Exploring the Dirac Equation: Positive & Negative Energy Solutions

[Fidelio]

Hi!

Homework Statement

1. Substituting an ansatz $\Psi(x)= u(p) e^{(-i/h) xp}$ into the Dirac equation and using $\{\gamma^i,\gamma^j\} = 2 g^{ij}$, show that the Dirac equation has both positive-energy and negative-energy solutions. Which are the allowed values of energy?

2. Starting from the DE, and using $\Psi(x) = e^{(1 /i \hbar)}(\psi_u(\vec{x}), \psi_l(\vec{x}))^T$, show that at the non-relativistic limit, the upper 2-component spinors, $\psi_u(\vec {x})$, for the positive-energy solutions fullfill the Schrödinger equation while the lower spinors, $\psi_l(\vec{x})$, vanish. Use the Dirac-Pauli representation.

Homework Equations

Dirac equation (covariant form) $(i \hbar \gamma^\mu \partial_\mu - mc) \Psi(x) = 0$
$\gamma^i = \beta \alpha_i$ and $\gamma^0 = \beta$

The Attempt at a Solution

I have no idea where to start. Any suggestions are welcome.

 

[HallsofIvy]

Well, I would suggest that you start by doing what you were told to do! If you substitute $u(p)e^{(-i/h)xp}$ into the Dirac equation, what do you get?