Einstein in 1916: "On the Quantum Theory of Radiation"

TL;DR

This paper examines Einstein's 1916 work on quantum radiation, highlighting its significance in advancing the understanding of radiation's quantum nature and its impact on quantum theory development.

Contribution

It analyzes Einstein's pioneering 1916 paper, emphasizing its novel approach to understanding radiation through quantum theory.

Findings

Einstein's 1916 paper marked a major step in quantum theory development.

The work provided key insights into the quantum nature of radiation.

It influenced subsequent research in quantum physics.

Abstract

In this contribution I discuss a peak in Einstein's endeavor to extract as much information as possible about the nature of radiation from the Planck distribution is his paper "On the Quantum Theory of Radiation" of 1916. This is one of the most important contributions of Einstein to quantum theory.

Equations

$$⟨(Mv-Rv\tau +\Delta {)}^2⟩=⟨(Mv{)}^2⟩.$$

$$⟨{\Delta }^2⟩=2R⟨M{v}^2⟩\tau =2RkT\tau .$$

$$S:=g_n{e}^{-E_n/kT}+g_m{e}^{-E_m/kT}+\cdot \cdot \cdot$$

$$\frac{1}{S}{g}_n{e}^{-E_n/kT}{B}_m^n{\rho }_{{\nu }_0}^{\prime }\frac{d{\Omega }^{\prime }}{4\pi },{\nu }_0=\frac{E_m-E_n}{h},$$

$$\frac{1}{S}{g}_m{e}^{-E_m/kT}{B}_n^m{\rho }_{{\nu }_0}^{\prime }\frac{d{\Omega }^{\prime }}{4\pi }.$$

$$-Rv=\frac{h{\nu }_0}{c}\frac{1}{S}{g}_n{B}_m^n\left(e^{-E_n/kT}-e^{-E_m/kT}\right)\times \int {\rho }_{{\nu }_0}^{\prime }({\theta }^{\prime },{\phi }^{\prime })\cos {\theta }^{\prime }\frac{d{\Omega }^{\prime }}{4\pi }.$$

$$\frac{{\rho }_{{\nu }^{\prime }}^{\prime }}{{\nu }^{\prime 3}}=\frac{{\rho }_{\nu }}{{\nu }^3}.$$

$${\rho }_{{\nu }^{\prime }}^{\prime }\simeq {\left(1+\frac{v}{c}\cos {\theta }^{\prime }\right)}^{-3}{\rho }_{[1+\frac{v}{c}\cos {\theta }^{\prime }]{\nu }^{\prime }},$$

$${\rho }_{{\nu }_0}^{\prime }$$

$$\int {\rho }_{{\nu }_0}^{\prime }\cos {\theta }^{\prime }\frac{d{\Omega }^{\prime }}{4\pi }=-\frac{v}{c}\left[{\rho }_{{\nu }_0}-\frac{1}{3}{\nu }_0{\left(\frac{\partial {\rho }_{\nu }}{\partial \nu }\right)}_{{\nu }_0}\right]=\frac{v}{c}{\left\{\frac{{\nu }^4}{3}\frac{\partial }{\partial \nu }\left(\frac{{\rho }_{\nu }}{{\nu }^3}\right)\right\}}_{\nu ={\nu }_0}.$$

$$R=\frac{h\nu }{c^2}\frac{1}{S}{g}_n{B}_m^n{e}^{-E_n/kT}\left(1-e^{-h\nu /kT}\right)\times \left[-\frac{{\nu }^4}{3}\frac{\partial }{\partial \nu }\left(\frac{{\rho }_{\nu }}{{\nu }^3}\right)\right].$$

$$R=\frac{1}{3}{\left(\frac{h\nu }{c}\right)}^2\frac{1}{kT}\frac{g_n}{S}{e}^{-E_n/kT}{B}_m^n{\rho }_{\nu }.$$

$$R=\frac{1}{3}{\left(\frac{h\nu }{c}\right)}^2\frac{1}{2kT}Z.$$

$$⟨{\Delta }^2⟩={\left(\frac{h\nu }{c}\right)}^2⟨{\cos }^2\vartheta {⟩}_{S^2}Z\tau .$$

Full text

Einstein in 1916: “On the Quantum Theory of Radiation”

Norbert Straumann, Uni Zürich

Introduction

A peak in Einstein’s endeavor to extract as much information as possible about the nature of radiation from the Planck distribution is his paper “On the Quantum Theory of Radiation” of 1916 [1].111The paper was first published in: Physikalische Gesellschaft Zürich. Mitteilungen 18 (1916): 47-62. The same paper was received 3 March and published 15 March 1917 in: Physikalische Zeitschrift 18 (1917): 121-128. In the first part he gives a derivation of Planck’s formula which has become part of many textbooks on quantum theory.

This part of the paper is now considered as the theoretical foundation of the laser, that was technically realized almost half a century later.

In the second part of his fundamental paper, Einstein discusses the exchange of momentum between atoms and radiation by making use of the theory of Brownian motion. Using a truly beautiful argument, which will be presented below, he shows that in every elementary process of radiation, and in particular in spontaneous emission, an amount $h\nu/c$ of momentum is emitted in a random direction and that the atomic system suffers a corresponding recoil in the opposite direction.

This recoil was first experimentally confirmed in 1933 by showing that a long and narrow beam of excited sodium atoms widens up after spontaneous emissions have taken place [2]. Einstein’s paper [1] ends with the following remarkable statement concerning the role of “chance” in his description of the radiation processes by statistical laws, to which Pauli [3] drew special attention:

“The weakness of the theory lies, on the one hand, in the fact that it does not bring us any closer to a merger with the undulatory theory, and, on the other hand, in the fact that it leaves the time and direction of elementary processes to ‘chance’; in spite of this I harbor full confidence in the trustworthiness of the path entered upon.”

