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Homework Help: Dirac equation

  1. Apr 30, 2012 #1

    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 Schrödinger 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.
    Last edited by a moderator: Apr 30, 2012
  2. jcsd
  3. Apr 30, 2012 #2


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    Science Advisor

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