The purpose of this post is to detail the derivation of Fermi’s golden rule, which incidentally was actually first due to Paul Dirac, not Enrico Fermi (see Stigler’s law of eponymy). First, one has to develop the general formalism of the interaction picture for working with time-dependent Hamiltonians \(H(t)\).
Interaction Picture
In general, given a time-dependent state \(|\psi(t)\rangle\) in the Schrodinger picture, recall that if the Hamiltonian \(H\) is time-independent \(\partial H/\partial t=0\) then one can always eliminate the time-dependence in \(|\psi(t)\rangle\) by setting it equal to its initial \(t=0\) value via the “backwards in time” evolution operator \(|\psi(0)\rangle=e^{iHt/\hbar}|\psi(t)\rangle\). Then, to ensure that expectations of observables are unaffected \(\langle\psi(0)|A'(t)|\psi(0)\rangle=\langle\psi(t)|A|\psi(t)\rangle\), it follows that all observables must also transform via \(A'(t)=e^{iHt/\hbar}Ae^{-iHt/\hbar}\). This is of course the usual Heisenberg picture, and by differentiating \(A'(t)\) one recovers the usual Heisenberg equation of motion \(\dot{A’}=\frac{i}{\hbar}[H,A’]+(\dot A)’\).
However, when \(H=H(t)\) becomes time-dependent, then \(e^{-iH(t)t/\hbar}\) is no longer the correct time evolution operator, so one cannot just apply the naive Heisenberg picture transformation to states and observables. However, if one is able to decompose the Hamiltonian \(H(t)=\tilde H+\Delta\tilde H(t)\) into a well-understood, time-independent piece \(\tilde H\) and a hopefully small but time-dependent perturbation \(\Delta\tilde H(t)\), then one can basically filter out the uninteresting dynamics due to \(\tilde H\) to focus just on the effects of the perturbation \(\Delta\tilde H(t)\) by employing a Heisenberg-like construction, namely acting with the reverse time evolution operator as pertains only to the unperturbed, time-independent Hamiltonian \(\tilde H\). This defines the (still time-dependent) interaction picture state \(|\psi_I(t)\rangle\):
\[|\psi_I(t)\rangle=e^{i\tilde Ht/\hbar}|\psi(t)\rangle\]
where \(|\psi(t)\rangle\) is the Schrodinger picture state. Again, for the same reason as above observables must transform accordingly \(A_I(t)=e^{i\tilde Ht/\hbar}Ae^{-i\tilde Ht/\hbar}\). However, the interaction picture is roughly in between the Schrodinger and Heisenberg pictures; both \(|\psi_I(t)\rangle\) and \(A_I(t)\) bear some of the time-dependence. To clarify how each depends on time \(t\), one can differentiate both with respect to \(t\) to obtain analogs of the Schrodinger equation and Heisenberg’s equation of motion, but now all within the interaction picture:
\[i\hbar\frac{\partial|\psi_I(t)\rangle}{\partial t}=\Delta\tilde H_I(t)|\psi_I(t)\rangle\]
\[\frac{dA_I(t)}{dt}=\frac{i}{\hbar}[\tilde H,A_I(t)]+\left(\frac{\partial A}{\partial t}\right)_I\]
So in the interaction picture, states evolve according to the perturbation Hamiltonian \(\Delta\tilde H_I(t)\) by design while observables evolve solely according to the unperturbed model Hamiltonian \(\tilde H\).
