Theorem Ehrenfest

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. 

Equivalently, expectation values of three-dimensional dynamical quantities may be evaluated for each dimension and then combined, if appropriate, into vector notation. For example, the two Ehrenfest theorems in three dimensions are... 

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

Doyen G (1993) Tunnel current and generalized Ehrenfest theorem. J Phys Condens Matter 5 3305... 

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... 

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