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## Homework Statement

Suppose a relativistic particle with spin 1/2 at rest. Show that if we apply an electrical field at t=0 there's a probability fot t>0 of finding the particle in a negative energy state if such negative energy states are assumed to be originally empty.

## Homework Equations

Dirac equation at rest:

[tex]

(\boldsymbol{\hat{\beta}} mc^2) \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi(\mathbf{x},t) }{\partial t}

[/tex]

Spinor solutions with positive energy for t<0

[tex]

$\Psi_{+}=\begin{pmatrix}

\psi^{(1)}\\ \psi^{(2)}

\end{pmatrix}={1\over\sqrt{V}}\begin{pmatrix}

1\\ 1\\ 0\\ 0

\end{pmatrix}e^{-imc^2t/\hbar}$\\

[/tex]

Applied electrical field and consecuent magnetic potential

[tex]

$\mathbf{E}=E_0 \cos(\omega t)\mathbf{u}_{x} \qquad \mathbf{A}=-\frac{E_0}{\omega} \sin(\omega t)\mathbf{u}_{x}$

[/tex]

## The Attempt at a Solution

I'm not sure how to solve this problem. Is it supposed that the particle at t>0 remains at rest? Or it has a negative energy E'?. First order perturbation theory must be used, but how about the final state of the particle?

Perturbed Hamiltonian should be

[tex]

$\displaystyle H'(t)=q c \boldsymbol{\hat{\alpha}} \mathbf{A}=q c\hat{\alpha}_{x}A_{x}$

[/tex]

Is it correct?

Thank you

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