Modelling one-dimensional crystal by using harmonic oscillator potential

Our recent developed filter method (Phys. Rev E 96(3), 033302, 2017) is applied here to investigate the energy spectrum and their corresponding wave function of one dimensional crystal. The periodic one dimensional potential is modelled by using one dimensional periodic harmonic oscillator, with variation on oscillator potential depth, quasi-potential depth, and crystal width. For energy less than the potential depth of the oscillator, the computational results reveal that the periodic harmonic oscillator produces a discrete spectrum, as the energy spectrum of a single harmonic potential. However, for energy almost equal to or greater than the depth of the potential oscillator, the periodic harmonic oscillator demonstrates the existence of pattern similar to energy band in crystal.