关键词: δ函数, 波函数, 数值解

Abstract: The delta potential well is an important idealized model in quantum mechanics, which has significant applications in understanding quantum tunneling phenomena and describing the scattering of electrons by impurities This article selects four common approximate expressions of the delta function in literature. Numerical methods are used to solve the Schrdinger equation corresponding to different function forms in the delta potential well. The numerical results are compared with the theoretical results in terms of wave function, energy eigenvalues, and wave function derivatives Research has found that among the four common approximate expressions of the δ function in literature, the function δ(x)=limk→∞kπe-(kx)2 can better replace the δ function for numerical operations. The wave function, energy eigenvalues, and wave function derivatives obtained from numerical calculations are very close to theoretical results, and the accuracy of the numerical solution corresponding to the δ function potential well approximation expression increases with the increase of parameter k. This proves that the method of numerically solving the Schrdinger equation under the δ potential well by constructing an approximation function is feasible.

Key words: δ function, wave function, numerical solution