Okay, now how does one solve such equations? One can first convert the Schrodinger-like differential equation into an integral equation:
\[|\psi_I(t)\rangle=|\psi(0)\rangle-\frac{i}{\hbar}\int_0^t\Delta\tilde H_I(t_1)|\psi_I(t_1)\rangle dt_1\]
Recursively substituting the equation into itself yields:
\[|\psi_I(t)\rangle=|\psi(0)\rangle-\frac{i}{\hbar}\int_0^t\Delta\tilde H_I(t_1)\left[|\psi(0)\rangle-\frac{i}{\hbar}\int_0^{t_1}\Delta\tilde H_I(t_2)|\psi_I(t_2)\rangle dt_2\right]dt_1\]
which simplifies to:
\[|\psi_I(t)\rangle=|\psi(0)\rangle-\frac{i}{\hbar}\int_0^t\Delta\tilde H_I(t_1)dt_1|\psi(0)\rangle+\left(-\frac{i}{\hbar}\right)^2\int_0^t\Delta\tilde H_I(t_1)\int_0^{t_1}\Delta\tilde H_I(t_2)|\psi_I(t_2)\rangle dt_2dt_1\]
From which one identifies the analog of the time evolution operator \(U_I(t)\) obeying \(|\psi_I(t)\rangle=U_I(t)|\psi_I(0)\rangle\) in the interaction picture given by the Dyson series:
\[U_I(t)=\sum_{n=0}^{\infty}\left(-\frac{i}{\hbar}\right)^n\int_0^tdt_1…\int_0^{t_{n-1}}dt_n\Delta \tilde H_I(t_1)…\Delta\tilde H_I(t_n)\]
where the order in which one performs the integrals is important because in general \([\Delta H_I(t_i),\Delta H_I(t_j)]\neq 0\). The abbreviation for this Dyson series expansion of the interaction picture time evolution operator \(U_I(t)\) is the time-ordered exponential \(U_I(t):=\mathcal Te^{-\frac{i}{\hbar}\int_0^t\Delta\tilde H_I(t’)dt’}\). Of course, this construction by itself is not limited to the interaction picture; back in the Schrodinger picture one also has \(|\psi(t)\rangle=U_S(t)|\psi(0)\rangle\) with the Schrodinger time evolution operator \(U_S(t)=\mathcal Te^{-\frac{i}{\hbar}\int_0^tH(t’)dt’}\) generated by the total Hamiltonian \(H(t)=\tilde H+\Delta\tilde H(t)\). Indeed they are clearly related by the usual transformation \(U_I(t)=e^{i\tilde Ht/\hbar}U_S(t)e^{-i\tilde Ht/\hbar}\) of operators between the two pictures.
Now, of course if \(\Delta\tilde H_I(t)=0\), then the interaction picture coincides with the Heisenberg picture so states remain “thermodynamically frozen” at their initial states like an ice cube. The intuition though is that once some external perturbation like a passing electromagnetic wave “turns on”, it will start to “melt” the state in the Heisenberg picture, and one would like to understand the “kinetics” of this melting. Although the above is the exact, analytical answer in the form of the Dyson series for \(U_I(t)\), in practice this is often hopeless to compute. Hence arises the need for time-dependent perturbation theory. First, let us Taylor expand (or should one say “Dyson expand”?) \(U_I(t)\approx 1-\frac{i}{\hbar}\int_0^t\Delta\tilde H_I(t’)dt’\) to first-order in the perturbation Hamiltonian \(\Delta\tilde H_I\) (one could go to higher-order if one wished, but for simplicity here it is customary to assume that the perturbation is small enough that first-order suffices). Then:
\[|\psi_I(t)\rangle\approx |\psi_I(0)\rangle-\frac{i}{\hbar}\int_0^t\Delta\tilde H_I(t’)dt’|\psi_I(0)\rangle\]
However, this is still a bit abstract right now, i.e. one would like to work in some concrete basis for state space \(\mathcal H\). The natural choice is the \(\tilde H\)-eigenbasis of unperturbed stationary states that one already understand well \(\tilde H|\tilde E_n\rangle=\tilde E_n|\tilde E_n\rangle\). Therefore, projecting onto one of these \(\tilde H\)-eigenspaces yields (after resolving the identity and reverting back to the Schrodinger picture):
