<p>This paper presents a numerical method for solving Schrödinger equations with arbitrary potentials with the adoption of the quantum diagonalization scheme and basis functions in the discrete wavelet transform. The quantum diagonalization scheme, developed to solve the Hubbard model and so on, is known to allow the diagonalization of a Hamiltonian matrix of a much large size, because it does not require to maintain the entire matrix in memory. And, the adoption of wavelet basis functions enables representing eigenstates efficiently using a number of bases much smaller than the number of grid points to integral and reducing the computation time to perform the diagonalization process. With the help of another additional techniques, the method allows to find solutions of a Schrödinger equation much exactly and efficiently. The validity of the method and its efficiency depending on several conditions are proven and surveyed by applying it to several problems for which the exact solutions are known.</p>