# How to numerically solve real-space 1D time-independent Schrodinger equation using 2D momentum-space Hamiltonian?

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.

• Hello @TribalChief, could you be more specific about solving "for this system along a 1D real space line"? The Hamiltonian you showed can be easily obtained from the Tight-Binding method, considering only first-neighbors hoppings with probabilities t, spanning all over the space. Therefore, the system to be solved in real space must take into account periodic boundary conditions. There are some methods to solve it without using DFT, such as Split Operator Method, Fourier Grid Method. However, to give you a more assertive answer, we should get to the point you want, and I really missed it! – Anibal Bezerra May 24 at 19:30
• @AnibalBezerra, sorry for the confusion. I am considering a 1D supercell in real space, along which the potential $V(x)$ varies. I have the given 2D k-space Hamiltonian, but wish to calculate the wavefunctions along the 1D real space line. In particular, I want to reproduce the well-known result that the wavefunction amplitude peaks at the 'domain wall' (where $V(x)$ changes drastically, as seen in the figure). This peak corresponds to the existence of an edge state at the domain wall. I am looking at doing this numerically using the k-space Hamiltonian, instead of analytical derivations. – TribalChief May 24 at 19:46
• I should observe that to simulate a quantum wire we have to take into account quantization in two dimensions, so the problem is in essence 2D - even in the real space. I took a quick look at the paper you cited, and the simulation they did was beyond the simple 1D tight-binding method. To my understanding, they used Green's functions and more specialized techniques. I'm not a specialist with topological phases, so I'm afraid to go beyond! However, the methods I've mentioned should do the task - not as easy as you want, but would fit! – Anibal Bezerra May 24 at 23:12

That said, given the amplitudes on the graphene lattice, you can create an approximate real-space wave function from an eigenstate of the lattice hamiltonian through linear combination of atomic orbitals: Place a carbon $$p_z$$ orbital on every site and simply sum them up with the corresponding amplitudes (which can easily be done analytically).