Section 42–5 Einstein’s laws of radiation

Spontaneous emission / Stimulated emission / Absorption

 

In this section, Feynman discusses spontaneous emission, stimulated emission, and absorption of radiation under the “Einstein’s laws of radiation,” based on Einstein’s (1917) paper On the Quantum Theory of Radiation. It is worth noting that Einstein himself did not use the term “spontaneous emission” and “stimulated emission”; instead, he described them as “without excitation from external causes” and “changes of state due to irradiation” respectively. Moreover, while Feynman refers only to “thermal equilibrium,” Einstein distinguished among three types of equilibrium: thermal, dynamical, and thermodynamic.

 

1. Spontaneous Emission

“Now what is the formula going to be for the rate of emission from m to n? Einstein proposed that this must have two parts to it. First, even if there were no light present, there would be some chance that an atom in an excited state would fall to a lower state, emitting a photon; this we call spontaneous emission (Feynman et al., 1963).”

 

Feynman explains spontaneous emission as the process by which an excited atom emits a photon even in the absence of external radiation (including light). This concept was introduced by Einstein in his 1917 paper: “According to Hertz, an oscillating Planck resonator radiates energy in the well-known way, regardless of whether or not it is excited by an external field.” In Einstein’s theory, spontaneous emission is treated as a random, inherent property of the atom—essentially a “black box” process. However, his treatment was phenomenological rather than microscopic: he modeled it as a fundamental statistical process without specifying the underlying physical mechanism. Consequently, the exact emission time, emission direction, and the particular atom within an ensemble that undergoes the transition are individually unpredictable and can only be described probabilistically for large collections of atoms.

 

“Thus the analog of spontaneous radiation of a classical system is that if the atom is in an excited state there is a certain probability A_mn, which depends on the levels again, for it to go down from m to n, and this probability is independent of whether light is shining on the atom or not (Feynman et al., 1963).”

 

Modern Quantum Electrodynamics (QED) shows that spontaneous emission is not “spontaneous” in the everyday sense of occurring without any physical interaction. Although the term remains useful phenomenologically, it is potentially misleading because the underlying mechanism is now understood differently. In QED, an excited atom interacts continuously with the electromagnetic field, including the zero-point (vacuum) fluctuations. These vacuum fluctuations can induce the atom to transition to a lower energy state, thereby emitting a photon. In this sense, spontaneous emission is not literally uncaused, but it arises from the interaction between matter and the quantum vacuum field. For this reason, some physicists have described the process as “vacuum-stimulated emission” (Milonni, 1984). In other words, spontaneous emission can be viewed as “stimulated emission driven by the zero-point fluctuations of the vacuum,” though this perspective remains a useful heuristic since vacuum fluctuations are virtual rather than real photons.

“Einstein assumed that Planck’s final formula was right, and he used that formula to obtain some new information, previously unknown, about the interaction of radiation with matter (Feynman et al., 1963).”

 

In his 1917 paper, Einstein writes, “Planck's formula could be derived in an astonishingly simple and general way. It was obtained from the condition that the internal energy distribution of the molecules demanded by quantum theory, should follow purely from an emission and absorption of radiation.” He then explains his assumption of isotropy (or pseudo-isotropy using time-averages), which allows the coefficients B_mn and B_nm to be independent of direction. Even if a molecule is anisotropic, the molecule’s orientation fluctuates randomly over time, so that directional effects vanish on average (pseudo-isotropy). The deeper significance of Einstein’s argument is that it implicitly anticipates the modern quantum concept of radiation as photons carrying momentum in definite directions. Once radiation is understood to transfer directional momentum to atoms during emission or absorption, pseudo-isotropy is no longer trivial: it becomes a statistical requirement ensuring that, in thermal equilibrium, no direction is privileged on average.

 

2. Stimulated emission

“an emission proportional to the intensity of light, called induced emission or sometimes stimulated emission… (Feynman et al., 1963).”

 

Stimulated emission can be understood in terms of external electromagnetic wave, thermal equilibrium, and Boltzmann distribution. Firstly, stimulated emission is a quantum mechanical process when an external electromagnetic wave—specifically a photon—interacts with an atom or molecule that is already in a higher-energy excited state and releases a second photon. This newly emitted photon is physically identical to the triggering photon—sharing the exact same frequency, phase, polarization, and direction of travel. Next, Einstein mathematically predicted this process by showing that it is a strict requirement for thermodynamic equilibrium, so that a system can maintain a stable energy balance according to Planck's radiation law. Under normal circumstances dictated by the Boltzmann distribution, the population density naturally favors the lower energy state. Ultimately, one "triggering" photon goes in, and two identical, correlated photons come out, effectively multiplying the light.

