Abstract: A unitary matrix is constructed by virtue of quadratic form theory for coordinates and momentums transformation. The Hamihonian of the n-D coupling harmonic oscillators is diagonalized by means of the transfor- mation. The exact solution of energy spectrum of n-D coupling quantum harmonic oscillator is given. It also offers a general mathematic method solving problems of this kind.

Key words: quadratic form, n-D coupling harmonic oscillator, unitary matrix, energy eigenvalue