Consider the usual simple 2-level graphene Hamiltonian with mass in momentum-space where:
$$ H(k,V)=-t \sum_{\delta} [\cos(k\cdot\delta)\sigma_x-\sin(k\cdot\delta)\sigma_y+V\sigma_z], $$ where $t$ is the hopping, $\sigma_i$ are Pauli matrices, $\delta$ lattice vectors and $V$ an on-site potential ($V=0$ for graphene, where the energy dispersion is degenerate/gapless at a Dirac point).
The parameter space above is 2D momentum/k-space, and the eigenvalue equation can be easily solved as I show in my answer here: How to create a 3D band structure from DFT band structure calculation?
Now, I want to numerically solve its real-space analog. I know k and real space are related to each other via Fourier transforms (https://quantummechanics.ucsd.edu/ph130a/130_notes/node82.html). However, I want to plot the wavefunction amplitudes of this system along a 1D real space line where the on-site potential $V(x)$ varies. So, I want to plot something like figure c) here: https://arxiv.org/pdf/2005.06096.pdf (reproduced below):
Is there a simple way to do this? I know I am being naive here, but would it be as simple as just taking the Fourier transforms of eigenvectors from the k-space method? This comes with the issue of choosing k-points corresponding to the real-space 1D line. I know that in theory, people usually change the Hamiltonian between spaces by inverting the creation/annihilation operators' spaces, etc. But is there something more straightforward numerically (something as silly as Fourier transforming the above k-space Hamiltonian)? There's also just replacing $k_x$ with $~\partial_x$ (as in section 3.5.7 in https://arxiv.org/abs/1310.0255), but I am just wondering whether there is something more straightforward due to numerics.
Or should I come up with some other way? I do not wish to use DFT and ab initio techniques, just simple numerical software like Python or MATLAB.