Exploring Dirac Hamiltonian with Matrices

[LagrangeEuler]

Homework Statement

Matrices
$\alpha_k=\gamma^0 \gamma^k$, $\beta=\gamma^0$ and $\alpha_5=\alpha_1\alpha_2\alpha_3 \beta$. If we know that for Dirac Hamiltonian
$$H_D\psi(x)=E \psi(x)$$
then show that
$$\alpha_5 \psi(x)=-E \psi(x)$$

Homework Equations

The Attempt at a Solution

I tried to multiply Gamma matrices from wikipedia link, but I am not sure how to work with that od the state $\psi(x)$? How to write that as column vector? I am not sure what to do here?

 

[eys_physics]

From which book is this problem? Should it be proved for a specific basis for the gamma matrices or in general? You can e.g. write the four vector in the form $\psi=\begin{pmatrix}\xi \\ \chi \end{pmatrix}$, where $\xi$ and $\chi$ are column vectors (containing to two elements).