Physical Observables and Phase Variables

At the beginning of any trajectory one must specify either the coordinates and the velocities [ 7o 4o] or the coordinates and the momenta [qo. Pol- t is physically desirable that the initial conditions describe an observable situation of the reactants. 

We denote by G the set of all the experimentally observable quantities (called physical observables) which must be reproduced. Such quantities are, for instance, the collision energy, the quantum numbers defining the intramolecular state (vibrations and the principal quantum number of rotation), the total angular momentum etc... However, there are other dynamical variables which have a clear meaning in Classical Mechanics but correspond to no physical observable because of the Uncertainty Principle. We call them phase variables and denote them globally by g. The phase variables must be given particular values to obtain, at given G, a particular trajectory. Such variables are, for instance, the various intramolecular normal vibrational phases, the intermolecular orientation, the secondary rotation quantum numbers, the impact parameter, etc... Thus we look for relationships of the type qo = qq (G, g) and either qo = qo (G, g) or po = Po (G, g) 

To reproduce the results of a given experiment whose theoretical specification is G we must obtain a set of trajectories for various g. Then the theoretical results (the energy transfer, the reaction probability, etc.. .) are obtained as averages overg. Thus an observable quantity F, observable at the end of the reaction, is calculated for given G as 

N is the total number of trajectories. Here g,- is implicitly selected pseudo random-jy251,252) 

Sometimes an additional average over G is required to obtain the theoretical value of a macroscopic experimental quantity. Then one uses a normalized distribution function i (G) of all the components of G, so that in application of the basic principles of Statistical Mechanics F = fi P(G)F(G)dG where F denotes the subspace of all the possible values of G under the given experimental conditions. 

The integral is most often transformed into a finite sum. If the selected values Gj of G are regularly distributed over F, then  

These expressions represent the most general theoretical results for whatever macroscopic experimental quantity F. 

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