D.42 De­riva­tion of the Ein­stein A co­ef­fi­cients

Ein­stein did not re­ally de­rive the spon­ta­neous emis­sion rate from rel­a­tivis­tic quan­tum me­chan­ics. That did not ex­ist at the time. In­stead Ein­stein used a dirty trick; he peeked at the so­lu­tion.

To see how, con­sider a sys­tem of iden­ti­cal atoms that can be in a low en­ergy state $\psi_{\rm {L}}$ or in an ex­cited en­ergy state $\psi_{\rm {H}}$. The frac­tion of atoms in the low en­ergy state is $P_{\rm {L}}$ and the frac­tion in the ex­cited en­ergy state is $P_{\rm {H}}$. Ein­stein as­sumed that the frac­tion $P_{\rm {H}}$ of ex­cited atoms would evolve ac­cord­ing to the equa­tion

$$\frac{{\rm d}P_{\rm {H}}}{{\rm d}t} = B_{\rm {L\to{H}}} \r... ...rho(\omega_0)\; P_{\rm {H}} - A_{\rm {H\to{L}}}\; P_{\rm {H}}$$

where $\rho(\omega)$ is the am­bi­ent elec­tro­mag­netic field en­ergy den­sity, $\omega_0$ the fre­quency of the pho­ton emit­ted in a tran­si­tion from the high to the low en­ergy state, and the $A$ and $B$ val­ues are con­stants. This as­sump­tion agrees with the ex­pres­sion (7.46) given in chap­ter 7.8.

Then Ein­stein de­manded that in an equi­lib­rium sit­u­a­tion, in which $P_{\rm {H}}$ is in­de­pen­dent of time, the for­mula must agree with Planck’s for­mula for the black­body elec­tro­mag­netic ra­di­a­tion en­ergy. The equi­lib­rium ver­sion of the for­mula above gives the en­ergy den­sity as

$$\rho(\omega_0) = \frac{A_{\rm {H\to{L}}}/B_{\rm {H\to{L}}}... ..._{\rm {L\to{H}}}P_{\rm {L}}/B_{\rm {H\to{L}}}P_{\rm {H}}) - 1}$$

Equat­ing this to Planck’s black­body spec­trum as de­rived in chap­ter 6.8 (6.11) gives

$$\frac{A_{\rm {H\to{L}}}/B_{\rm {H\to{L}}}} {(B_{\rm {L\to{... ...{\pi^2c^3} \frac{\omega_0^3}{e^{\hbar\omega_0/{k_{\rm B}}T}-1}$$

The atoms can be mod­eled as dis­tin­guish­able par­ti­cles. There­fore the ra­tio $P_{\rm {H}}$$\raisebox{.5pt}{$/$}$​$P_{\rm {L}}$ can be found from the Maxwell-Boltz­mann for­mula of chap­ter 6.14; that gives the ra­tio as $e^{-(E_{\rm {H}}-E_{\rm {L}})/{k_{\rm B}}T}$, or $e^{-\hbar\omega_0/{k_{\rm B}}T}$ in terms of the pho­ton fre­quency. It then fol­lows that for the two ex­pres­sions for $\rho(\omega_0)$ to be equal,

$$B_{\rm {L\to{H}}} = B_{\rm {H\to{L}}} \qquad \frac{A_{\rm... ...to{L}}}}{B_{\rm {H\to{L}}}} = \frac{\hbar\omega_0^3}{\pi^2c^3}$$

That $B_{\rm {L\to{H}}}$ must equal $B_{\rm {H\to{L}}}$ is a con­se­quence of the sym­me­try prop­erty men­tioned at the end of chap­ter 7.7.2. But it was not self-ev­i­dent when Ein­stein wrote the pa­per; Ein­stein re­ally in­vented stim­u­lated emis­sion here.

The valu­able re­sult for this book is the for­mula for the spon­ta­neous emis­sion rate $A_{\rm {H\to{L}}}$. With $B_{\rm {H\to{L}}}$ given by (7.47), it de­ter­mines the spon­ta­neous emis­sion rate. So it has been ob­tained with­out us­ing rel­a­tivis­tic quan­tum me­chan­ics. (Or at least not ex­plic­itly; there sim­ply are no non­rel­a­tivis­tic pho­tons.)