User:WillowW/Semiclassical radiation

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[edit] Semiclassical approach to radiation

Main article: Perturbation theory (quantum mechanics)

Einstein's coefficients Bij 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 Aij 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 E = hν 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 H = -2 \mathbf{d} \cdot \mathbf{\mathcal{E}_{0}} \cos \omega t, where \mathbf{d} is the material system's electric dipole moment and where \mathbf{\mathcal{E}_{0}} and ω represent the electric field and angular frequency of the incoming radiation, respectively. The probability per unit time wji of the radiation inducing a transition between discrete energy levels Ei and Ej may be computed using time-dependent perturbation theory

w_{ji} = \frac{2\pi}{\hbar^{2}} \left| \langle \phi_{j} | \mathbf{d} \cdot \mathbf{\mathcal{E}_{0}} | \phi_{i} \rangle \right|^{2} \delta(\omega_{ij} - \omega)

where ωij is defined by \omega_{ij} \equiv \left( E_{i} - E_{j} \right)/\hbar, and where φi and φj represent the unperturbed eigenstates of energy Ei and Ej, respectively. Assuming that the polarization vector \mathbf{\mathcal{E}_{0}} of the incoming radiation is oriented randomly relative to the dipole moment \mathbf{d} of the material system, the corresponding Bij rate constants can be computed

B_{ji} = \frac{8\pi^{2}}{3\hbar^{2}} \left| \langle \phi_{j} | \mathbf{d} | \phi_{i} \rangle \right|^{2}

from which Bji = Bij. Thus, if the two states φi and φj do not result in a net dipole moment (i.e., if \langle \phi_{j} | \mathbf{d} | \phi_{i} \rangle = 0), the absorption and induced emission are said to be "disallowed".