Quantum dynamics of the free particles

A. Mauger and N. Pother, Aging effects in the quantum dynamics of a dissipative free particle Non-Ohmic case. Phys. Rev. E 65, 056107 (2002). 

Similar considerations may be applied to free transport of protons (cf. Fig.5.11). For dilute solutions of protons in an oxide essentially all nearest neighbour oxygen ions are available, and thus in this case Nd is unity. However, the specification of Z, s and co is not straightforward in this case. The dynamics of free proton diffusion in oxides are complicated by 1) the multistep process (jump+rotation), 2) the dependency on the dynamics of the oxygen ion sublattice, and 3) the quantum mechanical behaviour of a light particle such as the proton. 

Postulate II For every dynamical variable, there is an associated mathematical operation. Furthermore, if there is an eigenvalue for such an operation and a particular wavefimc-tion, that eigenvalue is the result that would be obtained from measuring that dynamical variable for the system having this particular wavefunction. This explains, in part, how we can glean a picture of a quantum mechanical system from its wavefunction. An example, so far, is that the operation associated with the square of the momentum has an eigenvalue of Qi/ky for the free particle wavefunction, A(x). We extract (h/Kf from the wavefunction by applying the associated operator. 

We use the term molecule in its original narrower meaning only for a structure of atoms that does not possess free valencies. In the spectroscopic and quantum-theoretical systems this boundary has become somewhat smudged. There we are normally interested in the total term system of a structure which is characterised in terms of its components by electrons and nuclei. In the system of the dynamic interactions it is important to define the molecule in such a fashion that it does not merely represent a collection of particles, but also has a dynamic aspect. 

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