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Let me begin by saying this question isn't related to GR.

I'm reading a paper (see https://arxiv.org/abs/0903.0798v1) that talks about deriving a Schrodinger equation for an electron confined on a curved surface and the result is

$$ -\frac{\hbar^2}{2m}\bigg[ \frac{1}{\sqrt{g}}\sum_{i,j=1}^{2} \frac{\partial}{\partial u^i}\sqrt{g}g^{ij}\frac{\partial}{\partial u^j} + (H^2 - K) \bigg]\Psi = E\Psi $$

where $g_{ij}$ is the metric tensor and $H$ and $K$ are both curvature related quantities that depend on $g_{ij}$. I'm interested on the form of this schrodinger equation for a surface described by the equation

$$z(x,y)= \alpha \cos{(\beta x)}\cos{(\gamma y)}.$$

How do I calculate for $g_{ij}$ in this case?

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A choice of basis vectors spanning the surface is $$\mathbf{e}_1 = \hat{x} + z_x \hat{z}, \quad \mathbf{e}_2 = \hat{y} + z_y \hat{z}$$ where I've defined $z_x = \partial z / \partial x$ and $z_y = \partial z / \partial y$. By definition, the elements of the metric satisfy $$g_{ij} = \mathbf{e}_i \cdot \mathbf{e}_j$$ where the dot product is taken under the metric of the embedding space. So $$g_{ij} = \begin{pmatrix} \mathbf{e}_1 \cdot \mathbf{e}_1 & \mathbf{e}_1 \cdot \mathbf{e}_2 \\ \mathbf{e}_2 \cdot \mathbf{e}_1 & \mathbf{e}_2 \cdot \mathbf{e}_2 \end{pmatrix} = \begin{pmatrix} 1 + z_x^2 & z_x z_y \\ z_x z_y & 1 + z_y^2 \end{pmatrix}.$$ If you want the metric in a different coordinate system, you can transform this the usual way.

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