Abstract: Based on quantum theory and numerical calculation, we have mainly studied the basic characteristic of n-dimensional linear harmonic oscillator, including wave function, energy, energy levels, degeneracy and probability density, and obtain four-dimensional figures of three-dimensional harmonic oscillator and 3D figures of two-dimensional harmonic oscillator. The results show that the energy of a linear harmonic oscillator is discrete and the energy is quantized. The energy of a harmonic oscillator is evenly distributed, two adjacent energy level spacing is ΔE=ħω. The degeneracy of two-dimensional linear harmonic oscillator is N+1, but the corresponding ground-state wave function has no degeneracy when N=0. The intersection line between wave function and the plane with Ψ = 0 is N. The probability density graph can more intuitively show the number and magnitude of the peak value of probability density, and the number of maximal value of probability density distribution is (n_x+1)(n_y+1). For the three-dimensional linear harmonic oscillator, based on the ground state wave function, within a certain range (−3 ,3), the slices numbers of the x, y and z axis are n_x+1, n_y+1, n_z+1, respectively. The total number of slices is n_x+n_y+n_z+3. In addition, this research has obtained the four-dimensional space slice diagrams of three-dimensional linear harmonic oscillator by MATLAB software. The results of this visualization have established consistence with the theoretical results.