  In this paper, we introduce and study a noncommutative extension of the Gross
Laplacian, called quantum Gross Laplacian. Then, applying the quantum Gross
Laplacian to the particular case where the operator is the multiplication
operator, we find a relation between classical and quantum Gross Laplacian. As
application, we give explicit solution of linear quantum white noise
differential equation. In particular, we give a explicit solution of the
quantum Gross heat equation.
