  The nonlocal nature of the Majorana zero modes implies an inherent
teleportation channel and unique transport signatures for Majorana
identification. In this work we make an effort to eliminate some
inconsistencies between the Bogoliubov-de Gennes equation based treatment and
the method using the associated regular fermion number states of vacancy and
occupation within the `second quantization' framework. We first consider a
rather simple `quantum dot--Majorana wire--quantum dot' system, then a more
experimentally relevant setup by replacing the quantum dots with transport
leads. For the latter setup, based on the dynamical evolution of electron-hole
excitations, we develop a single-particle-wavefunction approach to quantum
transport, which renders both the conventional quantum scattering theory and
the steady-state nonequilibrium Green's function formalism as its stationary
limit. Further, we revisit the issue of Majorana tunneling spectroscopy and
consider in particular the two-lead coupling setup. We present comprehensive
discussions with detailed comparisons, and predict a zero-bias-limit
conductance of $e^2/h$ (for symmetric coupling to the leads),which is a half of
the popular result of the zero-bias-peak, or, the so-called Majorana quantized
conductance ($2e^2/h$). The present work may arouse a need to reexamine some
existing studies and the proposed treatment is expected to be involved in
analyzing future experiments in this fast developing field.
