User:WillowW/Semiclassical radiation

Semiclassical approach to radiation
Einstein's coefficients $$B_{ij}$$ for induced transitions can be computed semiclassically, i.e., by treating the electromagnetic radiation classically and the material system quantum mechanically. However, this semiclassical approach does not yield the coefficients $$A_{ij}$$ 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 $$E=h \nu$$ 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. This approach is called second quantization or quantum field theory  ; 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 $$\omega$$ represent the electric field and angular frequency of the incoming radiation, respectively. The probability per unit time $$w_{ji}$$ of the radiation inducing a transition between discrete energy levels $$E_{i}$$ and $$E_{j}$$ 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 $$\omega_{ij}$$ is defined by $$\omega_{ij} \equiv \left( E_{i} - E_{j} \right)/\hbar$$, and where $$\phi_{i}$$ and $$\phi_{j}$$ represent the unperturbed eigenstates of energy $$E_{i}$$ and $$E_{j}$$, 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 $$B_{ij}$$ 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 $$B_{ji} = B_{ij}$$. Thus, if the two states $$\phi_{i}$$ and $$\phi_{j}$$ 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".