Properly implementing boundary conditions on a periodic 2d system

Question

I'm trying to read the following paper: https://arxiv.org/abs/0903.4271. In this paper, they showed how to write a Schrodinger equation for a quantum particle constrained on a corrugated 2D surface.

Here $z=a\cos{\gamma x}$. The Schrodinger equation for a particle constrained on such a surface is derived to be

$$\frac{-\hbar^2}{2m}\left[ \frac{1}{w(x)} \frac{\partial}{\partial x} \left( \frac{1}{w(x)}\frac{\partial}{\partial x}\right) + \frac{\partial^2}{\partial y^2}\right]\psi(x,y) + U(x,y)\psi(x,y) = E\psi(x,y) $$ where $w(x) = \sqrt{1+(a\gamma\sin{\gamma x})^2}$ and the geometric potential $U(x,y)$ is

$$U(x,y) = -\frac{\hbar^2}{8m}\frac{\left[a\gamma^2\cos{\gamma x}\right]^2}{\left[1+(a\gamma\sin{\gamma x})^2\right]^3} $$

I've been able to follow through these derivations but what I'm not sure of is to how to set-up the boundary conditions if I want to solve numerically for the eigenstates and eigenvalues?

Since the corrugation on the surface is periodic (and so is $U(x,y)$), I'm assuming Bloch theorem is applicable so that $\psi(x,y) = e^{i(k_x x+k_y y)}f(x,y)$ where $f$ has the same period as the surface. So can I use $\psi(x,y) = \psi(x+P,y)$ where P means one periodic interval along $x$? What boundary conditions do I need here?

Answer 1

From Bloch theorem, the wave function $\psi(x, y)$ will get a phase factor $e^{ika}$ by shifting the position $x$ a period length $a$:

$$ \tag{1} \psi_k(x+a, y) = e^{ika} \psi_k(x, y) $$

The parameter $k$ denotes the phase change, the wave vector. Therefore, the wave function is charaterised by the wave vector, and also its eigen energy $E(k)$, knowing as the band energy structure.

In solving the wave function your have first to specific a value of $k$ within the first Brillouin zone, i.e. $ -\frac{\pi}{a} \le k \le \frac{\pi}{a}$. Then solve the eigen function $\psi_k$ and eigen energy $E(k)$ using boundary condition Eq.(1).

Since the potential is independent of $y$, you may first separate

$$\psi_{k_x, k_y} = \psi_{k_x}(x) e^{i k_y y}$$

The kinetic energy in $y$ component will added to the eigen energy as:

$$ E(kx, ky) = E(k_x) + \frac{\hbar^2 k_y^2}{2 m} $$

Where $E(k_x)$ is the numerical solved energy with $k_x$ to specify the phase factor in boundary condition, Eq.(1).