  We study the Laughlin wave function on the cylinder. We find it only
describes an incompressible fluid when the two lengths of the cylinder are
comparable. As the radius is made smaller at fixed area, we observe a
continuous transition to the charge density wave Tao-Thouless state. We also
present some exact properties of the wave function in its polynomial form. We
then study the edge excitations of the quantum Hall incompressible fluid
modeled by the Laughlin wave function. The exponent describing the fluctuation
of the edge predicted by recent theories is shown to be identical with
numerical calculations. In particular, for $\nu=1/3$, we obtain the occupation
amplitudes of edge state $n(k)$ for 4-10 electron size systems. When plotted as
a function of the scaled wave vector they become essentially free of
finite-size effects. The resulting curve obtains a very good agreement with the
appropriate infinite-size Calogero-Sutherland model occupation numbers.
Finally, we numerically obtain $n(k)$ of the edge excitations for some pairing
states which may be relevant to the $\nu=5/2$ incompressible Hall state.
