Section 41–3 Equipartition and the quantum oscillator

Planck’s Quantum Hypothesis / Cutoff factor / Johnson noise

 

In this section, Feynman discusses Planck's Quantum Hypothesis and the resulting “cutoff factor,” which are fundamental to understanding both blackbody radiation and Johnson (thermal) noise. Thus, the section could be aptly titled “blackbody radiation and Johnson noise” to reflect the concepts between electromagnetic emission in a cavity and electronic fluctuations in a resistor. Both phenomena are unified by the Fluctuation-Dissipation Theorem—a principle linking thermal fluctuations to energy dissipation—which Feynman elaborates in the subsequent section.

 

1. Planck's Quantum Hypothesis

“Planck studied this curve. He first determined the answer empirically, by fitting the observed curve with a nice function that fitted very well. ... In other words, he had the right formula instead of kT, and then by fiddling around he found a simple derivation for it which involved a very peculiar assumption. That assumption was that the harmonic oscillator can take up energies only ℏω at a time. The idea that they can have any energy at all is false (Feynman et al., 1963, p. 41-6).”

 

In a 1931 letter to the American physicist Robert W. Wood, Planck wrote: “… one finds that the continuous loss of energy into radiation can be prevented by assuming that energy is forced, at the onset, to remain together in certain quanta. This was a purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result.” He admitted that this mathematical method was an act of desperation—a way to force the equations to match experimental data. However, there is no clear evidence that Planck initially embraced the physical reality of quantized energy. It was Einstein, five years later, who took the quantum hypothesis seriously, proposing that light itself consists of discrete packets of energy, or photons. In doing so, Einstein initiated the revolution that Planck had inadvertently made possible—but was ultimately reluctant to accept (Kragh, 2000).

 

Strictly speaking, Planck’s derivation of his radiation law did not rely solely on the idea that a harmonic oscillator could only possess energies in discrete multiples of ℏω. A key nuance is that this quantization of energy applies specifically to oscillators confined within a cavity. From the perspective of quantum physics, such confinement imposes boundary conditions on the wave function, leading to discrete standing waves and quantized energy levels. This principle is clarified by the counterexample of a free particle. In Vol. 1, Ch. 38*, Feynman explains that an electron that is not bound by a potential well can possess a continuous spectrum of energies. This distinction highlights that quantization is not an intrinsic property of energy, but a consequence of physical constraints imposed by the system’s environment. In the case of the blackbody radiation, the oscillators are bound to the cavity walls, restricting energy exchange to discrete "quantized" amounts. This mechanism naturally suppresses the emission of high-frequency radiation, thereby resolving the ultraviolet catastrophe problem.

 

*In section 38, Feynman mentions: “[w]hen the electron is free, i.e., when its energy is positive, it can have any energy; it can be moving at any speed. But bound energies are not arbitrary (Feynman et al., 1963, p. 38-7).”

 

2. Cutoff factor

“This is the famous cutoff factor that Jeans was looking for, and if we use it instead of kT in (41.13), we obtain for the distribution of light in a black box I(ω)dω = ℏω³dω/π²c²(e^ℏω/kT−1). We see that for a large ω, even though we have ω³ in the numerator, there is an e raised to a tremendous power in the denominator, so the curve comes down again and does not ‘blow up’—we do not get ultraviolet light and x-rays where we do not expect them! (Feynman et al., 1963, p. 41-7).”

 

Feynman mistakenly credited Sir James Jeans with introducing the cutoff factor. In 1900, Rayleigh proposed an exponential function to suppress the unphysical divergence of radiation energy at high frequencies. His initial formula took the form: r(ν, T) = c₁ν²Te^(-c₂ν/T). This exponential factor, similar to the one in Wien’s formula, was intended to better fit with short-wavelength experimental data. However, in 1905, Rayleigh re-derived the formula without this exponential factor, obtaining an expression closer to the modern Rayleigh–Jeans law. He also calculated the coefficient c₁, but his value was eight times larger than the accepted one. Later in 1905, Jeans identified an error in Rayleigh’s derivation and corrected the coefficient, arriving at the familiar form: u(l, T) = 8pkT/l⁴. Despite this correction, the Rayleigh–Jeans formula did not gain recognition, as Planck’s (1900) blackbody law provided a better fit to empirical data.

