The Ehrenfest Theorem

In extension of Eq. (3.46) we can then express the first-order correction to a field-dependent expectation value of the general but spin-free operator O with the first-order 

In Section 3.3 we looked at the dependence of an expectation value on a perturbing field P and expanded the expectation value in powers of this perturbation. In this section, we want to study now the time evolution of an expectation value of an arbitrary operator P. Finally, in the section on time-dependent response theory, Section 3.11, we will combine both and study the effects of a time-dependent perturbation Pa. .if)-Let us study the time dependence of an expectation value by deriving an expression for the time derivative of an expectation value, i.e. an equation of motion for the expectation value of the operator P 

The time derivative of the wavefunction is given by the time-dependent electronic 

The second term vanishes, if the operator, P, itself is independent of time, as in all cases we consider here. We arrive thus at the Ehrenfest theorem 

One should note that the Ehrenfest theorem is derived solely by apphcation of the time-dependent Schrodinger equation. It contains therefore the same information and is often applied as an alternative to the time-dependent Schrodinger equation, when 

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The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Classical dynamics is studied as a special case by analyzing the Ehrenfest theorem, coherent states (16) and systems with quasi classical dynamics like the rigid rotor for molecules (17) and the oscillator (18) for various particle systems and for EM field in a laser. 

Derive both of the Ehrenfest theorems using equation (3.72). 

We derive the response functions using the Ehrenfest theorem for a time-independent one-electron operator Q... 

The Kn(t) parameters are obtained through the Ehrenfest theorem which can be written as [23]... 

Collecting all first-order terms obtained through the BCH expansion of the Ehrenfest theorem given in Eq. (74) leads to a system of differential equations... 

The derivation of cubic response is analogous to what we have seen so far. We use the BCH expansion of the Ehrenfest theorem but now collect terms that are third order in the perturbation. The resulting matrix equation is... 

We have resorted to an approximate technique which attempts to include the above mentioned main quantum effects via the construction of effective potentials V. Basically, each pmticle is represented by a single particle wavefunction tmd the Ehrenfest theorem is applied. Similar ideas have been used with good success ev( n for quantum solids like hydrogmi [38]. Effective quantum potentials ajx also among the results of the Feynman-Hibbs treatment [12] which have been apjjlied to pure neon clusters in the past [34]. 

Overall, the Ehrenfest theorem shows that quantum description is compatible with classical mechanics, under the expectation values of its main operators, i.e., the space, momentum andeneigy (Hamiltonian). Moreover, it says us that what we can know fi om quantum mechanical description of... 

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. 

What there was actually proofed is that the quadratic chemical reactivity equations for total energy in both finite and differential fashions may be derived employing the Kohn-Sham (as Schrodinger reminiscence) equation for chemical potential eigenvalue combined with the chemical quantum version of the Ehrenfest theorem involving the force concept and its active-reactive peculiar property for the chemical potential and... 

A second very useful theorem can be derived from the Ehrenfest theorem, if one considers the wavefunctions of two different stationary states and the unperturbed Hamiltonian instead of a general time-dependent wavefunction (Chen, 1964)... 

Contrary to response theory for exact states, in Section 3.11, or for coupled cluster wavefunctions, in Section 11.4, in MCSCF response theory the time dependence of the wavefunction is not determined directly from the time-dependent Schrodinger equation in the presence of the perturbation H t), Eq. (3.74). Instead, one applies the Ehrenfest theorem, Eq. (3.58), to the operators, which determine the time dependence of the MCSCF wavefunction, i.e. the operators hj ... 

Exercise 11.5 Derive the first-order equation Eq. (11.41) from the Ehrenfest theorem... 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

Exercise 11.7 Derive the SOPPA Hessian and overlap matrices using the Ehrenfest theorem Eq. (11.55). 

The general case was already known to Ehrenfest who showed in 1927 that the mean position and momentum of a wave function strictly follow what look like classical equations of motion, Ax t)/At = p(t), Ap(t)/At = —AV (x)/Ax. The quantal nature of the dynamics enters in the equation of motion for the momentum. For a classical trajectory x(t), pit) the expected result is A p(t)/At = —AV(x)/Ax. For a harmonic potential, where V(x) = lo , the two results are the same but, in general, the average (over the wave function) of the force, —AV x)/Ax, is not equal to the force at the average position. The Ehrenfest theorem is powerful but can be potentially misleading it is valid, say, if the wave function bifurcates into two components. Under such circumstances, the system is very much not at its average position. 

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