External Fields and Negative Energy Transitions In Dirac Particle

[javiergra24]

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:
$$(\boldsymbol{\hat{\beta}} mc^2) \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi(\mathbf{x},t) }{\partial t}$$
Spinor solutions with positive energy for t<0
$$\Psi_{+}=\begin{pmatrix} \psi^{(1)}\\ \psi^{(2)} \end{pmatrix}={1\over\sqrt{V}}\begin{pmatrix} 1\\ 1\\ 0\\ 0 \end{pmatrix}e^{-imc^2t/\hbar}\\$$
Applied electrical field and consecuent magnetic potential
$$\mathbf{E}=E_0 \cos(\omega t)\mathbf{u}_{x} \qquad \mathbf{A}=-\frac{E_0}{\omega} \sin(\omega t)\mathbf{u}_{x}$$

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

Thank you

 

[fzero]

The perturbed Hamiltonian acts on the spinor wavefunction. You'll need to use the Dirac matrices to write things in terms of the positive and negative energy components.

 

[javiergra24]

But the final state would be an antiparticle at rest? Or with some energy?

 

[fzero]

But the final state would be an antiparticle at rest? Or with some energy?

Oh I missed that the spinor is relativistic. A relativistic particle is not at rest, so neither the initial or final state are at rest.

 

[javiergra24]

Umm a relativistic particle can be at rest, with E=mc^2. The exact problem is:

"Consider a positive energy spin-1/2 particle at rest. Suppose that at t=0 we apply an external
(classical) vector potential (see mi first post)
which corresponds to an electric field of the form
Show that for there exists a finite probability of finding the particle in a negative energy state if
such negative energy states are assumed to be originally empty. In particular, work out quantitatively
the two cases: w>>mc^2 and w=mc^2 and and comment."