δ势阱是量子力学中一个重要的理想化模型，在理解量子隧穿现象和描述杂质对电子的散射等方面有着重要应用.本文选取了文献中常见的四种δ函数近似表达形式，利用数值方法求解了不同函数形式对应的δ势阱下的薛定谔方程，在波函数、能量本征值和波函数导数等多个方面将数值结果与理论结果进行了对比.研究发现，在文献中常见的四种δ函数近似表达形式中，δ(x)=limk→∞kπe-(kx)2函数能更好地替代δ函数进行数值运算，数值计算所得波函数、能量本征值和波函数导数都与理论结果非常接近，且δ函数势阱近似表达式对应的数值解的精度随着参数k的增加而提高，证明了通过构造近似函数来数值求解δ势阱下薛定谔方程的方法是可行的.

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.