Satisfying Dirac equation, how to make equal to zero.

[rwooduk]

Homework Statement

Homework Equations

Dirac equation: $$(\gamma ^{\mu}\rho_{\mu}-mc)\psi=0$$

The Attempt at a Solution

If we multiply out the Dirac equation by inserting all it's components we get:

which if I've multiplied it correctly gives $$
\begin{bmatrix}
\frac{E}{c}-mc\\ 0
\\ \rho_{z}
\\ \rho_{x}+i\rho_{y}

\end{bmatrix}e^{-\frac{iEt}{\hbar}}=0$$

I'm not sure how to show the left hand side is equal to zero?

edit just realized that it is at rest so all the $\rho$ components would be zero, but then I still don't see how E/c - mc multiplied by the exponential would give zero?

Any help / ideas would really be appreciated.

 

[mfb]

If the particle is at rest, what is its relation between energy and mass?

 

[rwooduk]

If the particle is at rest, what is its relation between energy and mass?

ahhhh E=mc^2 so the term in the matrix would be zero, many thanks!