Weak Form of the Effective Mass Schrodinger Equation

  1. Hi,

    I am numerically solving the 2D effective-mass 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]
     
    Last edited: Apr 29, 2014
  2. jcsd
  3. Greg Bernhardt

    Staff: Admin

    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
     
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