Editing Photon (section)

== Stimulated and spontaneous emission ==
{{Main|Stimulated emission|Laser}}
[[File:Stimulatedemission.png|thumb|upright=1.75|[[Stimulated emission]] (in which photons "clone" themselves) was predicted by Einstein in his kinetic analysis, and led to the development of the [[laser]]. Einstein's derivation inspired further developments in the quantum treatment of light, which led to the statistical interpretation of quantum mechanics.]]

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 <math>\rho(\nu)</math> of photons with frequency <math>\nu</math> (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''.<ref name="Einstein1916a">{{cite journal |last=Einstein |first=Albert |author-link=Albert Einstein |year=1916 |title=Strahlungs-emission und -absorption nach der Quantentheorie |journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]] |language=de |volume=18 |pages=318–323 |bibcode=1916DPhyG..18..318E}}</ref>

Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, the rate <math>R_{ji}</math> for a system to ''absorb'' a photon of frequency <math>\nu</math> and transition from a lower energy <math>E_{j}</math> to a higher energy <math>E_{i}</math> is proportional to the number <math>N_{j}</math> of atoms with energy <math>E_{j}</math> and to the energy density <math>\rho(\nu)</math> of ambient photons of that frequency,
: <math>
R_{ji}=N_{j} B_{ji} \rho(\nu) \!
</math>
where <math>B_{ji}</math> 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 <math>R_{ij}</math> for the emission of photons of frequency <math>\nu</math> and transition from a higher energy <math>E_{i}</math> to a lower energy <math>E_{j}</math> is
: <math>
R_{ij}=N_{i} A_{ij} + N_{i} B_{ij} \rho(\nu) \!
</math>
where <math>A_{ij}</math> is the rate constant for [[spontaneous emission|emitting a photon spontaneously]], and <math>B_{ij}</math> is the rate constant for emissions in response to ambient photons ([[stimulated emission|induced or stimulated emission]]). In thermodynamic equilibrium, the number of atoms in state <math>i</math> and those in state <math>j</math> must, on average, be constant; hence, the rates <math>R_{ji}</math> and <math>R_{ij}</math> must be equal. Also, by arguments analogous to the derivation of [[Boltzmann statistics]], the ratio of <math>N_{i}</math> and <math>N_{j}</math> is <math>g_i/g_j\exp{(E_j-E_i)/(kT)},</math> where <math>g_i</math> and <math>g_j</math> are the [[degenerate energy level|degeneracy]] of the state <math>i</math> and that of <math>j</math>, respectively, <math>E_i</math> and <math>E_j</math> their energies, <math>k</math> the [[Boltzmann constant]] and <math>T</math> the system's [[temperature]]. From this, it is readily derived that
: <math>g_iB_{ij}=g_jB_{ji}</math>
and
: <math>A_{ij}=\frac{8 \pi h \nu^{3}}{c^{3}} B_{ij}.</math>
The <math>A_{ij}</math> and <math>B_{ij}</math> are collectively known as the ''Einstein coefficients''.<ref>{{cite book |last1=Wilson |first1=J. |title=Lasers: Principles and Applications |last2=Hawkes |first2=F. J. B. |publisher=Prentice Hall |year=1987 |isbn=978-0-13-523705-2 |location=New York |at=Section 1.4 |language=en-us}}</ref>

