Suppose we have equation of the form:

$$H \Psi = E \Psi $$

where $H$ is Dirac Hamiltonian (also my question can be answered by people who are not familiar with Dirac Hamiltonian but familiar with numerical solution of different eigenvalue problems):

$$ H = \vec{\alpha} \vec{\nabla} + V(r) = H_0 + V(r)$$

and $\vec{\alpha}$ and $\beta$ are $4 \times 4$ Dirac matrices:

$$\vec{\alpha} = \left( \begin{matrix} 0 & \vec{\sigma} \\ -\vec{\sigma} & 0 \end{matrix} \right) \ , \ \ \beta = \left( \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right)$$

$\vec{\sigma}$ are $2 \times 2$ Pauli matrices.

Here $H_0$ is free Dirac Hamiltonian (without potential, only kinetic term is preserved).

I think that the best way to start solving such an equation numerically is to introduce plane-wave basis (which diagonalizes free Hamiltonian $H_0$):

$$\Psi = \sum_n c_n \psi_n $$

Then we have a problem like:

$$H \sum_n c_n \psi_n = E \sum_n c_n \psi_n $$

My question is: what are the most commonly used ways to solve this equation numerically for arbitrary potential? Just to compute matrix elements $H_{nm}$ in this plane-wave basis, to find eigenvalue $E$ and coefficients $C_n$ from this computation? Or maybe there are some better methods?

  • $\begingroup$ For the general potential you might want to use a different base, I suppose that you might want to deal with unbounded domains, then a plane-wave basis is not the best choice. Maybe you can try a Least-Square Finite Element Method, check this reference. $\endgroup$ – nicoguaro Oct 27 '16 at 16:31

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.