D.41 De­riva­tion of the Ein­stein B co­ef­fi­cients

The pur­pose of this note is to de­rive the Ein­stein $B$ co­ef­fi­cients of chap­ter 7.8. They de­ter­mine the tran­si­tion rates be­tween the en­ergy states of atoms. For sim­plic­ity it will be as­sumed that there are just two atomic en­ergy eigen­states in­volved, a lower en­ergy one $\psi_{\rm {L}}$ and an higher en­ergy one $\psi_{\rm {H}}$. It is fur­ther as­sumed that the atoms are sub­ject to in­co­her­ent am­bi­ent elec­tro­mag­netic ra­di­a­tion. The en­ergy in the am­bi­ent ra­di­a­tion is $\rho(\omega)$ per unit vol­ume and unit fre­quency range. Fi­nally it is as­sumed that the atoms suf­fer fre­quent col­li­sions with other atoms. The typ­i­cal time be­tween col­li­sions will be in­di­cated by $t_{\rm {c}}$. It is small com­pared to the typ­i­cal de­cay time of the states, but large com­pared to the fre­quency of the rel­e­vant elec­tro­mag­netic field.

Un­like what you may find else­where, it will not be as­sumed that the atoms are ei­ther fully in the high or fully in the low en­ergy state. That is a highly un­sat­is­fac­tory as­sump­tion for many rea­sons. For one thing it as­sumes that the atoms know what you have se­lected as $z$-​axis. In the de­riva­tion be­low, the atoms are al­lowed to be in a lin­ear com­bi­na­tion of the states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$, with co­ef­fi­cients $c_{\rm {L}}$ and $c_{\rm {H}}$.

Since both the elec­tro­mag­netic field and the col­li­sions are ran­dom, a sta­tis­ti­cal rather than a de­ter­mi­nate treat­ment is needed. In it, the prob­a­bil­ity that a ran­domly cho­sen atom can be found in the lower en­ergy state $\psi_{\rm {L}}$ will be in­di­cated by $P_{\rm {L}}$. Sim­i­larly, the prob­a­bil­ity that an atom can be found in the higher en­ergy state $\psi_{\rm {H}}$ will be in­di­cated by $P_{\rm {H}}$. For a sin­gle atom, these prob­a­bil­i­ties are given by the square mag­ni­tudes of the co­ef­fi­cients $c_{\rm {L}}$ and $c_{\rm {H}}$ of the en­ergy states. There­fore, $P_{\rm {L}}$ and $P_{\rm {H}}$ will be de­fined as the av­er­ages of $\vert c_{\rm {L}}\vert^2$ re­spec­tively $\vert c_{\rm {H}}\vert^2$ over all atoms.

It is as­sumed that the col­li­sions are glob­ally elas­tic in the sense that they do not change the av­er­age en­ergy pic­ture of the atoms. In other words, they do not af­fect the av­er­age prob­a­bil­i­ties of the eigen­func­tions $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$. How­ever, they are as­sumed to leave the wave func­tion of an in­di­vid­ual atom im­me­di­ately af­ter a col­li­sion in some state $c_{\rm {L,0}}\psi_{\rm {L}}+c_{\rm {H,0}}\psi_{\rm {H}}$ in which $c_{\rm {L,0}}$ and $c_{\rm {H,0}}$ are quite ran­dom, es­pe­cially with re­spect to their phase. What is now to be de­ter­mined in this note is how, un­til the next col­li­sion, the wave func­tion of the atom will de­velop un­der the in­flu­ence of the elec­tro­mag­netic field and how that changes the av­er­age prob­a­bil­i­ties $\vert c_{\rm {L}}\vert^2$ and $\vert c_{\rm {H}}\vert^2$.