\[\langle\tilde E_n|\psi_I(t)\rangle\approx \langle\tilde E_n|\psi_I(0)\rangle-\frac{i}{\hbar}\sum_{m}\langle \tilde E_m|\psi_I(0)\rangle\int_0^te^{i(\tilde E_n-\tilde E_m)t’/\hbar}\langle\tilde E_n|\Delta\tilde H(t’)|\tilde E_m\rangle dt’\]
This then is the key equation of first-order time-dependent perturbation theory. In practice, what often happens is that the initial state \(|\psi_I(0)\rangle=|\tilde{E}_0\rangle\) is often some \(\tilde H\)-eigenstate such as the ground state prior to the perturbation \(\Delta\tilde H(t)\) “kicking in”, so for this specific initial condition one has:
\[\langle\tilde{E}_n|\psi_I(t)\rangle\approx \delta_{n0}-\frac{i}{\hbar}\int_0^te^{i(\tilde{E}_n-\tilde{E}_0)t’/\hbar}\langle\tilde {E}_n|\Delta\tilde H(t’)|\tilde E_0\rangle dt’\]
In particular, the transition probability at time \(t\) from \(|\tilde E_0\rangle\) into another \(\tilde H\)-eigenstate \(|\tilde E_n\rangle\neq|\tilde E_0\rangle\) is the square \(|\langle\tilde{E}_n|\psi_I(t)\rangle|^2\) of the transition amplitude \(\langle\tilde{E}_n|\psi_I(t)\rangle\) as prescribed by Born’s rule:
\[|\langle\tilde E_n|\psi_I(t)\rangle|^2\approx\frac{1}{\hbar^2}\left|\int_0^t e^{i(\tilde{E}_n-\tilde{E}_0)t’/\hbar}\langle\tilde{E}_n|\Delta\tilde H(t’)|\tilde{E}_0\rangle dt’\right|^2\]
where of course because the interaction and Schrodinger pictures are unitarily equivalent, their predictions for this probability agree \(|\langle\tilde E_n|\psi_I(t)\rangle|^2=|\langle\tilde E_n|e^{i\tilde Ht/\hbar}|\psi(t)\rangle|^2=|e^{i\tilde E_nt/\hbar}\langle\tilde E_n|\psi(t)\rangle|^2=|\langle\tilde E_n|\psi (t)\rangle|^2\) so henceforth it will be convenient to drop the \(I\)-subscript. At this point, in order to make further progress on computing the transition probability \(|\langle\tilde E_n|\psi (t)\rangle|^2\) from \(|\psi (t)\rangle\mapsto|\tilde E_n\rangle\) at time \(t\geq 0\), it is necessary to specify a precise form of the perturbation Hamiltonian \(\Delta\tilde H(t’)\) (in the Schrodinger picture). Consider for instance applying a monochromatic electromagnetic wave of angular frequency \(\omega_{\text{ext}}\) that passes the system at time \(t=0\). Then one might intuitively expect the perturbation to be harmonic, of the form \(\Delta\tilde H(t)=\Delta\tilde H_0e^{i\omega_{\text{ext}}t}+\Delta\tilde H_0^{\dagger}e^{-i\omega_{\text{ext}}t}\) for \(t\geq 0\) (and \(\Delta\tilde H(t)=0\) for \(t<0\), where it is understood that \(\Delta\tilde H_0\) is a time-independent amplitude of the electromagnetic wave and the need for both of these terms is due to the Hermiticity \(H^{\dagger}=H\) of the Hamiltonian. Inserting this result into the formula for the transition probability yields:
\[|\langle\tilde E_n|\psi (t)\rangle|^2=\left|\frac{e^{i(\tilde E_n-\tilde E_0+\hbar\omega_{\text{ext}})t/\hbar}-1}{\tilde E_n-\tilde E_0+\hbar\omega_{\text{ext}}}\langle\tilde E_n|\Delta\tilde H_0|\tilde E_0\rangle+\frac{e^{i(\tilde E_n-\tilde E_0-\hbar\omega_{\text{ext}})t/\hbar}-1}{\tilde E_n-\tilde E_0-\hbar\omega_{\text{ext}}}\langle\tilde E_0|\Delta\tilde H_0|\tilde E_n\rangle\right|^2\]
Now suppose the light being shone happens to have precisely the right frequency \(\omega_{\text{ext}}\) needed to promote the quantum system across some energy gap \(\tilde E_n-\tilde E_0\) of the unperturbed Hamiltonian \(\tilde H\). In this case the second term in the formula dominates and the quantum system is said to absorb the light at this resonant frequency \(\hbar\omega_{\text{ext}}\approx \tilde E_n-\tilde E_0\):
