# Homework Help: Probability current of dirac equation with vector potential

1. Jul 23, 2010

### tobias_

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

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

2. Relevant equations

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

3. The attempt at a solution

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