## ‘Strahlungs-Emission und Absorption nach der Quantentheorie’, pp. 318-323 in: Verhandlungen der Deutschen Physikalischen Gesellschaft, Jahrg. 18, Nr. 13/14, 30 July 1916.

Braunschweig: Druck und Verlag von Friedr. Vieweg and Son, 1916.

First edition, complete journal issue in original printed wrappers, **inscribed by Einstein to fellow Nobel Laureate Walther Bothe**. “This work represents a major step forward in quantum theory” (Calaprice, p. 297). It introduced the concept of stimulated emission of radiation, the theoretical basis for the laser; it also contained a new derivation of Planck’s radiation law which provided, as a by-product, a justification of the frequency rule forming the basis of Bohr’s theory of atomic spectra. “According to Albert Einstein, when more atoms occupy a higher energy state than a lower one under normal temperature equilibrium, it is possible to force atoms to return to an unexcited state by stimulating them with the same energy as would be emitted naturally” (Britannica). This is ‘stimulated emission.’ “To claim that Einstein almost invented the laser would be an exaggeration, but the laser’s underlying mechanism, stimulated emission of radiation, was a creation of his radiation theory” (Kleppner, pp. 32-33). “During the summer of 1916, less than a year after he had completed the general theory of relativity, Einstein made a new, major contribution to the quantum theory. The two papers he wrote then deal with the quantum theory of radiation by arguments that do not depend on the classical electromagnetic theory, as had all earlier treatments of Planck's radiation law … When Einstein returned to the radiation problem in 1916, the quantum theory had undergone a major change. Niels Bohr’s papers had opened a new and fertile domain for the application of quantum concepts – the explanation of atomic structure and atomic spectra. In addition, Bohr's work and its generalizations by Arnold Sommerfeld and others constituted a fresh approach to the foundations of the quantum theory of matter" (*DSB*). In this paper, “Einstein considers a system of atoms in equilibrium with an external radiation field. An atom can change its internal energy state by absorbing or emitting radiation. Einstein introduces three basic assumptions about these exchanges of energy between matter and field. First, the probability of absorption of radiation is proportional to the radiation density. Second, there are two kinds of emission processes: one – spontaneous – following a law like that of radioactive decay; the other – stimulated – induced by the radiation field and with probability proportional to the radiation density. Third, at equilibrium the atoms are distributed among their internal states according to the Boltzmann distribution law. From these assumptions Planck's law follows in a simple way. Einstein was very pleased with his derivation, which he characterized in a letter to Besso: ‘An amazingly simple derivation of Planck's formula, I should like to say *the* derivation.’ As a bonus from his derivation Einstein found that the energy difference between two internal energy states of the atom had to be equal to *hv*, with *v* the frequency of the radiation absorbed or emitted in transitions between these two states, thus confirming one of the postulates of Niels Bohr's theory of spectra” (*Papers* 6, xxiii-xxiv). “Einstein meant the second part of this study, a proof of the oriented character of the emission process, to be his most essential contribution to quantum radiation theory [this second paper was published later in 1916 as ‘Zür Quantentheorie der Strahlung’]. Instead, Bohr gave more importance to the new deduction of the blackbody law; for this deduction reinforced the basic assumptions of his atomic theory and completed them with a statistical description of radiation processes” (Darrigol, p. 120).

*Provenance*: Inscribed by Einstein on front wrapper, “f[ür]. Dr Bothe”, i.e., Walther Bothe (1891-1957). “In 1929, in collaboration with W. Kolhörster, Bothe introduced a new method for the study of cosmic and ultraviolet rays by passing them through suitably arranged Geiger counters, and by this method demonstrated the presence of penetrating charged particles in the rays and defined the paths of individual rays. For his discovery of the ‘method of coincidence’ and the discoveries subsequently made by it, which laid the foundations of nuclear spectroscopy, Bothe was awarded, jointly with Max Born, the Nobel Prize in Physics 1954” (nobel.org).