\[|\langle\tilde E_n|\psi (t)\rangle|^2\approx\frac{4|\langle\tilde E_0|\Delta\tilde H_0|\tilde E_n\rangle|^2}{(\tilde E_n-\tilde E_0-\hbar\omega_{\text{ext}})^2}\sin^2\frac{\tilde E_n-\tilde E_0-\hbar\omega_{\text{ext}}}{2\hbar}t\]
In fact, one can do a bit more, by first rewriting the transition probability in terms of the unnormalized \(\text{sinc}\) function:
\[|\langle\tilde E_n|\psi (t)\rangle|^2\approx\frac{|\langle\tilde E_0|\Delta\tilde H_0|\tilde E_n\rangle|^2}{\hbar^2}t^2\text{sinc}^2\frac{\tilde E_n-\tilde E_0-\hbar\omega_{\text{ext}}}{2\hbar}t\]
This means that after applying the external electromagnetic wave \(\omega_{\text{ext}}\) for a long time \(t\to\infty\), the transition probability \(|\langle\tilde E_n|\psi (t)\rangle|^2\) graphed as a function of \(\omega_{\text{ext}}\) will essentially peak around the absorption resonance \(\hbar\omega_{\text{ext}}=\tilde E_n-\tilde E_0\) with height \(\sim t^2\) and width \(\sim 1/t\) as standard properties of the \(\text{sinc}\) function. In the true steady state \(t\to\infty\), one therefore expects the transition probability to approach the delta distribution:
\[|\langle\tilde E_n|\psi (t)\rangle|^2\approx\frac{2\pi|\langle\tilde E_0|\Delta\tilde H_0|\tilde E_n\rangle|^2}{\hbar}\delta(\tilde E_n-\tilde E_0-\hbar\omega_{\text{ext}})t\]
by virtue of the Dirichlet-like integral \(\int_{-\infty}^{\infty}\text{sinc}^2(x)dx=\pi\).
In physics the word “rate” generally means \(\frac{\partial}{\partial t}\). In this spirit, define the transition rate \(\Gamma_{|\tilde E\rangle\to|\tilde E’\rangle}\) to be the steady-state \(t\to\infty\) temporal density of the transition probability \(|\langle\tilde E_n|\psi(t)\rangle|^2\) from an initial \(\tilde H\)-eigenstate \(|\tilde E\rangle\) to a final \(\tilde H\)-eigenstate \(|\tilde E’\rangle\):
\[\Gamma_{|\tilde E\rangle\to|\tilde E’\rangle}:=\lim_{t\to\infty}\frac{\partial}{\partial t}|\langle\tilde E’|\psi(t)\rangle|^2\]
where the initial condition \(|\psi(0)\rangle=|\tilde E\rangle\) is implied. By the computations above, the transition rate \(\Gamma_{|\tilde E_0\rangle\to|\tilde E_n\rangle}\) is therefore constant (i.e. \(t\)-independent) and given by:
\[\Gamma_{|\tilde E_0\rangle\to|\tilde E_n\rangle}\approx\frac{2\pi|\langle\tilde E_0|\Delta\tilde H_0|\tilde E_n\rangle|^2}{\hbar}\delta(\tilde E_n-\tilde E_0-\hbar\omega_{\text{ext}})\]
By repeating an analogous analysis of the transition amplitude when the quantum system performs stimulated emission \(\tilde E_n-\tilde E_0\approx -\hbar\omega_{\text{ext}}\), the transition rate \(\Gamma_{|\tilde E_0\rangle\to|\tilde E_n\rangle}\) in this case is just:
\[\Gamma_{|\tilde E_0\rangle\to|\tilde E_n\rangle}\approx\frac{2\pi|\langle\tilde E_n|\Delta\tilde H_0|\tilde E_0\rangle|^2}{\hbar}\delta(\tilde E_n-\tilde E_0+\hbar\omega_{\text{ext}})\]
Together, these two transition probability formulas (for absorption and stimulated emission) are called Fermi’s golden rules. Although the need for both terms in the perturbation Hamiltonian \(\Delta\tilde H(t)\) were initially motivated by the Hermiticity of \(H\), it is evident now that each of them also bears physical significance; the perturbation \(\Delta\tilde H_0^{\dagger}e^{-i\omega_{\text{ext}} t}\) corresponds to absorption while the perturbation \(\Delta\tilde H_0e^{i\omega_{\text{ext}} t}\) corresponds to stimulated emission.