 

In Pais’ (1979) own words, “In order to suppress the catastrophic high-frequency behavior, he introduced next an ad hoc exponential cutoff factor and proposed the overall radiation law r(n, T) = c₁n²Te^-c₂n/T. This expression became known as the Rayleigh law (p. 872).” The use of the term cutoff could be attributed to Pais instead of Jeans or Rayleigh, but it is somewhat misleading. This exponential factor does not act as a sharp, abrupt cutoff; instead, it gradually reduces (or suppresses) the contribution of high frequency-modes. A more precise term, such as suppression factor or correction factor, better shows its role in correcting the unphysical high-frequency divergence predicted the classical theory. It is worth noting that this mathematical approach was not entirely new. A similar exponential factor had been employed earlier by Wilhelm Wien in his displacement law, which was used to fit the data for blackbody radiation (See below).

Source: Wien approximation - Wikipedia

“This, then, was the first quantum-mechanical formula ever known, or ever discussed, and it was the beautiful culmination of decades of puzzlement. Maxwell knew that there was something wrong, and the problem was, what was right? Here is the quantitative answer of what is right instead of kT. This expression should, of course, approach kT as ω→0 or as T→∞. See if you can prove that it does—learn how to do the mathematics (Feynman et al., 1963, p. 41-7).”

 

Feynman could have clarified the low-frequency and high-frequency behavior of Planck's radiation law. We can analyze the behavior of the intensity in two extreme limits. The formula we are analyzing is: I(ω) = ℏω³/π²c²(e^ℏω/kT−1).

1. Low-Frequency Behavior (ℏω << kT)

When the frequency is low (or high temperature), the energy of a single photon (ℏω) is much smaller than the average thermal energy (kT).

• Approximation: The exponential term can be expanded: e^x » 1 + x for small x. Setting x = ℏω/kT gives: e^ℏω/kT - 1 » 1 + (ℏω/kT) - 1 = ℏω/kT
• Result: Substituting this into Planck’s Law yields:

I(ω) = ℏω³/π²c²(e^ℏω/kT−1) » ω² kT/π²c²

• Significance: Planck’s law approaches the Rayleigh-Jeans Law. It reduces to classical physics at low frequencies where quantum effects are negligible.

2. High-Frequency Behavior (ℏω >> kT)

When the frequency is high (or low temperature), the energy required to excite a single oscillator (ℏω) is much larger than thermal fluctuations typically provide.

• Approximation: Since ℏω/kT is very large, e^ℏω/kT >> 1, making the "-1" in the denominator negligible: e^ℏω/k -1 » e^ℏω/kT
• Result: I(ω) = ℏω³/π²c²(e^ℏω/kT−1) » ℏω³(e^−ℏω/kT)/π²c²
• Significance: Planck’s Law approaches Wien’s Approximation. Specifically, the exponential term e^−ℏω/kT acts as a suppression factor that reduces the intensity due to ω³.

Feynman used the modern notation of ℏ (reduced Planck’s constant), where the quantum of energy is ℏω. This is equivalent to Planck’s original formulation, hn, since angular frequency w = 2pn. The crucial feature of Planck's formula is the exponential factor, which causes the spectral intensity to decay rapidly at high frequencies, thereby resolving the classical “ultraviolet catastrophe.” This term was popularized by Paul Ehrenfest in 1911, the same year the first Solvay Conference was convened to address the crisis in radiation theory. However, the status of Planck’s constant was not resolved during the meeting and Einstein wrote: “…the h-disease looks ever more hopeless.” Planck’s later reflection—that “science advances one funeral at a time”—seems a fitting description of the transition of classical physics to quantum physics.

 

3. Johnson Noise

“What is the origin of the generated power P(ω) if the resistance R is only an ideal antenna in equilibrium with its environment at temperature T? It is the radiation I(ω) in the space at temperature T which impinges on the antenna and, as “received signals,” makes an effective generator (Feynman et al., 1963, p. 41-8).”

