Weak Form of the Effective Mass Schrodinger Equation

In summary, the conversation discusses the numerical solution of the 2D effective-mass Schrodinger equation, which includes a term for the effective mass matrix. The weak form for when the effective mass is isotropic is also mentioned. The conversation ends with a question about the correct form of the matrix.
  • #1
Morberticus
85
0
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]
 
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  • #2
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?
 

What is the Weak Form of the Effective Mass Schrodinger Equation?

The Weak Form of the Effective Mass Schrodinger Equation is a mathematical representation of the behavior of a quantum particle in a potential field. It takes into account the effective mass of the particle, which can vary depending on the material it is in, and allows for the calculation of the particle's wave function and energy levels.

Why is the Weak Form of the Effective Mass Schrodinger Equation important?

The Weak Form of the Effective Mass Schrodinger Equation is important because it allows for the accurate prediction of the behavior of quantum particles in a variety of materials. This is crucial in understanding the properties of materials and developing new technologies.

How is the Weak Form of the Effective Mass Schrodinger Equation derived?

The Weak Form of the Effective Mass Schrodinger Equation is derived from the Schrodinger Equation, which is a fundamental equation in quantum mechanics. It is modified to incorporate the effective mass of the particle, which is a concept introduced in the band theory of solids.

What are the limitations of the Weak Form of the Effective Mass Schrodinger Equation?

One limitation of the Weak Form of the Effective Mass Schrodinger Equation is that it assumes a single-band model, meaning it does not account for interactions between different bands in a material. It also does not take into account relativistic effects, which may be significant for high-energy particles.

How is the Weak Form of the Effective Mass Schrodinger Equation used in research and applications?

The Weak Form of the Effective Mass Schrodinger Equation is used in a variety of research fields, including materials science, condensed matter physics, and semiconductor device engineering. It is also used in the development of new technologies such as transistors, lasers, and solar cells.

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