Abstract:

Using quantum theory derivation and numerical calculation methods，the diagram and model

of one- dimensional potential box function are firstly obtained. Then the wave function，energy

level and probability density of N-dimensional potential box functions in quantum mechanics are

studied comprehensively and systematically. Fi- nally，all characteristics of the potential box

functions are simulated using MATLAB software. We find that the ener- gy of particles is quantized

and discontinuous in N-dimensional potential box. The quantum number n cannot be ze- ro，and the

higher the quantum number n is，the higher the energy level is，while the higher the mass

m is，the lower the energy level is. Generally，the larger one-dimensional potential box length a

is，the lower the correspond- ing energy level is. The number of nodes is N-1，and the more

the nodes are，the shorter the wavelength is，the higher the frequency is，the higher the energy

level is. For two-dimensional potential box function，the number of peaks is nₓ ×ny ，and the

intersection numbers of wave function and a plane of Ψ = 0 is nₓ ×ny ，too. For degeneracy， in

general，the degeneracy of particles in the two-dimensional potential box model is uncertain. But

for the two di- mensional square potential box function model，the energy level degeneracy of the

particles in the box can be divid- ed into general and special cases. The degeneracy of

three-dimensional potential box function are nx +ny +nz . The vi- sualization results are in

complete agreement with the theoretical results. Finally，the color function of MATLAB

software is firstly used to realize the four-dimensional representation，and the four-dimensional

slice diagram of the three-dimensional potential box function are obtained. The results of

this visualization are in complete agreement with the theoretical results，and it is also of

great significance for the understanding of the abstract concept.

Key words:  N - dimensional potential box function, quantum theory, wave function, probability density, energy level, simulation