators and this is done by applying the Ehrenfest s equation for the evolution of an expectation value of an operator, X... 

The next step utilizes the time transformed operators and the Ehrenfest s equation where the object of the game is to establish building blocks for how the time transformed operators are changed due to the external perturbation. 

A different theory of local control has been derived from the viewpoint of global optimization, applied to finite time intervals [58-60]. This approach can also be applied within a classical context, and local control fields from classical dynamics have been used in quantum problems [61]. In parallel, Rabitz and coworkers developed a method termed tracking control, in which Ehrenfest s equations [26] for an observable is used to derive an explicit expression for the electric field that forces the system dynamics to reproduce a predefined temporal evolution of the control observable [62, 63]. In its original form, however, this method can lead to singularities in the fields, a problem circumvented by several extensions to this basic idea [64-68]. Within the context of ground-state vibration, a procedure similar to tracking control has been proposed in Ref. 69. In addition to the examples already mentioned, the different local control schemes have found many applications in molecular physics, like population control [55], wavepacket control [53, 54, 56], control within a dissipative environment [59, 70], and selective vibrational excitation or dissociation [64, 71]. Further examples include isomerization control [58, 60, 72], control of predissociation [73], or enantiomer control [74, 75]. 

Exercise. The following modifications of Ehrenfest s urn model is nonlinear.510 Two urns each contain a mixture of black and white balls. Every second I draw with one hand a ball from one urn and with the other a ball from the other urn, and transfer both. Write the difference equation for the probability pn(t) of having n white balls in the left urn. 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

Applying Ehrenfest s theorem to the nuclear KS equation (68), the classical trajectory... 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

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. 

The similarity of these classical mechanical Hamilton s equations of motion to their quantum mechanical Ehrenfest s Theorem counterpart, (see Section 9.1.7), is an expression of the Correspondence Principle equivalence of a quantum mechanical commutator, [A, B], to a classical mechanical Poisson bracket,... 

If we consider a classical-mechanical particle, its wave function will be large only in a very small region corresponding to its position, and we may then drop the averages in (7.114) to obtain Newton s second law. Hius classical mechanics is a special case of quantum mechanics. Equation (7.114) is known as Ehrenfest s theorem, after the physicist who derived it in 1927. 

Another important result governing the mechanics of an atom in a molecule is obtained from equation 16 when the operator F is set equal to the momentum for an electron. The result in this case is an expression for the force acting on the electrons in an atom, the Ehrenfest force, a force not to be confused with the Hellmann-Feynmann force acting on a nucleus. The expression for the Ehrenfest force is equivalent to having Newton s equation of motion for an atom in a molecule, as it determines all of the mechanical properties of the atom. The force F Q) is determined entirely by the pressure acting on the surface of the atom [Pg.44]

Finally, we consider in this section the force law in quantum mechanics (Ehrenfest, 1927). From Heisenberg s equation of motion employing the Schrodinger Hamiltonian we have... 

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