  A gauge-invariant quantum theory of the Friedmann-Robertson-Walker (FRW)
universe with dust is studied in terms of the Ashtekar variables. We use the
reduced phase space quantization which has following advantages: (i)
fundamental variables are all gauge invariant, (ii) there exists a physical
time evolution of gauge-invariant quantities, so that the problem of time is
absent and (iii) the reduced phase space can be quantized in the same manner as
in ordinary quantum mechanics. In the FRW model, the dynamical components of
the Ashtekar variables are given by a single quantity $p$ and its conjugate
momentum, where $p$ is related to the scale factor $a$ as $a\propto \sqrt{|p|}$
and its sign gives the orientation of triads. We solve a scattering problem in
terms of ingoing and outgoing energy eigenstates. We show that the incident
wave is reflected in rate $1/4$ and transmitted in rate $3/4$ at the classical
singularity $p=0$. Analyzing the dynamics of a wave packet, we show that the
classical initial singularity is replaced by a big bounce in quantum theory. A
possible interpretation of the result is that the wave function of the universe
has been in a superposition of states representing right-handed and left-handed
systems before the big bounce.
