  The fractional quantum Hall effect was experimentally discovered in 1982. It
was observed that the Hall conductivity $\sigma_{yx}$ of a two-dimensional
electron system is quantized, $\sigma_{yx}=e^2/3h$, in the vicinity of the
Landau level filling factor $\nu=1/3$. In 1983, Laughlin proposed a trial
many-body wave function, which he claimed described a ``new state of matter''
-- a homogeneous incompressible liquid with fractionally charged
quasiparticles. Here I develop an exact diagonalization theory that allows
calculation of the energy and other physical properties of the ground and
excited states of a system of $N$ two-dimensional Coulomb interacting electrons
in a strong magnetic field. I analyze the energies, electron densities, and
other physical properties of the systems with $N\le 7$ electrons, continuously
as a function of magnetic field in the range $1/4\lesssim\nu<1$. The results
show that both the ground and excited states of the system resemble a sliding
Wigner crystal, whose parameters are influenced by the magnetic field. Energy
gaps in the many-particle spectra appear and disappear as the magnetic field
changes. I also calculate the physical properties of the $\nu=1/3$ Laughlin
state for $N\le 8$ and show that neither this state nor its fractionally
charged excitations describe the physical reality. The results obtained shed
new light on the nature of the ground and excited states in the fractional
quantum Hall effect.