The evo­lu­tion equa­tions of the co­ef­fi­cients $\bar{c}_{\rm {L}}$ and $\bar{c}_{\rm {H}}$, in be­tween col­li­sions, were given in chap­ter 7.7.2 (7.42). They are in terms of mod­i­fied vari­ables $\bar{c}_{\rm {L}}$ and $\bar{c}_{\rm {H}}$. How­ever, these vari­ables have the same square mag­ni­tudes and ini­tial con­di­tions as $c_{\rm {L}}$ and $c_{\rm {H}}$. So it re­ally does not make a dif­fer­ence.

Fur­ther, be­cause the equa­tions are lin­ear, the so­lu­tion for the co­ef­fi­cients $\bar{c}_{\rm {L}}$ and $\bar{c}_{\rm {H}}$ can be writ­ten as a sum of two con­tri­bu­tions, one pro­por­tional to the ini­tial value $\bar{c}_{\rm {L,0}}$ and the other to $\bar{c}_{\rm {H,0}}$:

\begin{displaymath}
\bar c_{\rm {L}} = \bar{c}_{\rm {L,0}} \bar c_{\rm {L}}^{\r...
...}}^{\rm {L}}
+ \bar{c}_{\rm {H,0}} \bar c_{\rm {H}}^{\rm {H}}
\end{displaymath}

Here $(\bar{c}_{\rm {L}}^{\rm {L}},\bar{c}_{\rm {H}}^{\rm {L}})$ is the so­lu­tion that starts out from the lower en­ergy state $(\bar{c}_{\rm {L}}^{\rm {L}},\bar{c}_{\rm {H}}^{\rm {L}})$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(1,0)$ while $(\bar{c}_{\rm {L}}^{\rm {H}},\bar{c}_{\rm {H}}^{\rm {H}})$ is the so­lu­tion that starts out from the higher en­ergy state $(\bar{c}_{\rm {L}}^{\rm {H}},\bar{c}_{\rm {H}}^{\rm {H}})$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(0,1)$.

Now con­sider what hap­pens to the prob­a­bil­ity of an atom to be in the ex­cited state in the time in­ter­val be­tween col­li­sions:

\begin{displaymath}
\vert\bar{c}_{\rm {H}}\vert^2 - \vert\bar{c}_{\rm {H,0}}\ve...
...rm {H}}^{\rm {H}})
- \bar{c}_{\rm {H,0}}^*\bar{c}_{\rm {H,0}}
\end{displaymath}

Here $\Delta{\bar{c}_{\rm {H}}^{\rm {L}}}$ in­di­cates the change in $\bar{c}_{\rm {H}}^{\rm {L}}$ in the time in­ter­val be­tween col­li­sions; in par­tic­u­lar $\Delta{\bar{c}_{\rm {H}}^{\rm {L}}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\bar{c}_{\rm {H}}^{\rm {L}}$ since this so­lu­tion starts from the ground state with $\bar{c}_{\rm {H}}^{\rm {L}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Sim­i­larly, the change $\Delta{\bar{c}_{\rm {H}}^{\rm {H}}}$ equals $\bar{c}_{\rm {H}}^{\rm {H}}-1$ since this so­lu­tion starts out from the ex­cited state with $\bar{c}_{\rm {H}}^{\rm {H}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.

Be­cause the typ­i­cal time be­tween col­li­sions $t_{\rm {c}}$ is as­sumed small, so will be the changes $\Delta{\bar{c}_{\rm {H}}^{\rm {L}}}$ and $\Delta{\bar{c}_{\rm {H}}^{\rm {H}}}$ as given by the evo­lu­tion equa­tions (7.42). Note also that $\Delta{\bar{c}_{\rm {H}}^{\rm {H}}}$ will be qua­drat­i­cally small, since the cor­re­spond­ing so­lu­tion starts out from $\bar{c}_{\rm {L}}^{\rm {H}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, so $\bar{c}_{\rm {L}}^{\rm {H}}$ is an ad­di­tional small fac­tor in the equa­tion (7.42) for $\bar{c}_{\rm {H}}^{\rm {H}}$.

