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External Fields and Negative Energy Transitions In Dirac Particle

  • #1

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
 
Last edited:

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
Gold Member
3,119
289
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.
 
Last edited:
  • #3
But the final state would be an antiparticle at rest? Or with some energy?
 
  • #4
fzero
Science Advisor
Homework Helper
Gold Member
3,119
289
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.
 
  • #5
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."
 

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