While Einstein commended Planck’s epoch-making derivation of his radiation law in 1900, which ushered in the quantum era, he had also noted its limitations. Einstein also saw inconsistencies in Planck’s derivation of his law. For Einstein this inconsistency was no reason to reject Planck’s quantum theory, but it was a reason to study the foundations of traditional radiation theory and if needed, revise them.

“As [Einstein] had noted in 1906, Planck’s derivation of the [Rayleigh-Jeans law]

*u _{ν}* = (8π

*ν*

^{2}/

*c*

^{3})

*kT*

between average resonator energy [*u _{ν}*] and radiation spectrum [ν] only applied to classical resonators [

*T*is the temperature,

*k*is Boltzmann’s constant]. A new, quantum-theoretical picture of the interaction between matter and radiation was needed. Einstein found it in the summer of 1916, after the completion of his general theory of gravitation left him more time for quantum meditation.

“The new picture presumably emerged from a combination of three elements: Einstein’s derivation of the law of photochemical equivalence, his analogy between quantum states and chemical species, and Niels Bohr’s theory of atomic spectra. According to Bohr, atoms and molecules can only exist in a series of quantum states S_{0}, S_{1}, . . . S_{n}, . . . with well-defined energies *E*_{0}, *E*_{1}, . . . *E _{n}*, . . . Their interaction with radiation occurs through quantum jumps with characteristic values of the frequency of the emitted or absorbed radiation. Regarding the quantum states as chemical species and remembering his photochemical reasoning, Einstein knew that he could derive Wien’s law by balancing the absorption process S

*+*

_{n}*hν*→ S

_{n}_{+1 }with the emission process S

_{n}_{+1 }→ S

*+*

_{n}*hν*and by making the probability of the first reaction proportional to the density of radiation at frequency

*ν*. Something in this reasoning needed to be altered in order to get Planck’s law instead of Wien’s.

“At this point Einstein appealed to an analogy between classical and quantum theory. According to classical theory, an oscillating dipole spontaneously emits radiation, whether or not radiation is initially present in its surroundings. When external radiation encounters this dipole, it may either be absorbed if the phase of the incoming wave agrees with that of the oscillator, or it may be amplified in the contrary case. In the quantum theory of radiation, Einstein similarly admitted the existence of three kinds of processes: spontaneous emission (*Ausstrahlung*), absorption (*negative Einstrahlung*), and stimulated emission (*positive Einstrahlung*). The modern terminology is Bohr’s. For the probability per time unit of the respective sorts of quantum jump, Einstein assumed the forms

*A _{n}^{m}*,

*ρ*,

_{ν}B_{n}^{m}*ρ*,

_{ν}B_{m}^{n}where *n* is the upper quantum state, *m* the lower one, and *ρ _{ν}* is the density of radiation at the frequency

*ν*.

“Einstein did not say much on the nature of the probabilities he thus introduced. He only commented that his theory had the weakness to leave to chance the instant and direction of the spontaneous emission of light. He also noted the similarity between spontaneous emission and radioactive decay. Undoubtedly, he would have preferred a theory in which the emission and absorption probabilities were deduced from an underlying deterministic theory. He nonetheless expressed his ‘full trust in the present way of reasoning’. The probabilistic description of the interaction was a natural counterpart of the discrete character of quantum states: if a quantum system evolves mostly through quantum jumps, then the probability of a quantum jump obviously is the main quantity of physical interest. Instead of speculating on the precise timing and fine structure of the jumps, Einstein proceeded to show what could be done by means of the new probability coefficients.

“At thermal equilibrium, Einstein reasoned, statistical mechanics requires the number of atoms in a quantum state *n* to be proportional to exp(−*E _{n }/kT*). The kinetic equilibrium between the atoms and surrounding radiation further requires that the number of quantum jumps from

*m*to

*n*should be equal to the number of reverse jumps:

*ρ _{ν}B_{n}^{m}* exp(−

*E*) = [

_{m }/kT*ρ*] exp(−

_{ν}B_{n}^{m }+ A_{n}^{m}*E*).

