Abstract: We present a simple scheme to calculate the energy eigenvalues of particle bounded in the polynomial potential by using the linear variational method. The wave functions of the harmonic oscillator (HO)including one parameter are chosen as the basis functions，and the general formula of the Hamiltonian matrix element (HME) of coordinate operator for the HO is utilized. The algebraic expression of the system’s HME is derived. The HO parameter is determined from the rule that the trace of the Hamiltonian matrix takes the minimum value relative to this parameter.

Key words: variational method, basis function, Hamiltonian matrix