In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in the present part II and the forthcoming part III of this work that there is a common structure behind these two different concepts. If
R is a von Neumann algebra, a selfadjoint element
A∈R induces a continuous function
fA:Q(P(R))→R defined on the \emph{Stone spectrum}
Q(P(R)) (\cite{deg3}) of the lattice
P(R) of projections in
R.
fA is called the observable function corresponding to
A. The aim of this part is to study observable functions and its various characterizations.
Abstract The effects of discontinuously time‐varying perturbations on the dynamics of a particle moving in harmonic, symmetric double well and symmetric triple well potentials, are investigated both classically and quantum mechanically. The quantum dynamics is followed using the time‐dependent Fourier grid Hamiltonian (TDFGH) method while the classical dynamics is analyzed within the framework of classical Hamiltonian mechanics. Depending on the spatial symmetry of the perturbation and the characteristic features of the reversal time , different types of ‘phase space’ structures are observed in each of the potentials. For symmetric double and triple well potentials, quantum dynamics reveals that complete destruction of tunnelling (CDT) can be achieved in the presence of a time‐dependent spatially asymmetric perturbing field that is continuous in time. Any discontinuity in time‐variation of the perturbation may induce over the barrier transition. The relevance of these results in the context of (i) tunnelling control and (ii) quantum computing with 3‐state or 2‐state quantum registers is briefly discussed.