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 [itex]k[/itex] to be quantized is boundary conditions. There are two different boundary conditions that are typically used with such a box:

1. [itex]\psi(\vec{r}) = 0[/itex] whenever [itex]\vec{r}[/itex] 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 [itex]V(\vec{r})[/itex] with [itex]V(\vec{r}) = 0[/itex] inside the box and [itex]V(\vec{r}) = \infty[/itex] outside the box.
2. [itex]\psi(\vec{r}) = \psi(\vec{r} + L \hat{i}) = \psi(\vec{r} + L \hat{j}) = \psi(\vec{r} + L \hat{k})[/itex], where [itex]\hat{i}, \hat{j}, \hat{k}[/itex] 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 [itex]\psi(x,y,z) = \sum_{n,l,m} C_{n,l,m} sin(k_n x) sin(k_l y) sin(k_m z)[/itex] where [itex]k_n = \frac{\pi n}{L}[/itex] (and similarly for [itex]k_l[/itex] and [itex]k_m[/itex])

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