 

Feynman’s explanation for the origin of resistor noise may seem counterintuitive because it reframes the conventional understanding of Johnson noise. Rather than treating the noise solely as a result of random electron motion in a resistor, he reinterprets the resistor as an antenna immersed in a thermal radiation field. In this view, the resistor is not merely ‘generating’ noise, but it is ‘listening’ to the thermal radiation of its surroundings. At equilibrium, the resistor’s ability to dissipate energy (its resistance) exactly balances its fluctuations; it is continuously absorbing and re-radiating radiation like a blackbody. This reveals a fundamental reciprocity at thermal equilibrium: the fluctuations we observe are inseparable from the resistor’s dissipation, both reflecting its continuous energy exchange with the surrounding radiation field.

 

“Now let us return to the Johnson noise in a resistor. We have already remarked that the theory of this noise power is really the same theory as that of the classical blackbody distribution…… The two theories (blackbody radiation and Johnson noise) are also closely related physically… (Feynman et al., 1963, p. 41-8).”

 

Feynman could have stated the Fluctuation-Dissipation Theorem (FDT), which provides the unifying framework for both phenomena by establishing a fundamental link: the spectrum of thermal fluctuations in any system at equilibrium is determined by its dissipative properties. In the case of Johnson noise, the dissipation is dependent on the electrical resistance. Applying the FDT yields the Nyquist formula for voltage (noise) fluctuations. For blackbody radiation, the dissipation arises from the absorption and re-emission of radiation by matter, quantified by radiation damping and it is related to the thermal fluctuations. Applying the FDT to the electromagnetic field modes in a cavity leads to the Planck distribution of energy. Thus, the effects of both phenomena are concrete realizations of the same principle: the random thermal fluctuations are quantitatively linked to the dissipation of energy. They are not merely analogous but are derived from the same fundamental equation of statistical physics.

 

Key takeaways:

1. Energy quantization and statistical suppression of high frequencies

When a harmonic oscillator is confined within a cavity, it can absorb or emit energy only in discrete quanta. The resolution of the ultraviolet catastrophe comes from the quantization of energy combined with statistical weighting: high-frequency modes are exponentially suppressed by the Boltzmann factor. This same statistical factor underlies both blackbody radiation and Johnson (thermal) noise—it determines the probability that a system occupies a given energy state at thermal equilibrium.

2. Johnson–Nyquist noise as a thermodynamic phenomenon

Johnson (or Johnson–Nyquist) noise refers to the random voltage and current fluctuations generated by the thermal agitation of charge carriers in any resistive conductor. Far from being mere “unwanted interference,” Johnson noise is an intrinsic property of resistor at finite temperature. Its existence was predicted by Einstein (1907) more than two decades before Johnson’s experimental measurements and is explained by the fluctuation–dissipation theorem: any system capable of dissipating energy must also exhibit corresponding thermal fluctuations.

 

The Moral of the Lesson:

1. Science advances one funeral at a time

Planck’s (1949) famous quote: “a new scientific truth does not triumph by convincing its opponents … but because its opponents eventually die” highlights the sociological dimension of scientific change, emphasizing the stubborn mindset of scientists. Scientific revolutions, on this view, proceed as entrenched conceptual commitments give way to new theoretical frameworks adopted by succeeding generations (Kuhn, 1962). Planck’s own career exemplifies this dynamic: his quantum hypothesis initially faced resistance from advocates of classical physics but gained acceptance as the scientific community evolved. Conversely, Planck himself remained skeptical of Einstein’s photon and later developments in quantum mechanics, illustrating how even pioneering figures may resist subsequent conceptual breakthroughs.

