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1. The problem statement, all variables and given/known data

1. Substituting an ansatz [itex]\Psi(x)= u(p) e^{(-i/h) xp} [/itex] into the Dirac equation and using [itex]\{\gamma^i,\gamma^j\} = 2 g^{ij}[/itex], 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 [itex]\Psi(x) = e^{(1 /i \hbar)}(\psi_u(\vec{x}), \psi_l(\vec{x}))^T[/itex], show that at the non-relativistic limit, the upper 2-component spinors, ##\psi_u(\vec {x})##, for the positive-energy solutions fullfill the SchrÃ¶dinger equation while the lower spinors, ##\psi_l(\vec{x})##, vanish. Use the Dirac-Pauli representation.

2. Relevant equations

Dirac equation (covariant form) [itex](i \hbar \gamma^\mu \partial_\mu - mc) \Psi(x) = 0 [/itex]

[itex] \gamma^i = \beta \alpha_i[/itex] and [itex]\gamma^0 = \beta[/itex]

3. The attempt at a solution

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

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# Homework Help: Dirac equation

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