In practice, such a monochromatic perturbation at some precise frequency \(\omega_{\text{ext}}\) cannot cause any transitions \(\Gamma_{|\tilde E_0\rangle\to|\tilde E_n\rangle}=0\) between \(|\tilde E_0\rangle\) and any discrete bound states \(|\tilde E_n\rangle\) because of the delta function. So in practice, Fermi’s golden rules are useful in \(2\) cases:
- Case #\(1\): don’t ask about the transition rate from the initial \(\tilde H\)-eigenstate \(|\tilde E_0\rangle\) to some other discrete bound \(\tilde H\)-eigenstate \(|\tilde E_n\rangle\), but rather to any one of uncountably infinitely many \(\tilde H\)-eigenstate \(|\tilde E_n\rangle\) in some range (see the diagram below). In this case, letting \(g(\tilde E_n)\) denote the density of such scattering states, it follows that the transition rate from \(|\tilde E_0\rangle\) to just somewhere, anywhere in this “band” of scattering states is (taking the case of absorption here for example): \[\Gamma_{|\tilde E_0\rangle\to g(\tilde E_n)}:=\int\Gamma_{|\tilde E_0\rangle\to|\tilde E_n\rangle}g(\tilde E_n)d\tilde E_n\approx \frac{2\pi|\langle\tilde E_0|\Delta\tilde H_0|\tilde E_0+\hbar\omega_{\text{ext}}\rangle|^2}{\hbar}g(\tilde E_0+\hbar\omega_{\text{ext}})\]
- An example of Case #\(1\) of Fermi’s golden rules is the photoelectric effect, where one can use it to compute the transition/ionization rate of the electron \(e^-\) from an initial \(s\)-wave bound state in a hydrogen atom into an asymptotically free \(e^-\) is. A qualitative sketch of the calculation: first, ignore quantization of the EM field (valid for e.g. a laser) by imposing a suitable vector potential such as \(\textbf A\propto e^{i(\textbf k\cdot\textbf x-\omega_{\text{ext}}t)}\) for the usual planar Maxwellian EM wave. The perturbation Hamiltonian \(\Delta\tilde H\) is determined by expanding to first-order the usual minimally-coupled Hamiltonian for a charge \(q=-e\) in an electromagnetic field. Since this is absorption rather than stimulated emission, compute the matrix element of the appropriate term between the hydrogen atom ground state \(|1,0,0\rangle\) and the free \(e^-\) state \(|\textbf k\rangle\) (where recall that the density of such free states is \(g(E)\propto\sqrt E\)).
- Case #\(2\): basically if the perturbation \(\Delta\tilde H(t)\) isn’t actually monochromatic, but polychromatic in a way that can be Fourier-expanded into monochromatic components. Then in this case, even transitions between discrete bound \(\tilde H\)-eigenstates are possible. An obvious example of this latter case of Fermi’s golden rules is an atom bathing in a photon \(\gamma\) gas. To compute the (stimulated) absorption \(\Gamma_{1\to 2}\) and (stimulated) emission \(\Gamma_{2\to 1}\) rates, it is common to employ a so-called dipole approximation (though in this context I feel a better name would be long-wavelength approximation or even just cold approximation; note also that this is a further approximation on top of all the other approximations that have already been made before this!) in which the typical wavelength \(\lambda\gg r_B\) of photons \(\gamma\) in the thermal bath is much longer than the Bohr radius \(r_B\) representing the typical length scale of atoms. Essentially what this means is that the vector potential \(\textbf A\sim e^{i(\textbf k_{\text{ext}}\cdot\textbf x-\omega_{\text{ext}} t)}\approx e^{-i\omega_{\text{ext}}t}\) is approximately spatially uniform for \(\textbf k_{\text{ext}}\cdot\textbf x\ll 1\) for \(\textbf x\) ranging over the atom, and therefore the time-dependent perturbation to the atom just has the form \(\Delta\tilde H(t)=e\textbf E_{\text{ext}}(t)\cdot\sum_{e^-}\textbf X_{e^-}\) where \(e\sum_{e^-}\textbf X_{e^-}\) is the net electric dipole moment of the atom, hence the name “dipole approximation”. Chugging this into Case #\(2\) of Fermi’s golden rules, one can compute the rates of stimulated absorption and emission to be equal and given by:
