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Semiclassical approach to radiation

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Einstein's coefficients for induced transitions can be computed semiclassically, i.e., by treating the electromagnetic radiation classically and the material system quantum mechanically[1]. However, this semiclassical approach does not yield the coefficients for spontaneous emission from first principles, although they can be calculated using the correspondence principle and the classical (low-frequency) limit of Planck's law of black body radiation (the Rayleigh-Einstein-Jeans law). The semiclassical approach does not require the introduction of photons per se, although their energy formula must be adopted. A true derivation from first principles was developed by Dirac that required the quantization of the electromagnetic field itself; in this approach, photons are the quanta of the field[2][3]. This approach is called second quantization or quantum field theory[4][5][6]; the earlier quantum mechanics (the quantization of material particles moving in a potential) represents the "first quantization".


The incoming radiation is treated as a sinusoidal electric field applied to the material system, with an small (perturbative) interaction energy , where is the material system's electric dipole moment and where and represent the electric field and angular frequency of the incoming radiation, respectively. The probability per unit time of the radiation inducing a transition between discrete energy levels and may be computed using time-dependent perturbation theory

where is defined by , and where and represent the unperturbed eigenstates of energy and , respectively. Assuming that the polarization vector of the incoming radiation is oriented randomly relative to the dipole moment of the material system, the corresponding rate constants can be computed

from which . Thus, if the two states and do not result in a net dipole moment (i.e., if ), the absorption and induced emission are said to be "disallowed".

  1. ^ Cite error: The named reference Dirac1926 was invoked but never defined (see the help page).
  2. ^ Cite error: The named reference Dirac1927a was invoked but never defined (see the help page).
  3. ^ Cite error: The named reference Dirac1927b was invoked but never defined (see the help page).
  4. ^ Heisenberg, W (1929). "Zur Quantentheorie der Wellenfelder". Zeitschrift für Physik. 56: 1. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) (in German)
  5. ^ Heisenberg, W (1930). "Zur Quantentheorie der Wellenfelder". Zeitschrift für Physik. 59: 139. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) (in German)
  6. ^ Fermi, E. (1932). "Quantum Theory of Radiation". Reviews of Modern Physics. 4: 87.