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Weak Form of the Effective Mass Schrodinger Equation 
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#1
Apr2914, 04:09 PM

P: 81

Hi,
I am numerically solving the 2D effectivemass Schrodinger equation [itex]\nabla \cdot (\frac{\hbar^2}{2} c \nabla \psi) + (U  \epsilon) \psi = 0[/itex] where [itex]c[/itex] is the effective mass matrix [itex]\left( \begin{array}{cc} 1/m^*_x & 1/m^*_{xy} \\ 1/m^*_{yx} & 1/m^*_y \\ \end{array} \right)[/itex] I know that, when the effective mass is isotropic, the weak form is [itex]\int \frac{\hbar^2}{2m^*}\nabla \psi \cdot \nabla v + U\psi vd\Omega = \int \epsilon \psi vd\Omega[/itex] The matrix is giving me trouble however. Is this the correct form? [itex]\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[/itex] 


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