Abstract: By constructing the supersymmetric system which Hamiltonian commutates with that of harmonic os-cillator, the properties of harmonic oscillator can be obtained through the investigation of supersymmetric system. Based on the property of supersymmetric system, the recursion relation between the expanding parameter in the ei-genfunction of harmonic oscillator is obtained without using the theory of Hermite polynomials, then the eigenfunc-tion can be directly found from the recursion relation. The normalized eigenfunction obtained is actually the same as that represented through Hermite polynomials, and the parity of the eigenfunction can be evidently exhibited.

Key words: quantum harmonic oscillator, Schrtidinger equation, eigenfunction, supersymmetric system