  In this paper we provide a novel and general way to construct the result of
the action of any bosonic or fermionic operator represented in second quantized
form on a state vector, without resorting to the matrix representation of
operators and even to its elements. The new approach is based on our proposal
to compactly enumerate the configurations (i.e., determinants for fermions,
permanents for bosons) which are the elements of the state vector. This
extremely simplifies the calculation of the action of an operator on a state
vector. The computations of statical properties and of the evolution dynamics
of a system become much more efficient and applications to systems made of more
particles become feasible. Explicit formulations are given for spin-polarized
fermionic systems and spinless bosonic systems, as well as to general
(two-component) fermionic systems, two-component bosonic systems, and mixtures
thereof.
