User:WillowW/Semiclassical radiation

Semiclassical approach to radiation

Main article: Perturbation theory (quantum mechanics)

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".