  Weak measurement is unique in enabling measurements of non-commuting
operators as well as otherwise-undetectable peculiar phenomena predicted by the
Two-State-Vector-Formalism (TSVF). This article, the first in two parts,
explores novel applications of weak measurement. We first revisit the basic
principles of quantum measurement with the aid of the Michelson interferometer.
Weak measurement is then introduced in a simple visualized manner by a specific
choice of the reflecting mirror's position and momentum uncertainties. Having
introduced the method, we proceed to its refinement for a single particle. We
consider a photon going back and forth inside the interferometer, oscillating
between a superposed and a localized state, while subjected to alternating
strong and weak measurements. This cyclic process enables directly measuring
both the photon's position ("which-path") and momentum (interference), without
disturbing either. An alternative explanation of this result, not invoking weak
values, is thoroughly considered and shown to be at odds with the experimental
data. Finally a practical application of this experiment is demonstrated, where
a single photon measures the various transmission coefficients of a multiport
beam-splitter yet remains superposed. This method is then generalized to
measurement of the wave-function itself, performed again on a single particle.