_{n }/kTIn the high temperature limit for which *ρ _{ν}* → ∞, this condition gives

*B _{n}^{m}* =

*B*.

_{m}^{n}Therefore, the equilibrium value *u _{ν}* of the density

*ρ*[is given by]

_{ν}*u _{ν}*[exp((

*E*−

_{n}*E*) – 1] =

_{m})/kT*A*/

_{n}^{m}*B*.

_{n}^{m}According to a thermodynamic theorem by Wien, *u _{ν}/ν*

^{3}must be a function of

*ν/T*only. Hence

*E*must be proportional to

_{n}− E_{m}*ν*. Einstein thus derived Bohr’s strange frequency rule

*ΔE = hν*with complete generality and without recourse to any of the empirical laws of spectra. He then required the expression of

*u*to agree with the Rayleigh-Jeans law in the low-frequency limit. The outcome was Planck’s law, as well as the relation

_{ν}*A _{n}^{m}* /

*B*= 8π

_{n}^{m}*hν*

^{3}/

*c*

^{3}

between Einstein’s two probability coefficients …

“Einstein’s new theory of radiation is now remembered for the introduction of stimulated emission, which famously permitted the conception of masers and lasers. For Einstein and for his contemporaries, the importance of these memoirs lay elsewhere. First, Einstein filled an important gap in the derivation of Planck’s law by means of a simple, statistical description of radiation processes. Second, he corroborated two basic assumptions of Bohr’s atomic theory: the existence of stationary states and the frequency rule. In this regard, it should be emphasized that before Einstein’s and Sommerfeld’s contributions of 1916, Bohr believed that his frequency rule only applied to strictly periodic systems. For instance, he regarded the Zeeman effect as a violation of this rule. Einstein’s new considerations established its complete generality” (Darrigol, in *Cambridge Companion to Einstein*, pp. 134-136).

“The implication [of Einstein’s theory of stimulated emission] was that, if one arranges for a large number of atoms to be in identical excited states, a stray photon of the right energy can stimulate one atom to emit another photon, which stimulates another… and all the atoms release their excess energy in a sudden cascade. What’s more, the photon released by stimulated emission will be in phase – coherent – with the one that stimulated it, and so all the light produced in the cascade will be coherent.

“In 1955 American physicist Charles Townes of Columbia University in New York, an expert in molecular spectroscopy, and his co-workers showed how stimulated emission could be used to make a device for generating or amplifying microwaves, which they called a maser (microwave amplified stimulated emission of radiation). Three years later Townes and Arthur Schawlow explained how to extend the idea to visible and infrared frequencies to make an ‘optical maser’ – in effect, the laser.

“They proposed using ordinary (incoherent) light to pump atoms into an excited state, setting up the ‘population inversion’ in which the atoms are primed to return to their ground state by emitting photons. And their design used an optical cavity – basically two mirrors between which photons would bounce – to trap the emitted photons while they stimulated more emission. The device, they explained, would generate ‘extremely monochromatic [single-wavelength] and coherent light’. Theodore Maiman of the Hughes Research Laboratories in Malibu, California, described such a device, using a ruby crystal (already used for masers) as the lasing medium, in 1960” (‘A century ago Einstein sparked the notion of the laser,’ *Physics World* History Blog, 31 August 2017).

Weil *85. Calaprice, *An Einstein Encyclopedia*, 2015. Darrigol, *From c-numbers to q-numbers*, 1992. Kleppner, ‘Rereading Einstein on radiation,’ Physics Today 58 (2005), pp. 30-33. Pais, Subtle is the Lord, 1982.

8vo (228 x 154 mm), pp. [315]-332. Original printed wrappers. A fine copy.

Item #6164

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Price:
$25,000.00
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