  For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a
canonical orthonormal basis in which a previously introduced unitary mapping of
H to a Hermitian Hamiltonian h takes a simple form. We use this basis to
construct the observables O of the quantum mechanics based on H. In particular,
we introduce pseudo-Hermitian position and momentum operators and a
pseudo-Hermitian quantization scheme that relates the latter to the ordinary
classical position and momentum observables. These allow us to address the
problem of determining the conserved probability density and the underlying
classical system for pseudo-Hermitian and in particular PT-symmetric quantum
systems. As a concrete example we construct the Hermitian Hamiltonian h, the
physical observables O, the localized states, and the conserved probability
density for the non-Hermitian PT-symmetric square well. We achieve this by
employing an appropriate perturbation scheme. For this system, we conduct a
comprehensive study of both the kinematical and dynamical effects of the
non-Hermiticity of the Hamiltonian on various physical quantities. In
particular, we show that these effects are quantum mechanical in nature and
diminish in the classical limit. Our results provide an objective assessment of
the physical aspects of PT-symmetric quantum mechanics and clarify its
relationship with both the conventional quantum mechanics and the classical
mechanics.