\[\Gamma_{1\to 2}=\Gamma_{2\to 1}=\frac{\pi e^2}{3\varepsilon_0\hbar^2}\left|\sum_{e^-}\langle 2|\textbf X_{e^-}|1\rangle\right|^2g(\Delta E_{12})\]
Note that non-relativistic quantum mechanics predicts that atoms can only undergo stimulated absorption or stimulated emission. In particular, non-relativistic quantum mechanics predicts that there are no such things as “spontaneous absorption” or “spontaneous emission” \(\Gamma_{1\to 2}^*=\Gamma_{2\to 1}^*=0\) where an atom can undergo a transition between \(\tilde H\)-eigenstates in the absence \(\textbf E_{\text{ext}}=\textbf B_{\text{ext}}=\textbf 0\) of external electromagnetic fields/photons (because the non-relativistic Schrodinger equation asserts that the \(t\)-evolution of \(\tilde H\)-eigenstates is to stay right where they are rather than hopping to other \(\tilde H\)-eigenstates). Indeed, it turns out there is no such thing as “spontaneous absorption” \(\Gamma_{1\to 2}^*\), but there can be spontaneous emission \(\Gamma_{2\to 1}^*\neq 0\). This phenomenon only arises once one quantizes the classical, smooth electromagnetic field \((\textbf E,\textbf B)\). When this is done, it is found that spontaneous emission is possible due to interactions between the atom and zero-point fluctuations of the quantum electromagnetic field, but to delve deeper would be the subject of quantum electrodynamics.
Nevertheless, Einstein was able to compute the rate of spontaneous emission \(\Gamma_{2\to 1}^*\) without knowing anything about the quantization of the electromagnetic field. Or more precisely, Einstein showed that if one could calculate the stimulated emission rate \(\Gamma_{2\to 1}\), then one would also get the spontaneous emission rate \(\Gamma_{2\to 1}^*\) “for free” since he found that the two were proportional to each other and what that proportionality constant was.
Einstein’s Statistical Argument for Spontaneous Emission
Consider any \(2\) energy levels \(E_1<E_2\) in an atom of degeneracies \(\Omega_1,\Omega_2\) with number of electrons \(N_1,N_2\) respectively.
Then in general, one heuristically expects probabilistic kinetics of the form (ASIDES: what happens if one also adds a “spontaneous absorption” term \(A_{12}N_1\) among the rate terms? And also, need to explicitly related the \(\Gamma\)-transition rates above in Fermi’s golden rule with these Einstein \(A,B\) coefficients):
\[\dot N_2=-\dot N_1=-B_{12}(\Delta\omega_{12})N_1\mathcal E_{\text{ext}}(\Delta\omega_{12})+B_{21}(\Delta\omega_{12})N_2\mathcal E_{\text{ext}}(\Delta\omega_{12})+A_{21}(\Delta\omega_{12})N_2\]
where the spontaneous emission term \(A_{21}(\Delta\omega_{12})N_2\) is not proportional to the spectral energy density of photons \(\mathcal E_{\text{ext}}(\Delta\omega_{12})\) with the correct frequency \(\Delta\omega_{12}=(E_2-E_1)/\hbar\) precisely because it’s spontaneous (i.e. not stimulated by the radiation and therefore independent of its “concentration”). Another way to see this is that, in the absence of any external radiation \(\mathcal E_{\text{ext}}(\Delta\omega_{12})=0\) with the correct frequency \(\Delta\omega_{12}\), the only operating mechanism is spontaneous emission in which the occupation number \(N_2(t)=N_2(0)e^{-A_{21}(\Delta\omega_{12})t}\) decays exponentially to \(N_2\to 0\) with lifetime \(\tau(\Delta\omega_{12})=1/A_{21}(\Delta\omega_{12})\) inversely correlated with the gap size \(\Delta\omega_{12}\) (since a larger gap size \(\Delta\omega_{12}\) signifies a more “unstable” state).
