Ehrenfest’s theorem

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Fig. 3.2. Two-dimensional potential energy surface V(R, 7) (dashed contours) for the photodissociation of C1CN, calculated by Waite and Dunlap (1986) the energies are given in eV. The closed contours represent the total dissociation wavefunction tot R,l E) defined in analogy to (2.70) in Section 2.5 for the vibrational problem. The energy in the excited state is Ef = 2.133 eV. The heavy arrow illustrates a classical trajectory starting at the maximum of the wavefunction and having the same total energy as in the quantum mechanical calculation. The remarkable coincidence of the trajectory with the center of the wavefunction elucidates Ehrenfest s theorem (Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Reprinted from Schinke (1990).

Nevertheless, this simple propagation method provides an intriguing picture of the evolution of the quantum mechanical wavepacket, at least for short times. It readily demonstrates that for short times the center of the wavepacket follows essentially a classical trajectory ( Ehrenfest s theorem, Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Figure 4.2 depicts an example the evolution of the two-dimensional wavepacket follows very closely the classical trajectory that starts initially with zero momenta at the Franck-Condon point. 

Classical trajectories are the backbones for the quantum mechanical wavefunctions ( Ehrenfest s theorem). If the dissociation is direct, a single trajectory, which starts near the equilibrium of the parent molecule, illustrates in a clear way the overall fragmentation mechanism. 

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. 

This is the correct form of Ehrenfest s theorem with damping appearing only for the motion as we should expect. 

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. 

For the time evolution of the density, we utilize Ehrenfest s theorem... 

Important for molecular science is how the many-electron probability density and current density can be defined. Following the lines of reasoning introduced in chapter 4, we derive the density and current-density expressions for the many-electron system according to Ehrenfest s theorem employing the many-electron Dirac-Coulomb-(Breit) Hamiltonian,... 

Figure 4 shows two typical trajectories which trace the quantum mechanical wave-packet motion. They are started on the symmetric stretch line with different initial momenta pointing into the exit channels. The two trajectories represent dissociation into H + OD and D + OH, respectively. What was found for the quantum dynamics is more clearly demonstrated by the classical trjectories the H + OD dissociation is faster and the oscillations of both trajectories around the minimum energy path clearly shows the vibrational excitation of the fragments. Indeed, if we compute the time-evolution of the bondlength expectation values th d) from the bifurcated packets it is found that they resemble closely the classical trajectories (as can be expected from Ehrenfest s theorem). The wave-packet motion shows that the dissociation proceeds as can be anticipated classically. 

Another rigorous constraint follows from Ehrenfest s theorem which relates the acceleration to the gradient of the external potential... 

In the same way we can write Ehrenfest s theorem for the Kohn-Sham system... 

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