Properly implementing boundary conditions on a periodic 2d system

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?

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).

• Will that boundary condition be enough or should I also assume that Equation (1) holds for the 1st derivatives as well?
– jboy
Mar 17, 2021 at 8:56
• Eq(1) is enough. If you use discretize meshes, the derivative of 1st point will involve the Cell in the left side, which will then transfer to the last point of this cell with an additional Bloch phase. Similar to the last point involving cell to its right. Effectively, this boundary condition places the Bloch phase in the most up-right and low-left corner of the matrix.
– ytlu
Mar 17, 2021 at 10:50