section_path: Stimulated and spontaneous emission
section_name: Stimulated and spontaneous emission
paragraph_text: Stimulated and spontaneous emission
  
  In 1916, Albert Einstein showed that Planck's radiation law could be derived from a semi-classical, statistical treatment of photons and atoms, which implies a link between the rates at which atoms emit and absorb photons. The condition follows from the assumption that functions of the emission and absorption of radiation by the atoms are independent of each other, and that thermal equilibrium is made by way of the radiation's interaction with the atoms. Consider a cavity in thermal equilibrium with all parts of itself and filled with electromagnetic radiation and that the atoms can emit and absorb that radiation. Thermal equilibrium requires that the energy density ρ (ν) {\displaystyle \rho (\nu)} of photons with frequency ν {\displaystyle \nu } (which is proportional to their number density) is, on average, constant in time; hence, the rate at which photons of any particular frequency are emitted must equal the rate at which they are absorbed.
  
  Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, the rate R j i {\displaystyle R_{ji}} for a system to absorb a photon of frequency ν {\displaystyle \nu } and transition from a lower energy E j {\displaystyle E_{j}} to a higher energy E i {\displaystyle E_{i}} is proportional to the number N j {\displaystyle N_{j}} of atoms with energy E j {\displaystyle E_{j}} and to the energy density ρ (ν) {\displaystyle \rho (\nu)} of ambient photons of that frequency,
  
  where B j i {\displaystyle B_{ji}} is the rate constant for absorption. For the reverse process, there are two possibilities: spontaneous emission of a photon, or the emission of a photon initiated by the interaction of the atom with a passing photon and the return of the atom to the lower-energy state. Following Einstein's approach, the corresponding rate R i j {\displaystyle R_{ij}} for the emission of photons of frequency ν {\displaystyle \nu } and transition from a higher energy E i {\displaystyle E_{i}} to a lower energy E j {\displaystyle E_{j}} is
  
  where A i j {\displaystyle A_{ij}} is the rate constant for emitting a photon spontaneously, and B i j {\displaystyle B_{ij}} is the rate constant for emissions in response to ambient photons (induced or stimulated emission). In thermodynamic equilibrium, the number of atoms in state i {\displaystyle i} and those in state j {\displaystyle j} must, on average, be constant; hence, the rates R j i {\displaystyle R_{ji}} and R i j {\displaystyle R_{ij}} must be equal. Also, by arguments analogous to the derivation of Boltzmann statistics, the ratio of N i {\displaystyle N_{i}} and N j {\displaystyle N_{j}} is g i / g j exp (E j − E i) / (k T), {\displaystyle g_{i}/g_{j}\exp {(E_{j}-E_{i})/(kT)},} where g i {\displaystyle g_{i}} and g j {\displaystyle g_{j}} are the degeneracy of the state i {\displaystyle i} and that of j {\displaystyle j}, respectively, E i {\displaystyle E_{i}} and E j {\displaystyle E_{j}} their energies, k {\displaystyle k} the Boltzmann constant and T {\displaystyle T} the system's temperature. From this, it is readily derived that
  
  g i B i j = g j B j i {\displaystyle g_{i}B_{ij}=g_{j}B_{ji}} and
  
  The A i j {\displaystyle A_{ij}} and B i j {\displaystyle B_{ij}} are collectively known as the Einstein coefficients.
  
  Einstein could not fully justify his rate equations, but claimed that it should be possible to calculate the coefficients A i j {\displaystyle A_{ij}}, B j i {\displaystyle B_{ji}} and B i j {\displaystyle B_{ij}} once physicists had obtained "mechanics and electrodynamics modified to accommodate the quantum hypothesis". Not long thereafter, in 1926, Paul Dirac derived the B i j {\displaystyle B_{ij}} rate constants by using a semiclassical approach, and, in 1927, succeeded in deriving all the rate constants from first principles within the framework of quantum theory. Dirac's work was the foundation of quantum electrodynamics, i.e., the quantization of the electromagnetic field itself. Dirac's approach is also called second quantization or quantum field theory; earlier quantum mechanical treatments only treat material particles as quantum mechanical, not the electromagnetic field.
  
  Einstein was troubled by the fact that his theory seemed incomplete, since it did not determine the direction of a spontaneously emitted photon. A probabilistic nature of light-particle motion was first considered by Newton in his treatment of birefringence and, more generally, of the splitting of light beams at interfaces into a transmitted beam and a reflected beam. Newton hypothesized that hidden variables in the light particle determined which of the two paths a single photon would take. Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation from quantum mechanics. Ironically, Max Born 's probabilistic interpretation of the wave function was inspired by Einstein's later work searching for a more complete theory.