More generally then, when there are photons with the right frequency \(\mathcal E_{\text{ext}}(\Delta\omega_{12})\neq 0\) for stimulated absorption/emission, the transient \(t\)-dependence is tricky to model, but the steady state \(t\to\infty\) where \(\dot N_1=\dot N_2=0\) is straightforward: the rates of stimulated absorption/emission and spontaneous emission must balance:
\[-B_{12}(\Delta\omega_{12})N_1\mathcal E_{\text{ext}}(\Delta\omega_{12})+B_{21}(\Delta\omega_{12})N_2\mathcal E_{\text{ext}}(\Delta\omega_{12})+A_{21}(\Delta\omega_{12})N_2=0\]
or isolating for the spectral energy density \(\mathcal E_{\text{ext}}(\Delta\omega_{12})\) of correct-frequency photons:
\[\mathcal E_{\text{ext}}(\Delta\omega_{12})=\frac{A_{21}(\Delta\omega_{12})}{B_{12}(\Delta\omega_{12})(N_1/N_2)-B_{21}(\Delta\omega_{12})}\]
But being in the \(t\to\infty\) steady state is synonymous with the atom being in thermodynamic equilibrium with the photon gas at some temperature \(T\) in the canonical ensemble, so statistical mechanics asserts that both \(\frac{N_1}{N_1+N_2}=\Omega_1e^{-E_1/kT}/Z\) and \(\frac{N_2}{N_1+N_2}=\Omega_2e^{-E_2/kT}/Z\) will be Boltzmann distributed while the spectral energy density of the radiation \(\mathcal E_{\text{ext}}(\Delta\omega_{12})=\frac{\hbar\Delta\omega_{12}^3}{\pi^2 c^3}\frac{1}{e^{\hbar\Delta\omega_{12}/kT}-1}\) will be Planck distributed. Plugging these distributions in leads to:
\[\frac{\hbar\Delta\omega_{12}^3}{\pi^2 c^3}\frac{1}{e^{\hbar\Delta\omega_{12}/kT}-1}=\frac{A_{21}(\Delta\omega_{12})\Omega_2}{B_{12}(\Delta\omega_{12})\Omega_1}\frac{1}{e^{\hbar\Delta\omega_{12}/kT}-B_{21}(\Delta\omega_{12})\Omega_2/B_{12}(\Delta\omega_{12})\Omega_1}\]
One can then just “pattern-match” (it is an encouraging sign that this is even possible!) because the relation must hold at arbitrary temperature \(T\in\textbf R\) to obtain:
\[\frac{B_{12}(\Delta\omega_{12})}{\Omega_2}=\frac{B_{21}(\Delta\omega_{12})}{\Omega_1}\]
\[\frac{\hbar\Delta\omega_{12}^3}{\pi^2 c^3}=\frac{A_{21}(\Delta\omega_{12})\Omega_2}{B_{12}(\Delta\omega_{12})\Omega_1}\]
The first relation is just reconfirming that \(\Gamma_{12}=\Gamma_{21}\). Morally, it makes sense that these stimulated rates coincide because absorption of a photon of energy \(\pm E\) is “equivalent” to emission of a photon of energy \(\mp E\). Mathematically, the time-dependent perturbation \(\Delta\tilde H(t)=\Delta\tilde H_0e^{i\omega t}+\Delta\tilde H_0^{\dagger}e^{-i\omega t}\) is Hermitian.
Plugging the first relation into the second, one obtains a direct relation between the rates of stimulated and spontaneous emission:
\[A_{21}(\Delta\omega_{12})=\frac{\hbar\Delta\omega_{12}^3}{\pi^2 c^3}B_{21}(\Delta\omega_{12})\]
At first, even after one has seen Einstein’s statistical argument for \(\Gamma_{1\to 2}=\Gamma_{2\to 1}\) and \(A_{21}(\Delta\omega_{12})\propto\Delta\omega_{12}^3B_{21}(\Delta\omega_{12})\), it is still not clear how the argument manages to “get away” with not knowing anything about the quantization of the electromagnetic field while still obtain the correct answer. It turns out to be implicit; one has already quantized the electromagnetic field by taking the Planck distribution as an input to the derivation.
Being thermodynamic in nature, whereas transition rates are inherently kinetic, Einstein’s statistical argument of course fails to actually yield an explicit expression for the various transition rates. This “kinetics” part was of course the whole point of the time-dependent perturbation theory calculation with the dipole approximation. Combining Einstein’s thermodynamic argument with a dose of quantum kinetics, the lifetime \(\tau\) of an isolated atom in an excited state before it spontaneously emits a photon \(\gamma\) in order to decay to a lower energy is roughly:
\[\tau\approx \frac{3m_{e^-}c^3}{4\hbar\Delta\omega_{12}^3a_0}\]