The convergence characteristics of Multi-scale Quantum Harmonic Oscillator Algorithm (MQHOA) prove that single scale convergence process cannot simultaneously get global search accuracy and local search accuracy.Only by multi-scale iteration can we gradually get the accurate position of the global optimum solution.MQHOA solves the optimization problem by two nested convergence processes:Quantum Harmonic Oscillator convergence process (QHO process) and Multi-scale convergence process (M process).QHO process shrinks the searching areas by the manner harmonic oscillator's wave function moving from high-energy state to low-energy state.M process shrinks the search areas by half cutting to improve searching precision.The wave function convergence theorem proves that sampling distribution is Gauss distribution when QHO process is convergent.By the wave function diagram in different energy level and scale

we can track the algorithm iterative process explicitly.The experiments demonstrate the shape of ground-state wave function

the existence of zero-point energy on the ground state

all of which exactly match the physical model of MQHOA.