Problem set 3

Solid State theory, spring '23

Due April 13, Th. Before class.

1. Effective-mass theory in an external potential

$\hbar = 1$ in this problem.

(1) Consider the following Hamiltonian for a 2-dimensional system

\begin{matrix} (1) & \begin{matrix} H = v (p_{x} σ^{x} + p_{y} σ^{y}) + m (y) σ^{z} = [\begin{array}{cc} m (y) & v p_{x} - i v p_{y} \\ v p_{x} + i v p_{y} & - m (y) \end{array}], \end{matrix} \end{matrix}

$v$ $p_a = -\text i \partial_a, a = x,y$ $m(y) = - m' y$ $m' = \text{constant} >0$ . Show that the Hamiltonian has eigenstates of the form

\begin{matrix} (2) & \begin{matrix} ψ_{k} (x, y) \propto e^{i k x} [\begin{matrix} φ (y) \\ φ (y) \end{matrix}], \end{matrix} \end{matrix}

$\varphi(y)$ $\varepsilon_k$ .

(2) Consider the Hamiltonian for a 2-dimensional system

\begin{matrix} (3) & H = \frac{1}{2} p^{2} + α σ \cdot \hat{z} \times p - β σ^{z}, \end{matrix}

$\alpha, \beta >0$ $p_a = -\text i \partial_a, a = x,y$ $k_x$ axis.

7.19;

10.2, 10.3, 10.5;

11.2, 11.4, 11.8.