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Momentum space

  1. Nov 17, 2009 #1
    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:

    [tex]\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).[/tex]

    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 Schrodinger'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.
     
  2. jcsd
  3. Nov 17, 2009 #2

    olgranpappy

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    The vector [itex]\bold{k}[/itex] 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 [itex]u[/itex]; if you act with the actual momentum operator [itex]-i\nabla[/itex] on [itex]\psi[/itex] you find a term proportional to [itex]\bold{k}[/itex], but also a term proportional to [itex]\nabla u[/itex].
     
  4. Dec 4, 2011 #3
    I'm having problems with the whole idea of momentum space, at the most elementary level. I would appreciate some help, please.
     
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