What is the Relationship Between K-Space and Momentum Space in Crystals?

[seang]

Hi,

I am having trouble understanding some things about k-space or momentum space in a crystal. The trouble began when I was first introduced to the Bloch theorem, a few weeks back.

It is:

$$\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).$$

In this equation, there is k, which is a vector in momentum space (?). Is it also a vector representing the quantum numbers? (i.e., are the momentum space vector components the quantum numbers of a particular solution of Schrödinger's equation?). Does this mean that in an E(k) plot, as you traverse a particular direction of k, you are traversing the quantum states of the system?

Also, the notation suggests that k is independent of r. In a crystal lattice with a periodic potential, if you move an electron around would not its momentum change also? I guess I'm not sure how the momentum and position of a particle in a crystal are independent of each other.

This MUST sound pretty vague, but there is something I'm not quite getting here. Thanks a lot for your help.

 

[olgranpappy]

Hi,

I am having trouble understanding some things about k-space or momentum space in a crystal. The trouble began when I was first introduced to the Bloch theorem, a few weeks back.

It is:

$$\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).$$

In this equation, there is k, which is a vector in momentum space (?). Is it also a vector representing the quantum numbers? (i.e., are the momentum space vector components the quantum numbers of a particular solution of Schrödinger's equation?). Does this mean that in an E(k) plot, as you traverse a particular direction of k, you are traversing the quantum states of the system?

Also, the notation suggests that k is independent of r. In a crystal lattice with a periodic potential, if you move an electron around would not its momentum change also? I guess I'm not sure how the momentum and position of a particle in a crystal are independent of each other.

This MUST sound pretty vague, but there is something I'm not quite getting here. Thanks a lot for your help.

The vector $\bold{k}$ is "crystal momentum" which is not the same as the actual momentum of a particle. The fact that the electron's real momentum will change as it moves about in the crystal is encoded in the periodic function $u$; if you act with the actual momentum operator $-i\nabla$ on $\psi$ you find a term proportional to $\bold{k}$, but also a term proportional to $\nabla u$.

 

[Endervhar]

I'm having problems with the whole idea of momentum space, at the most elementary level. I would appreciate some help, please.