Question

2. Consider a tight-binding model of a one-dimensional crystal with two atoms A and B per unit cell (see diagram on previous page) and lattice constant a. The atoms in unit cell $n$ have associated electronic states $|A, n\rangle$ and $|B, n\rangle$; these states are orthonormal. The atoms are both of the same type, so the on-site matrix element of the Hamiltonian $H$ is the same for each: $\langle A, n|H|A, n\rangle = \langle B, n|H|B, n\rangle = -\beta$ The bonds within each unit cell are strong, so that the inter-site Hamiltonian matrix element within each unit cell is $\langle A, n|H|B, n\rangle = -\gamma_1$, while the bonds between each unit cell and the next are weaker, so the inter- site Hamiltonian between each unit cell and the next is $\langle B, n - 1|H|A, n\rangle = \langle B, n|H|A, n + 1\rangle = -\gamma_2$, with $\gamma_1 > \gamma_2 > 0$. You may assume that the Hamiltonian matrix elements be- yond nearest neighbours are zero. (a) Assuming that a Bloch state of wave-number $k$ can be written as $\left|\psi_k\right\rangle = \frac{1}{\sqrt{N}} \sum_n (u_0|A, n\rangle + v_0|B, n\rangle)e^{ikna}$ (where $u_0$ and $v_0$ are constant coefficients), find the action of $H$ on $|\psi_k\rangle$ and hence show that the time-independent Schrödinger equation can be written $\begin{pmatrix} -\beta & -\gamma_1 - \gamma_2e^{-ika} \\\ -\gamma_1 - \gamma_2e^{ika} & -\beta \end{pmatrix} \begin{pmatrix} u_0 \\\ v_0 \end{pmatrix} = \epsilon \begin{pmatrix} u_0 \\\ v_0 \end{pmatrix}.$

          2. Consider a tight-binding model of a one-dimensional crystal with two atoms A
and B per unit cell (see diagram on previous page) and lattice constant a. The
atoms in unit cell $n$ have associated electronic states $|A, n\rangle$ and $|B, n\rangle$; these
states are orthonormal. The atoms are both of the same type, so the on-site
matrix element of the Hamiltonian $H$ is the same for each:
$\langle A, n|H|A, n\rangle = \langle B, n|H|B, n\rangle = -\beta$
The bonds within each unit cell are strong, so that the inter-site Hamiltonian
matrix element within each unit cell is
$\langle A, n|H|B, n\rangle = -\gamma_1$,
while the bonds between each unit cell and the next are weaker, so the inter-
site Hamiltonian between each unit cell and the next is
$\langle B, n - 1|H|A, n\rangle = \langle B, n|H|A, n + 1\rangle = -\gamma_2$,
with $\gamma_1 > \gamma_2 > 0$. You may assume that the Hamiltonian matrix elements be-
yond nearest neighbours are zero.
(a) Assuming that a Bloch state of wave-number $k$ can be written as
$\left|\psi_k\right\rangle = \frac{1}{\sqrt{N}} \sum_n (u_0|A, n\rangle + v_0|B, n\rangle)e^{ikna}$
(where $u_0$ and $v_0$ are constant coefficients), find the action of $H$ on $|\psi_k\rangle$
and hence show that the time-independent Schrödinger equation can be
written
$\begin{pmatrix} -\beta & -\gamma_1 - \gamma_2e^{-ika} \\\ -\gamma_1 - \gamma_2e^{ika} & -\beta \end{pmatrix} \begin{pmatrix} u_0 \\\ v_0 \end{pmatrix} = \epsilon \begin{pmatrix} u_0 \\\ v_0 \end{pmatrix}.$

2. Consider a tight-binding model of a one-dimensional crystal with two atoms A
and B per unit cell (see diagram on previous page) and lattice constant a. The
atoms in unit cell n have associated electronic states |A, n⟩ and |B, n⟩; these
states are orthonormal. The atoms are both of the same type, so the on-site
matrix element of the Hamiltonian H is the same for each:
⟨ A, n|H|A, n⟩ = ⟨ B, n|H|B, n⟩ = -β
The bonds within each unit cell are strong, so that the inter-site Hamiltonian
matrix element within each unit cell is
⟨ A, n|H|B, n⟩ = -γ1,
while the bonds between each unit cell and the next are weaker, so the inter-
site Hamiltonian between each unit cell and the next is
⟨ B, n - 1|H|A, n⟩ = ⟨ B, n|H|A, n + 1⟩ = -γ2,
with γ1 > γ2 > 0. You may assume that the Hamiltonian matrix elements be-
yond nearest neighbours are zero.
(a) Assuming that a Bloch state of wave-number k can be written as
|⟩ = (1)/(√(N)) (u0|A, n⟩ + v0|B, n⟩)e^ikna
(where u0 and v0 are constant coefficients), find the action of H on |⟩
and hence show that the time-independent Schrödinger equation can be
written
< p m a t r i x >

< p m a t r i x >
= ϵ
< p m a t r i x >
.

Added by Bailey I.

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