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Fidelio
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Hi!
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.
Dirac equation (covariant form) (i \hbar \gamma^\mu \partial_\mu - mc) \Psi(x) = 0
\gamma^i = \beta \alpha_i and \gamma^0 = \beta
I have no idea where to start. Any suggestions are welcome.
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.
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