运用分数维空间方法理论研究了GaAs/AlGaAs无限深和有限深方形量子阱中激子效应对 三次谐波产生的影响。利用分数维空间模型获得波函数和束缚能级为空间维度的函数，而空间维度数是阱宽的函数。无限深方阱的维度数随着 阱宽的减小从三维极限过渡到二维;而在有限深阱中，当维度数达到一个极值后，维度数随阱宽的减小而增大。采用密度矩阵和迭代法导出三次 谐波的表达式。数值结果表明，考虑激子效应的三次谐波系数比只考虑电子状态的系数增大40%左右，并且三次谐波系数大小依赖于激子的受限 程度。结果还表明在弛豫率较小情况下可以获得较大的三次谐波系数。

Exciton effects on the third-harmonic generation in GaAs/AlGaAs infinite and finite square quantum wells are calculated within the framework of the fractional-dimensional space approach. The wave functions and bound energies are obtained as functions of spatial dimensionality and the dimension is a function of the well width. For an infinite confining potential the dimension (D) has a transition from the 3D limit to 2D limit when the well width decreases. However, in a finite well, when the well width decreases below a given value, the dimension increases. The analytical expression of the third-harmonic generation is described using the compact density method and the iterative procedure. The numerical results show that the third-harmonic generation coefficient with considering exciton effects is 40% greater than the one without considering exciton effects and it is very sensitively dependent on the exciton confinement. In addition, the smaller the relaxation constant is, the larger the third-harmonic generation will be.