 

Although the underlying mechanism of a laser is simulated emission, its operating conditions differ fundamentally from the thermal equilibrium assumed in Einstein’s (1917) theory of radiation. In Einstein’s framework, atomic populations obey the Boltzmann distribution, so lower energy states are naturally more populated than excited states (N₁ > N₂). A laser, however,  operates far from thermal equilibrium. An external pump source continuously supplies energy to the lasing medium, driving the atoms into a non-equilibrium condition known as population inversion (N₂ > N₁). Thus, ordinary stimulated emission and laser action are “same same but different”: both arise from the same quantum transition process, but population inversion in a laser helps to sustain stimulated emission to achieve coherent light amplification. This inversion is possible by using a lasing medium (e.g., a gas or semiconductor) with metastable states, whose relatively longer lifetimes allow excited atoms to accumulate rather than decaying immediately by spontaneous emission.

 

3. Absorption

“Thus Einstein assumed that there are three kinds of processes: an absorption proportional to the intensity of light, an emission proportional to the intensity of light, called induced emission or sometimes stimulated emission, and a spontaneous emission independent of light (Feynman et al. 1963).”

 

Absorption occurs when an atom or molecule in a lower energy state absorbs a photon whose energy matches the energy difference between two energy levels, thereby transitioning to a higher energy state. In Einstein’s theory, the absorption rate is proportional to the intensity (spectral density) of the radiation field. Microscopically, absorption arises from the interaction between the electromagnetic field and the atom’s charged constituents, which transfer quantized energy to the atom. In reality, absorption is not perfectly monochromatic but occurs over a finite frequency range depending on thermal motion (Doppler broadening) and collisional effects. The probability of a transition is constrained by selection rules and depends on photon polarization and molecular orientation. For the system to remain in dynamic equilibrium, the overall rates of upward and downward transitions must balance exactly:

Rate of upward transitions (1 ® 2) = Rate of downward transitions (2 ® 1)

This condition is known as the principle of detailed balance, which states that at equilibrium, every microscopic process transferring energy in one direction must be balanced by the corresponding reverse process occurring at the same average rate.

 

“This is not the only way one can arrange to keep the numbers of atoms in the various levels constant, but it is the way it actually works. That every process must, in thermal equilibrium, be balanced by its exact opposite is called the principle of detailed balancing (Feynman et al., 1963).”

 

In his 1917 paper, Einstein did not explicitly use the term “principle of detailed balancing,” but mentions the photochemical principle of equivalence, Doppler Principle, and Boltzmann’s principle. The photochemical law of equivalence states that each absorbed photon activates one atom or molecule in a photochemical reaction. Einstein also invoked the Doppler principle to account for frequency shifts caused by the thermal motion of atoms, thereby broadening spectral lines. At the same time, he employed Boltzmann’s principle to relate the relative populations of energy levels at thermal equilibrium. However, Einstein effectively imposed a condition of dynamic equilibrium, which was later formalized and named as the “principle of detailed balancing” by Richard C. Tolman. By combining these principles, Einstein showed that the interplay between absorption, spontaneous emission, and stimulated emission must reproduce the Planck’s blackbody spectrum while maintaining thermodynamic equilibrium.

 

Perhaps Feynman could have concluded the section with the famous dialogue with his father, because it captures a genuine conceptual puzzle about spontaneous emission: how can a photon be created during an atomic transition if it was not already “inside” the atom beforehand? This exchange reveals not a failure of physics, but the limits of classical intuition when applied to quantum processes.

 

You might wonder what he got out of it all. I went to MIT. I went to Princeton. I came home, and he said, "Now you've got a science education. I have always wanted to know something that I have never understood, and so, my son, I want you to explain it to me."

I said yes.

He said, "I understand that they say that light is emitted from an atom when it goes from one state to another, from an excited state to a state of lower energy.

I said, "That's right."

"And light is a kind of particle, a photon, I think they call it."

"Yes."

"So if the photon comes out of the atom when it goes from the excited to the lower state, the photon must have been in the atom in the excited state."

I said, "Well, no."

He said, "Well, how do you look at it so you can think of a particle photon coming out without it having been in there in the excited state?"

I thought a few minutes, and I said, "I'm sorry; I don't know. I can't explain it to you." He was very disappointed after all these years and years of trying to teach me something, that it came out with such poor results (Feynman, 1969).

 

Feynman’s admission that he could not provide an intuitive picture of the process exposes a deeper truth. Spontaneous emission, stimulated emission, and absorption all resist simple classical visualization. In spontaneous emission, a photon appears without an obvious physical cause; in stimulated emission, one photon appears to generate another identical photon; and in absorption, a photon seems to “disappear” into the atom.

His father’s seemingly simple question exposes the limits of everyday mental models; even after mastering the mathematics of Einstein’s coefficients, we cannot give a fully intuitive “picture” of these processes, only a consistent set of probabilistic rules that work. Thus, the dialogue serves as a humbling reminder that even when quantum phenomena can be controlled and exploited—in lasers, solar cells, and quantum optics—the underlying nature of what “really happens” during a quantum transition still challenges human intuition.

