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Homework Help: Probability current of dirac equation with vector potential

  1. Jul 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Given the probability/energyprobability current of the dirac equation
    [tex]j^\mu=\Psi^{+}\gamma^{0}\gamma^{\mu}\Psi[/tex] with continuity equation [tex]\partial_\mu j^\mu = 0[/tex]
    I need to find the current when there is an additional vector potential, introduced via minimal substitution [tex]\partial_{\mu}\rightarrow\partial_{\mu}+\frac{ie}{\hbar}A_{\mu}[/tex]

    2. Relevant equations

    * Dirac Equation [tex](i\hbar\gamma^{\mu}\partial_{\mu}-mc)\Psi=0[/tex]
    * Probability Current [tex]j^{\mu}=\Psi^{+}\gamma^{0}\gamma^{\mu}\Psi[/tex]
    * Continuity Equation [tex]\partial_\mu j^\mu = 0[/tex]
    * Minimal Substitution [tex]\partial_{\mu}\rightarrow\partial_{\mu}+\frac{ie}{\hbar}A_{\mu}[/tex]

    3. The attempt at a solution

    I tried to make an ansatz [tex]j^{\mu}=\Psi^{+}(\gamma^{0}\gamma^{\mu}+\alpha A^{\mu})\Psi[/tex] that didn't really work out. So I hope someone can help :)
  2. jcsd
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