Abstract: Constructing a unitary matrix for transformation of coordinates by means of quadratic form theory, the Hamiltonian of the 3D coordinate-momentum coupling harmonic oscillator is transformed into diagonalmatrix, it not only offer a general mathematic method for solving this kind of problems, but also have an active effect upon upgrading students＇ ability to solve physics problems by mathematical theory.

Key words: quadratic form, 3D quantum harmonic oscillator with coordinate-momentum coupling, unitary matrix, diagonalizing