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snickersnee
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(maybe this should go in the math section?)
I'm supposed to derive the k.p form of the Schroedinger equation by plugging in the Bloch wavefunction expansion. But my actual question is just about the math.
So when I plugged Bloch into the SE, one of the terms has this Laplacian:
[itex]\nabla^2[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})][/itex]
The Laplacian is the divergence of the gradient, so first I need to show this:
[itex]\nabla[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})]={([\nabla u_{n\mathbf{k}}]+i\mathbf{k}u_{n\mathbf{k}}(\mathbf{r})})\exp{(i \mathbf{k} \cdot \mathbf{r})}[/itex]
Then the next step would be this:
[itex]\nabla^2[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})]={{\nabla^2 u_{n\mathbf{k}}(\mathbf{r})+2i\mathbf{k}\cdot \nabla u_{n\mathbf{k}}(\mathbf{r})-k^2 u_{n\mathbf{k}}(\mathbf{r})}}\exp(i\mathbf{k}\cdot \mathbf{r})[/itex]
I'm also given the following hints:
In addition, I'll probably need these:
[itex]\text{Gradient } \nabla V=\mathbf{\hat{x}}\frac{\partial V}{\partial x}+\mathbf{\hat{y}}\frac{\partial V}{\partial y}+\mathbf{\hat{z}}\frac{\partial V}{\partial z}\\
\text{Divergence } \nabla \cdot V=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\\
\text{Laplacian } \nabla^2 V=\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}+\frac{\partial^2 V}{\partial z^2} [/itex]
So I start by taking the gradient:
[itex]\nabla[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})]=u_{n\mathbf{k}}(\mathbf{r}) \nabla [\exp(i\mathbf{k}\cdot \mathbf{r})] +\exp(i\mathbf{k}\cdot \mathbf{r}) \nabla u_{n\mathbf{k}}(\mathbf{r})[/itex]
Apparently, the first term on the right hand side works out to be this:
[itex]i\mathbf{k}u_{n\mathbf{k}}(\mathbf{r})\exp{(i \mathbf{k}\cdot \mathbf{r})}[/itex]
but how do you come up with that? It would really help me if someone could please work it out explicitly.
Homework Statement
I'm supposed to derive the k.p form of the Schroedinger equation by plugging in the Bloch wavefunction expansion. But my actual question is just about the math.
Homework Equations
So when I plugged Bloch into the SE, one of the terms has this Laplacian:
[itex]\nabla^2[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})][/itex]
The Laplacian is the divergence of the gradient, so first I need to show this:
[itex]\nabla[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})]={([\nabla u_{n\mathbf{k}}]+i\mathbf{k}u_{n\mathbf{k}}(\mathbf{r})})\exp{(i \mathbf{k} \cdot \mathbf{r})}[/itex]
Then the next step would be this:
[itex]\nabla^2[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})]={{\nabla^2 u_{n\mathbf{k}}(\mathbf{r})+2i\mathbf{k}\cdot \nabla u_{n\mathbf{k}}(\mathbf{r})-k^2 u_{n\mathbf{k}}(\mathbf{r})}}\exp(i\mathbf{k}\cdot \mathbf{r})[/itex]
I'm also given the following hints:
- k vector is independent of position
- u_nk(r) is a function of position only
- so divergence of k vector is 0
- (the above is used when calculating the laplacian)
In addition, I'll probably need these:
[itex]\text{Gradient } \nabla V=\mathbf{\hat{x}}\frac{\partial V}{\partial x}+\mathbf{\hat{y}}\frac{\partial V}{\partial y}+\mathbf{\hat{z}}\frac{\partial V}{\partial z}\\
\text{Divergence } \nabla \cdot V=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\\
\text{Laplacian } \nabla^2 V=\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}+\frac{\partial^2 V}{\partial z^2} [/itex]
The Attempt at a Solution
So I start by taking the gradient:
[itex]\nabla[u_{n\mathbf{k}}(\mathbf{r})\exp(i\mathbf{k}\cdot \mathbf{r})]=u_{n\mathbf{k}}(\mathbf{r}) \nabla [\exp(i\mathbf{k}\cdot \mathbf{r})] +\exp(i\mathbf{k}\cdot \mathbf{r}) \nabla u_{n\mathbf{k}}(\mathbf{r})[/itex]
Apparently, the first term on the right hand side works out to be this:
[itex]i\mathbf{k}u_{n\mathbf{k}}(\mathbf{r})\exp{(i \mathbf{k}\cdot \mathbf{r})}[/itex]
but how do you come up with that? It would really help me if someone could please work it out explicitly.
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