- #1
Morberticus
- 82
- 0
Hi,
I am numerically solving the 2D effective-mass Schrödinger 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}<br /> 1/m^*_x & 1/m^*_{xy} \\<br /> 1/m^*_{yx} & 1/m^*_y \\<br /> \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
I am numerically solving the 2D effective-mass Schrödinger 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}<br /> 1/m^*_x & 1/m^*_{xy} \\<br /> 1/m^*_{yx} & 1/m^*_y \\<br /> \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
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