Quantum object expectation values

Optimal control theory A method for determining the optimum laser field used to maximize a desired product of a chemical reaction. The optimum field is derived by maximizing the objective function, which is the sum of the expectation value of the target operator at a given time and the cost penalty function for the laser field, under the constraint that quantum states of the reactants satisfy the Schrodinger equation. 

Statistical mechanics is the branch of physical science that studies properties of macroscopic systems from the microscopic starting point. For definiteness we focus on the dynamics ofan A-particle system as our underlying microscopic description. In classical mechanics the set of coordinates and momenta, (r, p ) represents a state of the system, and the microscopic representation of observables is provided by the dynamical variables, v4(r, p, Z). The equivalent quantum mechanical objects are the quantum state [/ ofthe system and the associated expectation value Aj = of the operator that corresponds to the classical variable A. The corresponding observables can be thought of as time averages... 

That is, suppose a quantum object A and the expectation value of a given property for this object (7tA), which can be associated with the experimental value of the given property of the object. The following approximate relationship can be found [59] ... 

The origin of the QQSPR equations is simply the quantum mechanical statistical calculation of observable expectation values see for instance Ref. [11] for more information. Indeed, as quantum mechanics admits, knowing the DF tag for a given molecular quantum object P Pp r), the observables, associated to some Hermitian operator Q(r) of such submi-croscopic object can be formally obtained as the expectation values ... 

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