# How to solve these coupled differential equations?

I am trying to solve for wavefunctions of 2D tilted Dirac systems, the Hamiltonian for which is: $$\hat H = v_{x}\sigma_{x}\hat p_{x}+v_{y}\sigma_{y}\hat p_{y}+I_{2}(v_{t}^{x}\hat p_{x}+v_{t}^{y}\hat p_{y})+VI_{2}$$ where \begin{align} V(x, y) = \left\{ \begin{array}{cc} V_{0} & \hspace{5mm} \text{if 0 \leq x \leq D} \\ 0 & \hspace{5mm} \text{otherwise} \end{array} \right. \end{align} and $$\sigma_{i}$$ are the Pauli matrices. $$I_{2}$$ is a $$2\times 2$$ Identity matrix. Also, the constants $$v_x$$ and $$v_y$$ are the $$x$$ and $$y$$ components of velocity and $$v_t^x$$ and $$v_t^y$$ are tilt parameters. The time-independent equation ($$H\Psi=E\Psi$$, where $$\Psi=(\psi_{1}\,\psi_{2})^{T}$$) gives the following two equations:

$$\begin{bmatrix} v_{t}^{x}\hat p_{x}+v_{t}^{y}\hat p_{y}+V & v_{x}\hat p_{x}-iv_{y}\hat p_{y}\\\ v_{x}\hat p_{x}+iv_{y}\hat p_{y} & v_{t}^{x}\hat p_{x}+v_{t}^{y}\hat p_{y}+V\end{bmatrix} \begin{bmatrix} \psi_1 \\\ \psi_2\end{bmatrix} = E\begin{bmatrix} \psi_1 \\\ \psi_2\end{bmatrix}$$

I can solve for the case when $$v_t^x=v_t^y=0$$ as that gives $$(v_{x}\hat p_{x}-iv_{y}\hat p_{y})\psi_2=(E-V)\psi_1$$ $$(v_{x}\hat p_{x}+iv_{y}\hat p_{y})\psi_1=(E-V)\psi_2$$ Now, these can be solved easily by substituting $$\psi_2$$ from 2nd equation in the first equation which gives me a equation in just $$\psi_1$$.

However, I have no ideas on how to decouple the two equations when either of $$v_{t}^{x}$$ and $$v_{t}^{y}$$ are not $$0$$. I'd appreciated any help on that part.

Thanks!

UPDATE 1: This paper solves a similar Hamiltonian.

• Wouldn't the answer be obtained simply by substituting $V \to V + \vec{v}_t \cdot \hat{\vec{p}}$ in your $\vec{v}_t = 0$ solution? – Michael Seifert Jun 20 at 14:13
• Can you "round up the usual suspects?" Is a perturbation solution acceptable? So you write it as some kind of perturbation of the v^x_t and v^y_t zero case. Or you write it as a perturb of a special case of those v's that you can solve. Is a variational solution acceptable? You give it some kind of parameter and find the minimum "badness" of the solution. Is some kind of series solution acceptable? Can you write a basis such as sinusoidal fcns or some such, that you can make a series solution that converges as you get higher orders. And so on. – puppetsock Jun 20 at 14:15
• @MichaelSeifert I don't think that would work since the action of $\hat p_y$ on wavefunction is different from that of $V$. Acting $V$ on $\psi$ just multiplies it by a scalar while acting $\hat p_y$ on $\psi$ takes a partial derivative with respect to $y$. This prevents me from substituting $\psi$ from one equation into the other. – hhsomething69 Jun 20 at 17:08
• @puppetsock Although I'm not sure, I don't think these techniques are being used to obtain the solution. I have seen papers on this and none of them mentions any of these techniques. – hhsomething69 Jun 20 at 17:10
• @hhsomething69: You could always work in momentum space, in which case you'd end up with an equation where some rational polynomial of $\vec{p}$ multiplying $\tilde{\psi}_1$ is equal to the Fourier transform of $(E - V) \psi_1$. But since the Fourier transform of $V \psi_1$ is not the product of their individual Fourier transforms, this might not help as much as I originally thought it would. – Michael Seifert Jun 20 at 19:04