The varying-mass Schr\"odinger equation (VMSE) has been successfully applied to model electronic properties of semiconductor hetero-structures, for example, quantum dots and quantum wells. In this paper, we consider VMSE with small random heterogeneities, and derive a radiative transfer equation as its asymptotic limit. The main tool is to systematically apply the Wigner transform in the classical regime when the rescaled Planck constant $\epsilon\ll 1$, and expand the Wigner equation to proper orders of $\epsilon$. As a proof of concept, we numerically compute both VMSE and its limiting radiative transfer equation, and show that their solutions agree well in the classical regime.
翻译:不同质量的Schr\'odinger 方程式( VMSE) 已经成功地应用到半导体异体结构的电子特性模型中, 比如量点和量子井。 在本文中,我们把VMSE视为小的随机异质,并得出一个辐射传输方程式作为它的无症状极限。 主要工具是系统应用传统体系中的维格变异,当重新标定的Planck 常数$\epsilon\ll 1$时,并将维格方程式扩大到适当的 $\ epsilon$。 作为概念的证明,我们从数字上计算VMSE及其限制的辐射转移方程式,并表明它们的解决方案在传统体系中是完全一致的。