Let K be a field. In this paper, we proved that if ch K =0 ,then the n-thWeyl algebra Aｎ(K) has no finite dimensional representation . And the classification of irreducible (and indecomposable) Harish-Chandra modules over the n-th Weyl algebra Aｎ(K) is given. If K is all algebraically closed field of characteristic ≠0, then the classification of finite dimensional irreducible Aｎ(K)-modules is got.