There­fore, if the change in prob­a­bil­ity $\vert\bar{c}_{\rm {H}}\vert^2$ above is mul­ti­plied out, ig­nor­ing terms that are cu­bi­cally small or less, the re­sult is, (re­mem­ber that for a com­plex num­ber $c$, $c+c^*$ is twice its real part):

\begin{displaymath}
\vert\bar{c}_{\rm {H}}\vert^2 - \vert\bar{c}_{\rm {H,0}}\ve...
...0}}\vert^2 2 \Re\left(\Delta\bar{c}_{\rm {H}}^{\rm {H}}\right)
\end{displaymath}

Now if this is av­er­aged over all atoms and time in­ter­vals be­tween col­li­sions, the first term in the right hand side will av­er­age away. The rea­son is that it has a ran­dom phase an­gle, for one since those of $\bar{c}_{\rm {L,0}}$ and $\bar{c}_{\rm {H,0}}$ are as­sumed to be ran­dom af­ter a col­li­sion. For a num­ber with a ran­dom phase an­gle, the real part is just as likely to be pos­i­tive as neg­a­tive, so it av­er­ages away. Also, for the fi­nal term, $2\Re(\Delta\bar{c}_{\rm {H}}^{\rm {H}})$ is the ap­prox­i­mate change in $\vert\bar{c}_{\rm {H}}^{\rm {H}}\vert^2$ in the time in­ter­val, and that equals $-\vert\Delta\bar{c}_{\rm {L}}^{\rm {H}}\vert^2$ be­cause of the nor­mal­iza­tion con­di­tion $\vert\bar{c}_{\rm {L}}^{\rm {H}}\vert^2+\vert\bar{c}_{\rm {H}}^{\rm {H}}\vert^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1. So the rel­e­vant ex­pres­sion for the av­er­age change in prob­a­bil­ity be­comes

\begin{displaymath}
\vert\bar{c}_{\rm {H}}\vert^2 - \vert\bar{c}_{\rm {H,0}}\ve...
...m {H,0}}\vert^2 \vert\Delta\bar{c}_{\rm {L}}^{\rm {H}}\vert^2
\end{displaymath}

Sum­ming the changes in the prob­a­bil­i­ties there­fore means sum­ming the changes in the square mag­ni­tudes of $\Delta\bar{c}_{\rm {H}}^{\rm {L}}$ and $\Delta\bar{c}_{\rm {L}}^{\rm {H}}$.

If the above ex­pres­sion for the av­er­age change in the prob­a­bil­ity of the high en­ergy state is com­pared to (7.46), it is seen that the Ein­stein co­ef­fi­cient $B_{\rm {L\to{H}}}$ is the av­er­age change $\vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}\vert^2$ per unit time. This is ad­mit­tedly the same an­swer you would get if you as­sumed that the atoms are ei­ther in the low en­ergy state or in the high en­ergy state im­me­di­ately af­ter each col­li­sion. But as noted, that as­sump­tion is sim­ply not rea­son­able.

Now the needed $\Delta\bar{c}_{\rm {H}}^{\rm {L}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\bar{c}_{\rm {H}}^{\rm {L}}$ may be found from the sec­ond evo­lu­tion equa­tion (7.42). To do so, you can con­sider $\bar{c}_{\rm {L}}^{\rm {L}}$ to be 1. The rea­son is that it starts out as 1, and it never changes much be­cause of the as­sumed short evo­lu­tion time $t_{\rm {c}}$ com­pared to the typ­i­cal tran­si­tion time be­tween states. That al­lows $\bar{c}_{\rm {H}}^{\rm {L}}$ to be found from a sim­ple in­te­gra­tion. And the sec­ond term in the mod­i­fied Hamil­ton­ian co­ef­fi­cient (7.44) can be ig­nored be­cause of the ad­di­tional as­sump­tion that $t_{\rm {c}}$ is still large com­pared to the fre­quency of the elec­tro­mag­netic wave. That causes the ex­po­nen­tial in the sec­ond term to os­cil­late rapidly and it does not in­te­grate to a siz­able con­tri­bu­tion.

