# Weak Form of the Effective Mass Schrodinger Equation

by Morberticus
Tags: effective, form, mass, schrodinger, weak
 P: 81 Hi, I am numerically solving the 2D effective-mass Schrodinger equation $\nabla \cdot (\frac{-\hbar^2}{2} c \nabla \psi) + (U - \epsilon) \psi = 0$ where $c$ is the effective mass matrix $\left( \begin{array}{cc} 1/m^*_x & 1/m^*_{xy} \\ 1/m^*_{yx} & 1/m^*_y \\ \end{array} \right)$ I know that, when the effective mass is isotropic, the weak form is $\int \frac{-\hbar^2}{2m^*}\nabla \psi \cdot \nabla v + U\psi vd\Omega = \int \epsilon \psi vd\Omega$ The matrix is giving me trouble however. Is this the correct form? $\int \frac{-\hbar^2}{2m^*_x}\frac{\partial u}{\partial x}\frac{ \partial v}{\partial x} + \frac{-\hbar^2}{2m^*_{xy}}\frac{\partial u}{\partial x}\frac{ \partial v}{\partial y} + \frac{-\hbar^2}{2m^*_{yx}}\frac{\partial u}{\partial y}\frac{ \partial v}{\partial x} + \frac{-\hbar^2}{2m^*_y}\frac{\partial u}{\partial y}\frac{ \partial v}{\partial y} + U\psi v d\Omega= \int \epsilon \psi v d\Omega$