# How to solve the Dirac Equation and find its eigenfunctions?

Imagine you want to solve the Dirac equation: $$(i\gamma^\mu \partial_\mu - m )\psi=0 \\$$ Where $$\gamma^\mu$$ are the $$4 \times 4$$ gamma matrices, and $$\psi$$ is a 4 component spinor. $$\psi = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix}$$ When you trying to solve this differential equation, the derivatives of the different spinor components would be mixed by the matrices, and appear in all equations. You will finally get, a system of 4 equations, all of which depends on all the 4 spinor components.

My question is, how would you even solve this? It is possible to arrive to a differential equation that depends, only, on one function at a time?

Solving the Dirac equation for a free particle (i.e. without any external fields) $$(i\gamma^\mu \partial_\mu - m )\psi=0 \\$$ which you have written down, is actually not that hard. Searching for "Dirac equation free particle" will give many hits.

You begin with the approach where the wave spinor field $$\psi(\mathbf{x},t)$$ is the product of a constant spinor $$u$$ and a plane wave $$\begin{pmatrix} \psi_1(\mathbf{x},t) \\ \psi_2(\mathbf{x},t) \\ \psi_3(\mathbf{x},t) \\ \psi_4(\mathbf{x},t) \end{pmatrix} =\begin{pmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{pmatrix} e^{i(\mathbf{p}\mathbf{x}-Et)}$$ and $$\mathbf{p}$$ and $$E$$ are still unknown constants (the momentum and energy of the particle). This will lead you to the constraint $$E^2=\mathbf{p}^2+m^2$$ (which of course is the relativistic energy-momentum relation) and 3 algebraic relations between the 4 numbers $$u_1,u_2,u_3,u_4$$. So you don't have differential equations anymore.

• What if you want to add a potential? Sep 20, 2023 at 18:54
• @ÁlvaroRodrigo Then this simple approach doesn't work anymore. Sep 20, 2023 at 18:58
• And what is the correct approach? Sep 20, 2023 at 19:35