Based on the systematic discussion of normalization and the matrix representation of operator, we use state transition matrix to derive the solutions for stationary and non-stationary Schrödinger equation. Unlike the matrix representation of eigenstate in quantum information, this paper presents a matrix solution theory for superposition state, and provides a theoretical basis for applying the means of control to quantum systems.

Purpose -This purpose of this paper is to present a new approach of solving quantum control problem. Design/methodology/approach -State-transition-matrix (STM) is used to solute quantum control problem. Findings -The paper obtains the matrix representation for wave function, normalization, and some other quantum operators, and then gives the possible methods to establish the STM model and solve it. Originality/value -Based on this, many tools of the widely used control theory can be adopted to denote or analyze the quantum system.