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We study the Laughlin wave function on the cylinder. We find it onlydescribes an incompressible fluid when the two lengths of the cylinder arecomparable. As the radius is made smaller at fixed area, we observe acontinuous transition to the charge density wave Tao-Thouless state. We alsopresent some exact properties of the wave function in its polynomial form. Wethen study the edge excitations of the quantum Hall incompressible fluidmodeled by the Laughlin wave function. The exponent describing the fluctuationof the edge predicted by recent theories is shown to be identical withnumerical calculations. In particular, for $\nu=1/3$, we obtain the occupationamplitudes of edge state $n(k)$ for 4-10 electron size systems. When plotted asa function of the scaled wave vector they become essentially free offinite-size effects. The resulting curve obtains a very good agreement with theappropriate infinite-size Calogero-Sutherland model occupation numbers.Finally, we numerically obtain $n(k)$ of the edge excitations for some pairingstates which may be relevant to the $\nu=5/2$ incompressible Hall state.Comment: 25 pages revtex, 9 uuencoded figures, submitted separately, also  available from first author. CSULA-94-1

Laughlin state on stretched and squeezed cylinders and edge excitations in the quantum Hall effect