 

2. Johnson noise as white noise

Johnson noise is effectively a kind of white noise over a broad frequency range. Tinnitus is sometimes known as the perceived internal "noise" or auditory hallucination, but white noise is an external sound used to manage it. While Feynman is not known to be a chronic tinnitus sufferer, he had a fascination with the subjective experience of “neural noise.” In his autobiography Surely You're Joking, Mr. Feynman!, he discusses the internal "noise" people experience, particularly when falling asleep or in sensory-deprivation tanks. Feynman was deeply protective of his "thinking machine" (his brain) and was terrified of anything that might interfere with his internal clarity. For a physicist, tinnitus can be particularly frustrating because it introduces "entropy" or "noise" into the very "quiet" environment required for deep mathematical focus. Currently, there is no effective pharmaceutical drugs to eliminate tinnitus.  Sound therapies using unstructured, random ("white") noise do not target the underlying neural mechanisms and may, in some cases, increase perceptual fatigue rather than provide relief.

 

3. The 17-years 'knowledge-to-action' duration

It takes an average of 17 years for a medical discovery to reach clinical practice (Balas & Boren, 2000). This 'knowledge-to-action' duration represents a significant failure in our healthcare system. The delay is driven not by generational resistance alone, but by layered institutional inertia, including regulatory constraints, misaligned incentives, and difficulties in translating controlled research into complex clinical settings (Morris et al., 2011). Tinnitus is an example of this delay: while many clinicians still rely strictly on medication, research suggests that non-pharmacological factors, such as cervical (neck) issues and metabolic health, are often the missing pieces of the puzzle (Michiels et al., 2015). While a definitive cure remains elusive, some patients may experience improvement through a combination of posture correction, stress management, and low-impact exercise (e.g., swimming and yoga), with the effectiveness of these strategies depending on individual medical conditions rather than the symptom alone.

 

Review questions:

1. How would you explain 'energy is quantized' is not a universal principle in quantum physics? (Hint: You may contrast the energy spectrum of a confined system, e.g., a harmonic oscillator in a cavity, with that of a free particle.)

2. How would you explain Feynman mistakenly credited Jeans with introducing the exponential cutoff factor in early blackbody theory? (Hint: Justify why this credit is incorrect by summarizing the contributions of Wilhelm Wien, Rayleigh, and Planck.)

3. Identify the fundamental theorem that unifies Johnson noise and blackbody radiation, and explain how it connects a system's dissipative property to the spectrum of its thermal fluctuations.

 

References:

Balas, E. A., & Boren, S. A. (2000). Managing clinical knowledge for health care improvement. In J. Bemmel & A. McCray (Eds.), Yearbook of Medical Informatics 2000 (pp. 65–70). Schattauer.

Ehrenfest, P. (1911). Welche Züge der Lichtquantenhypothese spielen in der Theorie der  Wärmstrahlung eine wesentliche Rolle? ​ Annalen Der Physik​, ​ 36 , 91–118. 

Einstein, A. (1907). Über die Gültigkeitsgrenze des Satzes vom thermodynamischen Gleichgewicht und über die Möglichkeit einer neuen Bestimmung der Elementarquanta. Annalen der Physik, 327(3), 569-572.

Feynman, R. P. (1985). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

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.

Kragh, H. (2000). Max Planck: the reluctant revolutionary. Physics World, 13(12), 31.

Michiels S, De Hertogh W, Truijen S, Van de Heyning P. (2015). Cervical spine dysfunctions in patients with chronic subjective tinnitus. Otol Neurotol., 36(4), 741-5.

Morris, Z. S., Wooding, S., & Grant, J. (2011). The answer is 17 years, what is the question: Understanding time lags in translational research. Journal of the Royal Society of Medicine, 104(12), 510–520.

Pais, A. (1979). Einstein and the quantum theory. Reviews of modern physics, 51(4), 863.

Planck, M. (1900). On the theory of the energy distribution law of the normal spectrum. Verh. Deut. Phys. Ges, 2(237), 237-245.

Planck, M. (1949). Scientific autobiography and other papers (F. Gaynor, Trans.). Philosophical Library.

Rayleigh, L. (1900). LIII. Remarks upon the law of complete radiation. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 49(301), 539-540.

Rayleigh, L. (1905). The dynamical theory of gases and of radiation. Nature, 72(1855), 54-55.