How the components of the wave vector k are quantized?

[Mrinmoy Naskar]

for a electron in a cubic box ( dimension of the box is L) problem how to show the components of the wave vector k are quantized...

 

[stevendaryl]

for a electron in a cubic box ( dimension of the box is L) problem how to show the components of the wave vector k are quantized...

Should this be in the homework section?

What forces $k$ to be quantized is boundary conditions. There are two different boundary conditions that are typically used with such a box:

1. $\psi(\vec{r}) = 0$ whenever $\vec{r}$ is on the boundary of the box. This is equivalent to assuming that the box is embedded in infinite space, but there is a potential energy $V(\vec{r})$ with $V(\vec{r}) = 0$ inside the box and $V(\vec{r}) = \infty$ outside the box.
2. $\psi(\vec{r}) = \psi(\vec{r} + L \hat{i}) = \psi(\vec{r} + L \hat{j}) = \psi(\vec{r} + L \hat{k})$, where $\hat{i}, \hat{j}, \hat{k}$ are unit normal vectors to the three sides of the box. This is called "periodic boundary conditions".

The first boundary condition leads to the conclusion that $\psi(x,y,z) = \sum_{n,l,m} C_{n,l,m} sin(k_n x) sin(k_l y) sin(k_m z)$ where $k_n = \frac{\pi n}{L}$ (and similarly for $k_l$ and $k_m$)

The second boundary condition leads to the conclusion that $\psi(x,y,z) = \sum_{n,l,m} C_{n,l,m} e^{i k_n x + k_l y + k_m z}$ where $k_n = \frac{2\pi n}{L}$ (and similarly for $k_l$ and $k_m$)