Derivation of the Planck distribution

In his novel derivation of the Planck distribution, Einstein added the hitherto unknown process of induced emission222Einstein’s derivation shows that without assuming a non-zero probability for induced emission one would necessarily arrive at Wien’s instead of Planck’s radiation law., next to the familiar processes of spontaneous emission and induced absorption. For each pair of energy levels he described the statistical laws for these processes by three coefficients (the famous $A$- and $B$-coefficients) and established two relations between these coefficients on the basis of his earlier correspondence argument in the classical Rayleigh-Jeans limit and Wien’s displacement law. In addition, the latter implies that the energy difference $\varepsilon_{n}-\varepsilon_{m}$ between two internal energy states of the atoms in equilibrium with thermal radiation has to satisfy Bohr’s frequency condition: $\varepsilon_{n}-\varepsilon_{m}=h\nu_{nm}$.

Einstein was very pleased by this derivation, about which he wrote on August 11, 1916 to Besso: “An amazingly simple derivation of Planck’s formula, I should like to say the derivation” (CPAE, Vol. 8, Doc. 250).

In Dirac’s 1927 radiation theory these results follow – without any correspondence arguments – from first principles.

Brownian motion of an atom in the thermodynamic radiation field

In the second part, Einstein regarded in his paper as more important, he investigates the exchange of momentum between atoms and radiation by studying the Brownian motion of molecules in the thermodynamic radiation field. A molecule experiences thereby two types of forces:

$\bullet$ A systematic drag force $Rv$, where $v$ is the velocity of the molecule, that leads in a small time interval $(t,t+\tau)$ to the momentum change $Rv\tau$.

$\bullet$ An irregular change of momentum $\Delta$ in the time $\tau$, due to fluctuations of the radiation pressure.

In thermal equilibrium the following equation holds:

[TABLE]

Assuming that $\langle v\cdot\Delta\rangle=0$ we obtain the important relation, repeatedly used by Einstein in his earlier work on Brownian motion,

[TABLE]

Calculation of the friction $R$.

The computation of $R$ is not simple. For interested readers we present the details of Einstein’s instructive calculation, thereby including some flesh of his profound investigation. (Others may prefer to jump directly to the simple result, given in equation (3) below.)

The moving atom (molecule) sees an anisotropic radiation field. The friction force is the result of absorption and induced emission processes333The spontaneous emission has in the average no preferred direction, and therefore does in the average not transmit momentum to the atom.. Let $K$ be the rest system of the radiation field, with respect to which the radiation intensity is isotropic. With $K^{\prime}$ we denote the rest system of the atom. Let $S$ denote the partition sum

[TABLE]

for the energies $E_{n}$ and degeneracies $g_{n}$ of the atom. The fraction of time in the state $n$ is equal to $g_{n}e^{-E_{n}/kT}/S$, and similarly for the state $m$. Therefore, the number of absorptions $n\rightarrow m$ per unit time into the solid angle $d\Omega^{\prime}$ is equal to

[TABLE]

where $\rho_{\nu}$ denotes the energy density of the radiation as a function of frequency $\nu$. Correspondingly, for induced emission $m\rightarrow n$ the number is

[TABLE]

With Einstein’s relation $g_{m}B^{m}_{n}=g_{n}B^{n}_{m}$ from the first part of the paper, the momentum transfer per unit time in the $x$-direction is

[TABLE]

Calculation of $\rho_{\nu_{0}}(\theta^{\prime},\varphi^{\prime})$.

With elementary calculations one can prove the following Lorentz invariance

[TABLE]

(This follows more directly from the fact that $\rho(\nu)$ is proportional to $\nu^{3}$ times the Lorentz invariant distribution function $f$.) Using also the Doppler shift to first order in $v/c$: $\nu=\nu^{\prime}\left(1+\frac{v}{c}\cos\theta^{\prime}\right)$. This gives

[TABLE]

thus

[TABLE]

To first order in $v/c$ we obtain

[TABLE]

Writing $\nu$ instead of $\nu_{0}$, we arrive at

[TABLE]

For the Planck distribution we obtain

[TABLE]

Note that $(g_{n}/S)\exp(-E_{n}/kT)B^{n}_{m}\rho_{\nu}$ is the number of absorptions per unit time. Since this is equal to 1/2 times the number $Z$ of elementary processes (induced and spontaneous emissions plus absorptions) per unit time, we find for the friction force the simple result:

[TABLE]

Interpretation

Using this in (1) we arrive at the crucial relation

[TABLE]

From classical electrodynamics (and experience) we know that to a pencil of light with energy $\varepsilon$ belongs a momentum $\varepsilon/c$ in the direction of the light beam. From this Einstein concluded that the atom receives in an absorption process the momentum $(E_{m}-E_{n})/c$ in the direction of the infalling radiation, and in an induced emission process the same momentum in the opposite direction. The crucial relation (4) then implies that spontaneous emission must also be directed in such a way that for every elementary process of radiation an amount $h\nu/c$ of momentum is emitted in a random direction, and that the atomic system suffers a corresponding recoil in the opposite direction. “There is no radiation in spherical waves”444“Ausstrahlung in Kugelwellen gibt es nicht”., Einstein adds.

Einstein regarded this conclusion as the main result of his paper. In a letter to Michele Besso he states:

“With this, the existence of light-quanta is practically assured.”555“Damit sind die Lichtquanten so gut wie gesichert” (6. September 1916).

Two years later he added:

“I do not doubt anymore the reality of radiation quanta, although I still stand quite alone in this conviction.”

We conclude with the following interesting remark of Einstein to O. Stern:

“Ich habe hundertmal mehr über Quantenprobleme nachgedacht, als über die allgemeine Relativitätstheorie.”