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Tight binding method for a 1D crystal with a diatomic basis

  1. Jan 6, 2012 #1
    Hi, I'd be most grateful for any help regarding the following problem:

    Consider a 1D crystal with 2 atoms in a primitive cell (let's call them atoms A and B). Each atom has only one valence orbital denoted as [itex]\left|\psi_A(n)\right>[/itex] and [itex]\left|\psi_B(n)\right>[/itex] respectively.

    Show that the superposition
    [itex]\left|k\right> = \sum_{n = -\infty}^{\infty}e^{ikna}\left[ c_A\left|\psi_A(n)\right> + c_B\left|\psi_B(n)\right>\right][/itex]
    satisfies the Bloch's theorem ([itex]c_A, c_B[/itex] are constant)

    I know that the theorem is usually formulated as follows:
    There is a [itex]\vec{k}[/itex] from the reciprocal lattice for each wavefunction so that
    [itex]\psi\left( \vec{r}+\vec{R}\right) = e^{i\vec{k}\vec{R}}\psi\left( \vec{r}\right)[/itex]
    [itex]\vec{R}[/itex] is a vector from the direct lattice. But I still don't know what to do.

    I am probably not very good with quantum mechanics yet (we had the solid state and QM simultaneously this semester) and I realize this may be very easy. But I have been stuck on this and will be grateful for any help.
  2. jcsd
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