  Quantum statistics is defined by Hilbert space products between the
eigenstates associated with state preparation and measurement. The same Hilbert
space products also describe the dynamics generated by a Hamiltonian when one
of the states is an eigenstate of energy E and the other represents an
observable B. In this paper, we investigate this relation between the
observable time evolution of quantum systems and the coherence of Hilbert space
products in detail. It is shown that the times of arrival for a specific value
of B observed with states that have finite energy uncertainties can be used to
derive the Hilbert space product between eigenstates of energy E and
eigenstates of the dynamical variable B. In these Hilbert space products,
quantum phases and interference effects appear in the form of an action that
relates energy to time in the experimentally observable dynamics of localized
states. Quantum effects emerge in the measurement statistics when the precise
control of energy in quantum state preparation results in a coherent
randomization of the dynamics, such that two different arrival times contribute
to the quantum statistics of the same measurement outcome B. The non-classical
features associated with quantum interference can thus be explained as a
consequence of quantum dynamics and its role in state preparation and
measurement, indicating that the apparent randomness of control described by
the energy-time uncertainties is not merely a technical problem but rather
originates from the fundamental nature of interactions between physical
systems.