 

Key Takeaways:

The key insight behind absorption, spontaneous emission, and stimulated emission is that they are three complementary quantum processes governing how matter exchanges energy with radiation through transitions between discrete energy levels. Absorption occurs when an atom or molecule in a lower energy state absorbs a photon whose energy matches the gap between two allowed levels, thereby moving to a higher state. Spontaneous emission occurs when an excited atom decays randomly to a lower state and emits a photon even in the absence of externally applied radiation; in modern Quantum Electrodynamics, this process is understood as arising from the interaction between the atom and vacuum fluctuations of the quantized electromagnetic field. Stimulated emission occurs when an incoming photon induces an excited atom to emit a second photon with the same frequency, phase, polarization, and direction as the first photon.

 

The Moral of the Lesson:

In the early 1950s, Charles H. Townes devoted enormous time, effort, and funding to developing the maser, a device designed to amplify microwaves through stimulated emission and the technology precursor to the laser. After nearly two years of relentless work without a functioning prototype, Townes was confronted by two of Columbia University’s most eminent physicists: Isidor Isaac Rabi and Polykarp Kusch.

 

Rabi and Kusch were leading authorities on molecular beam techniques—the very experimental methods Townes was relying on—and both later received the Nobel Prize in Physics.  According to Townes, they bluntly advised him to abandon his project: “You should stop the work you are doing. It isn’t going to work. You know it’s not going to work. We know it’s not going to work. You’re wasting money. Just stop! (Townes, 1999, p. 65).” They were concerned that his prolonged failure would affect their research funding from the same source. However, Townes trusted his calculations and persisted. Just three months later, his team successfully built the first operating maser.

 

Remarkably, Rabi and Kusch were not the only skeptics. Prominent physicists such as Niels Bohr and John von Neumann also doubted the maser’s feasibility, while some questioning whether it violated Heisenberg’s uncertainty principle. Townes’s success remains as a timeless lesson in science: even the judgment of the most celebrated experts must ultimately be tested against experimental evidence. Scientific authority can guide inquiry, but nature itself remains the final arbiter.

 

Hindsight Bias and the Illusion of Simplicity

Years later, Townes recalled a conversation with Feynman about the laser. Feynman remarked that the hallmark of a truly great idea is that when people hear it, they respond by saying, “Gee, I could have thought of that (Townes, 1999, p. 10).” In retrospect, the laser appears elegant and inevitable, yet this great idea was initially dismissed by many prominent physicists, including Nobel laureates. Hindsight tends to compress intellectual struggle into inevitability, making revolutionary ideas appear far simpler than they actually were at the time of discovery.

 

The Hidden Genius of the Fabry-Pérot Cavity

Part of the skepticism is understandable when one recognizes that the laser was far more than a synthesis of principles such as the photochemical principle of equivalence and Boltzmann’s principle. One of the breakthroughs lay in engineering: scaling the maser from microwaves to visible light required confining light within a Fabry‑Perot cavity. This cavity uses two nearly parallel, partially reflective mirrors to bounce light back and forth repeatedly, producing multiple‑beam interference. Its effectiveness is not obvious: one might naïvely think that multiple reflections would likely scatter or weaken light, instead of sharpening it coherently. Moreover, the mirrors must be kept parallel to within a fraction of a wavelength—a requirement far more stringent than ordinary experience would suggests. By transforming a poorly understood optical phenomenon into a practical tool, Townes and team did more than invent a new device. They showed that a revolutionary scientific idea does not necessarily appear revolutionary at first; more often, it may initially seem impractical, counterintuitive, or even impossible.

 

Review Questions:

1. How would you explain spontaneous emission: as “stimulated emission driven by vacuum fluctuations” (Modern view) or as “emission without excitation from external causes” (Einstein’s view)?

2. What are the key similarities and differences between stimulated emission as described in Einstein’s 1917 radiation theory and the conditions required for laser operation?

3. Using everyday language that Feynman’s father could understand, how would you explain the absorption of radiation—that is, how an atom absorbs a photon and jumps to a higher energy state?

P.S. The evolution of the laser from Einstein’s 1917 theory of radiation to modern semiconductor lithography represents a premier arc in applied physics. Today's advanced chip manufacturing relies primarily on laser-produced plasma (LPP) extreme ultraviolet (EUV) lithography. In this process, high-power pulsed lasers vaporize tens of thousands of tin droplets per second, generating a hot plasma that emits a broad spectrum, from which the 13.5 nm wavelength is selectively collected by multilayer mirrors. This ultra-short wavelength is the enabling technology required to pattern the microscopic features on modern microchips. What began as a quantum insight is now the physical foundation driving global computing, artificial intelligence, and the digital economy. 

References:

Einstein, A. (1917). On the quantum theory of radiation. Physikalische Zeitschrift, 18(121), 167-83.

Feynman, R. P. (1969). What Is Science?. The Physics Teacher, 7(6), 313–320.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Milonni, P. W. (1984). Why spontaneous emission? American Journal of Physics, 52, 340-343.

Townes, C. H. (1999). How the laser happened: adventures of a scientist. Oxford University Press.