What is left is

\begin{displaymath}
\Delta\bar{c}_{\rm {H}}^{\rm {L}} = \frac{{\cal E}_{\rm {f}...
...frac{e^{-{\rm i}(\omega-\omega_0)t} - 1}{2(\omega-\omega_0)} %
\end{displaymath} (D.25)

and $\Delta\bar{c}_{\rm {L}}^{\rm {H}}$ is given by a vir­tu­ally iden­ti­cal ex­pres­sion. How­ever, since it is as­sumed that the atoms are sub­ject to in­co­her­ent ra­di­a­tion of all wave num­bers ${\vec k}$ and po­lar­iza­tions $p$, the com­plete $\Delta\bar{c}_{\rm {H}}^{\rm {L}}$ will con­sist of the sum of all their con­tri­bu­tions:

\begin{displaymath}
\Delta\bar{c}_{\rm {H}}^{\rm {L}}
= \sum_{{\vec k},p} \Delta\bar{c}_{\rm {H}}^{\rm {L}}({\vec k},p)
\end{displaymath}

(This re­ally as­sumes that the par­ti­cles are in a very large pe­ri­odic box so that the elec­tro­mag­netic field is given by a Fourier se­ries; in free space you would need to in­te­grate over the wave num­bers in­stead of sum over them.) The square mag­ni­tude is then

\begin{displaymath}
\vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}\vert^2 =
\sum_{{\ve...
...},p} \vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}({\vec k},p)\vert^2
\end{displaymath}

where the fi­nal equal­ity comes from the as­sump­tion that the ra­di­a­tion is in­co­her­ent, so that the phases of dif­fer­ent waves are un­cor­re­lated and the cor­re­spond­ing prod­ucts av­er­age to zero.

The bot­tom line is that square mag­ni­tudes must be summed to­gether to find the to­tal con­tri­bu­tion of all waves. And the square mag­ni­tude of the con­tri­bu­tion of a sin­gle wave is, ac­cord­ing to (D.25) above,

\begin{displaymath}
\vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}({\vec k},p)\vert^2 =...
...ight)}
{{\textstyle\frac{1}{2}}(\omega-\omega_0)t}
\right)^2
\end{displaymath}

Now broad­band ra­di­a­tion is de­scribed in terms of an elec­tro­mag­netic en­ergy den­sity $\rho(\omega)$; in par­tic­u­lar $\rho(\omega){\,\rm d}\omega$ gives the en­ergy per unit vol­ume due to the elec­tro­mag­netic waves in an in­fin­i­tes­i­mal fre­quency range ${\rm d}\omega$ around a fre­quency $\omega$. For a sin­gle wave, this en­ergy equals $\frac12\epsilon_0{\cal E}_{\rm {f}}^2$, chap­ter 13.2 (13.11). And the square am­pli­tudes of dif­fer­ent waves sim­ply add up to the to­tal en­ergy; that is the so-called Par­se­val equal­ity of Fourier analy­sis. So to sum the ex­pres­sion above over all the fre­quen­cies $\omega$ of the broad­band ra­di­a­tion, make the sub­sti­tu­tion ${\cal E}_{\rm {f}}^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\rho(\omega){\,\rm d}\omega$$\raisebox{.5pt}{$/$}$$\epsilon_0$ and in­te­grate:

\begin{displaymath}
\vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}\vert^2
=
\frac{\ve...
...}{2}}(\omega-\omega_0)t_{\rm {c}}}
\right)^2
{\,\rm d}\omega
\end{displaymath}

If a change of in­te­gra­tion vari­able is made to $u$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\textstyle\frac{1}{2}}(\omega-\omega_0)t_{\rm {c}}$, the in­te­gral be­comes

\begin{displaymath}
\vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}\vert^2
=
\frac{\ve...
...+2(u/t_{\rm {c}}))
\left(\frac{\sin u}{u}\right)^2 {\,\rm d}u
\end{displaymath}

Re­call that a start­ing as­sump­tion un­der­ly­ing these de­riva­tions was that $\omega_0t_{\rm {c}}$ was large. So the lower limit of in­te­gra­tion can be ap­prox­i­mated as $\vphantom{0}\raisebox{1.5pt}{$-$}$$\infty$.

