  We present a quantized hydrodynamic theory and its applications of
one-dimensional hard-core bosons in a harmonic trap. Quantizing the Hamiltonian
of a trapped hard-core bosons and diagonalize it in terms of the phase and
density fluctuations associated with the Bose field operator. As an
applications, we calculate discrete energy spectrum, dynamic structure factor,
momentum transferred by a two-photon Bragg pulse, single-particle density
matrix, momentum distribution, and two-particle correlation function. The
dynamic structure factor has multiple peaks due to the discrete nature of the
energy spectrum which can be observed by a two-photon Bragg pulse with long
duration. The coherence length at zero temperature is very small due to the
"fermionization" of the system. We also compute the momentum distributions
which has oscillatory behavior at large momentum. The pair distribution
function shows that there are many deep valleys at various relative separations
which implies shell structure due to the Pauli blocking in real space.