Einstein could not fully justify his rate equations, but claimed that it should be possible to calculate the coefficients <math>A_{ij}</math>, <math>B_{ji}</math> and <math>B_{ij}</math> once physicists had obtained "mechanics and electrodynamics modified to accommodate the quantum hypothesis".<ref>{{cite journal |last=Einstein |first=Albert |author-link=Albert Einstein |year=1916 |title=Strahlungs-emission und -absorption nach der Quantentheorie |journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]] |language=de |volume=18 |pages=318–323 |bibcode=1916DPhyG..18..318E |quote=p. 322: Die Konstanten <math>A^n_m</math> and <math>B^n_m</math> würden sich direkt berechnen lassen, wenn wir im Besitz einer im Sinne der Quantenhypothese modifizierten Elektrodynamik und Mechanik wären."}}</ref> Not long thereafter, in 1926, [[Paul Dirac]] derived the <math>B_{ij}</math> rate constants by using a semiclassical approach,<ref name="Dirac1926">{{cite journal |last=Dirac |first=Paul A. M. |author-link=Paul Dirac |year=1926 |title=On the Theory of Quantum Mechanics |journal=Proceedings of the Royal Society A |volume=112 |issue=762 |pages=661–677 |bibcode=1926RSPSA.112..661D |doi=10.1098/rspa.1926.0133 |doi-access=free}}</ref> and, in 1927, succeeded in deriving ''all'' the rate constants from first principles within the framework of quantum theory.<ref name="Dirac1927a">{{cite journal |last=Dirac |first=Paul A. M. |author-link=Paul Dirac |year=1927 |title=The Quantum Theory of the Emission and Absorption of Radiation |journal=Proceedings of the Royal Society A |volume=114 |issue=767 |pages=243–265 |bibcode=1927RSPSA.114..243D |doi=10.1098/rspa.1927.0039 |doi-access=free}}</ref><ref name="Dirac1927b">
{{cite journal |last=Dirac |first=Paul A. M. |author-link=Paul Dirac |year=1927b |title=The Quantum Theory of Dispersion |journal=Proceedings of the Royal Society A |volume=114 |issue=769 |pages=710–728 |bibcode=1927RSPSA.114..710D |doi=10.1098/rspa.1927.0071 |doi-access=free}}</ref> 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]];<ref name="Heisenberg1929">
{{cite journal |last1=Heisenberg |first1=Werner |author-link=Werner Heisenberg |last2=Pauli |first2=Wolfgang |author-link2=Wolfgang Pauli |year=1929 |title=Zur Quantentheorie der Wellenfelder |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=56 |issue=1–2 |page=1 |bibcode=1929ZPhy...56....1H |doi=10.1007/BF01340129 |s2cid=121928597}}</ref><ref name="Heisenberg1930">
{{cite journal |last1=Heisenberg |first1=Werner |author-link=Werner Heisenberg |last2=Pauli |first2=Wolfgang |author-link2=Wolfgang Pauli |year=1930 |title=Zur Quantentheorie der Wellenfelder |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=59 |issue=3–4 |page=139 |bibcode=1930ZPhy...59..168H |doi=10.1007/BF01341423 |s2cid=186219228}}</ref><ref name="Fermi1932">{{cite journal |last=Fermi |first=Enrico |author-link=Enrico Fermi |year=1932 |title=Quantum Theory of Radiation |journal=[[Reviews of Modern Physics]] |volume=4 |issue=1 |page=87 |bibcode=1932RvMP....4...87F |doi=10.1103/RevModPhys.4.87}}</ref> 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 [[Isaac Newton|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.<ref name="Newton1730" /> Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation<ref name="Pais1982" /> from quantum mechanics. Ironically, [[Max Born]]'s [[probability amplitude|probabilistic interpretation]] of the [[wave function]]<ref name="Born1926a">
{{cite journal |last=Born |first=Max |author-link=Max Born |year=1926 |title=Zur Quantenmechanik der Stossvorgänge |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=37 |issue=12 |pages=863–867 |bibcode=1926ZPhy...37..863B |doi=10.1007/BF01397477 |s2cid=119896026}}</ref><ref name="Born1926b">{{cite journal |last=Born |first=Max |author-link=Max Born |year=1926 |title=Quantenmechanik der Stossvorgänge |journal=[[European Physical Journal|Zeitschrift für Physik]] |language=de |volume=38 |issue=11–12 |page=803 |bibcode=1926ZPhy...38..803B |doi=10.1007/BF01397184 |s2cid=126244962}}</ref> was inspired by Einstein's later work searching for a more complete theory.<ref name="ghost_field">{{cite book|last=Pais|first=A.|author-link=Abraham Pais|year=1986|url={{google books |plainurl=y |id=mREnwpAqz-YC|page=260}}|page=260|title=Inward Bound: Of Matter and Forces in the Physical World|publisher=Oxford University Press|isbn=978-0-19-851997-3}} Specifically, Born claimed to have been inspired by Einstein's never-published attempts to develop a "ghost-field" theory, in which point-like photons are guided probabilistically by ghost fields that follow Maxwell's equations.</ref>