Note that this is es­sen­tially the same analy­sis as the one for Fermi’s golden rule, ex­cept for the pres­ence of the given field strength $\rho$. How­ever, here the math­e­mat­ics can be per­formed more straight­for­wardly, us­ing in­te­gra­tion rather than sum­ma­tion.

Con­sider for a sec­ond the lim­it­ing process that the field strength $\rho$ goes to zero, and that the atom is kept iso­lated enough that the col­li­sion time $t_{\rm {c}}$ can in­crease cor­re­spond­ingly. Then the term $2u$$\raisebox{.5pt}{$/$}$$t_{\rm {c}}$ in the ar­gu­ment of $\rho$ will tend to zero. So only waves with the ex­act fre­quency $\omega$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\omega_0$ will pro­duce tran­si­tions in the limit of zero field strength. That con­firms the ba­sic claim of quan­tum me­chan­ics that only the en­ergy eigen­val­ues are mea­sur­able. In the ab­sence of an elec­tro­mag­netic field and other dis­tur­bances, the en­ergy eigen­val­ues are purely the atomic ones. (Also re­call that rel­a­tivis­tic quan­tum me­chan­ics adds that in re­al­ity, the elec­tric field is never zero.)

In any case, while the term $2u$$\raisebox{.5pt}{$/$}$$t_{\rm {c}}$ may not be ex­actly zero, it is cer­tainly small com­pared to $\omega_0$ be­cause of the as­sump­tion that $\omega_0t_{\rm {c}}$ is large. So the term may be ig­nored any­way. Then $\rho(\omega_0)$ is a con­stant in the in­te­gra­tion and can be taken out. The re­main­ing in­te­gral is in ta­ble books, [40, 18.36], and the re­sult is

\begin{displaymath}
\vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}\vert^2
=
\frac{\pi...
..._{\rm {H}}\rangle\vert^2}{\hbar^2\epsilon_0}
\rho(\omega_0) t
\end{displaymath}

This must still be av­er­aged over all di­rec­tions of wave prop­a­ga­tion and po­lar­iza­tion. That gives:

\begin{displaymath}
\vert\Delta\bar{c}_{\rm {H}}^{\rm {L}}\vert^2
=
\frac{\pi...
...ngle\vert^2}
{3\hbar^2\epsilon_0}
\rho(\omega_0) t_{\rm {c}}
\end{displaymath}

where

\begin{displaymath}
\vert\langle\psi_{\rm {L}}\vert e{\skew0\vec r}\vert\psi_{\...
...langle\psi_{\rm {L}}\vert ez\vert\psi_{\rm {H}}\rangle\vert^2.
\end{displaymath}

To see why, con­sider the elec­tro­mag­netic waves prop­a­gat­ing along any axis, not just the $y$-​axis, and po­lar­ized in ei­ther of the other two ax­ial di­rec­tions. These waves will in­clude $ex$ and $ey$ as well as $ez$ in the tran­si­tion prob­a­bil­ity, mak­ing the av­er­age as shown above. And of course, waves prop­a­gat­ing in an oblique rather than ax­ial di­rec­tion are sim­ply ax­ial waves when seen in a ro­tated co­or­di­nate sys­tem and pro­duce the same av­er­age.

The Ein­stein co­ef­fi­cient $B_{\rm {L\to{H}}}$ is the av­er­age change per unit time, so the claimed (7.47) re­sults from di­vid­ing by the time $t_{\rm {c}}$ be­tween col­li­sions. There is no need to do $B_{\rm {H\to{L}}}$ sep­a­rately from $\Delta\bar{c}_{\rm {L}}^{\rm {L}}$; it fol­lows im­me­di­ately from the sym­me­try prop­erty men­tioned at the end of chap­ter 7.7.2 that it is the same.