I study integrable systems. I am especially interested in the Toda lattice. Although the Toda lattice is an infinite dimensional integrable system, essentially we can reduce to the finite dimensional case. There exist n conservation laws of the Toda lattice. We call the algebraic variety defined by the n equations “conservation quantity = const” an iso-level set of the Toda lattice. The fact that the iso-level set of the Toda lattice is homeomorphic to the flag variety is already known. I give another proof of this fact in a more differential geometric way. I consider the fiber bundle of eigenvectors of the Lax operator. I apply these results to the study of quantum cohomology. We define the quantized Lax operator of the Toda lattice. In analogy with the classical case, we consider eigenvectors of the quantized Lax operator. The quantum analogy of the 1-st Chern classes are no longer mere scalars but solutions of certain differential equations. I will study the quantum cohomology from the point of view of analysis.