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Electromagnetism Quantization

  1. Dec 16, 2014 #1

    naima

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    I start from these formulas(transverse electric and magnetic fields)
    ## E_\perp(r) = \Sigma_i i \mathscr E_{\omega_i}\epsilon_i [\alpha_i e^{i k_i . r}- \alpha^\dagger_i e^{-i k_i . r}]##
    and
    ##B(r) = \Sigma_i i(1/c) \mathscr E_{\omega_i}(\kappa_i \times \epsilon_i) [\alpha_i e^{i k_i . r}- \alpha^\dagger_i e^{-i k_i . r}]##
    where epsilon is a unit vector orthogonal to ##\kappa_i = k_i/|k_i|##.
    The authors compute their commutator and write it as
    ##[E_{x \perp}(r),B_y(r')] = -i (\hbar/\epsilon_0) \partial_z \delta(r - r')##

    I do not see where this ##\partial_z## comes from.
    Have you an idea?

    Is it related to FT(f'(z)) = i k FT(f(k))
     
    Last edited: Dec 16, 2014
  2. jcsd
  3. Dec 16, 2014 #2

    naima

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    I think that the solution is closer.
    As ##\mathscr(E)_{\omega_i} = \sqrt{\frac{\hbar \omega_i}{2 \epsilon_0 L}}## and ##h \omega_i = |k_i| c##
    I get in the commutator a ##-1/ \epsilon_0## and the ##|k_i|## disappears in ##\kappa_i## and i come closer to a "i z FT(f(z)" term.
    Help will be appreciated!
     
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