Section 38–4 The size of an atom

Bohr radius / Rydberg energy / Stability of matter

 

In this section, Feynman discusses the Bohr radius, Rydberg energy, and stability of matter, extending beyond the simple topic of “the size of an atom.” A more precise title could be “Three Applications of the Uncertainty Principle” since these concepts were explained using the uncertainty principle. However, the section is also related to the stability of the hydrogen atom and the stability of matter.

 

1. Bohr radius

“This particular distance is called the Bohr radius, and we have thus learned that atomic dimensions are of the order of angstroms, which is right: This is pretty good—in fact, it is amazing, since until now we have had no basis for understanding the size of atoms! (Feynman et al., 1963, p. 38-6).”

 

Feynman estimated the order of magnitude of the Bohr radius using the uncertainty principle and Planck’s constant (h) instead of the reduced Planck constant (ℏ). While this method obtains the approximate atomic scale, it does not give the exact numerical factor. In contrast, Bohr introduced the quantization of angular momentum, postulating that an electron in a hydrogen atom follows discrete orbits: mvr = nℏ where n is an integer and ℏ is the reduced Planck constant. Although this assumption successfully explained atomic spectra, it lacked a deeper theoretical justification. Moreover, the Bohr radius is not a directly measurable quantity; it represents the most probable distance between the electron and nucleus in the ground state of hydrogen.

 

A more rigorous derivation arises from solving the Schrödinger equation for an electron in a Coulomb potential, leading to quantized energy levels and the Bohr radius as a fundamental length scale. Unlike Bohr’s model, which assumes quantization, this approach derives the Bohr radius naturally from the boundary conditions imposed on the electron’s wavefunction. While the uncertainty principle and Bohr’s quantization provide insights into atomic structure, Schrödinger’s equation offers a more consistent framework, revealing the Bohr radius as an intrinsic property of quantum wave behavior. More importantly, in quantum mechanics, the electron does not follow a definite trajectory; instead, its position is governed by a probability distribution described by its wavefunction. This understanding is closely linked to the stability of atoms, as electrons do not spiral inward but instead occupy discrete, quantized energy levels, preventing atomic collapse.

 

2. Rydberg energy

“However, we have cheated, we have used all the constants in such a way that it happens to come out the right number! This number, 13.6 electron volts, is called a Rydberg of energy; it is the ionization energy of hydrogen (Feynman et al., 1963, p. 38-6).”

 

In the Audio Recordings [at the end of this lecture (first try), 56 min: 05 sec], Feynman says something like this: “… This is just an order of magnitude. Actually, I’ve cheated you—I put the constant just where I want… at the right place and this does come out as the mean radius of the hydrogen atom and this does come out as the actual binding energy of hydrogen but we have no right to believe that. Thank you.” In a sense, Feynman’s derivation of the Rydberg energy relied on a “working backward” approach, using the known value of the Bohr radius. However, historically, the Bohr radius was derived from the Rydberg energy (or equivalently, the Rydberg constant), not the other way around. Therefore, while this application of the uncertainty principle provides a useful heuristic, it should not be taken seriously as a formal derivation.

 

The Rydberg energy (the ionization energy of hydrogen, 13.6 eV) was not initially derived from theory but was instead determined empirically from atomic spectra. Key contributors to this discovery included Johannes Rydberg and earlier spectroscopists such as Balmer, Ångström, and Paschen. Rydberg established the “Rydberg constant” by analyzing spectral data, without fully understanding its deeper significance. Bohr later provided a theoretical explanation for the Rydberg formula using his semiclassical model, making a crucial step in confirming the quantization of energy levels. However, the modern determination of the Rydberg constant relies on high-precision spectroscopy and least-squares data fitting, rather than a direct measurement from hydrogen spectra.

 

3. Stability of matter

“So we now understand why we do not fall through the floor. As we walk, our shoes with their masses of atoms push against the floor with its mass of atoms. In order to squash the atoms closer together, the electrons would be confined to a smaller space and, by the uncertainty principle, their momenta would have to be higher on the average, and that means high energy; the resistance to atomic compression is a quantum-mechanical effect and not a classical effect (Feynman et al., 1963, p. 38-6).”

 

Feynman’s explanation of why we do not fall through the floor could incorporate the term Pauli exclusion principle and Coulomb force. If an electron were confined to a smaller region near a nucleus, its position uncertainty (Δx) would decrease. By the uncertainty principle, this would necessitate an increase in momentum uncertainty (Δp), leading to higher kinetic energy. This increase in energy counterbalances the attractive Coulomb force, preventing the collapse of atom. Additionally, Coulomb repulsion between the negatively charged electron clouds of adjacent atoms further resists compression. As electrons are forced closer together, their wavefunctions would overlap (or antisymmetric). Due to the Pauli exclusion principle, which does not allow two electrons from occupying the same quantum state, it effectively provides an additional mechanism that prevents matter from collapsing. This quantum mechanical effect, along with Coulomb repulsion —explain the stability of matter.

 

Mathematical Proof of Stability

Feynman’s question, “Why do we not fall through the floor?”, is related to the second kind of stability, now commonly known as the stability of matter. This problem was first mathematically solved in 1967 by Freeman Dyson and Andrew Lenard, about five years after this lecture of Feynman. Their analysis showed that the stability of matter relies on the Pauli exclusion principle. Building on this work, Elliott Lieb and Walter Thirring refined Dyson and Lenard’s approach by introducing the Lieb-Thirring inequality, providing a more elegant and conceptually clear proof. Thus, the stability of matter—why we do not fall through the floor—can be explained through a combination of the Pauli exclusion principle and Coulomb repulsion.

 

Note: In the preface of the book titled The Stability of Matter: From Atoms to Stars, Dyson writes: “Lenard and I found a proof of the stability of matter in 1967. Our proof was so complicated and so unilluminating that it stimulated Lieb and Thirring to find the first decent proof. (...) Why was our proof so bad and why was theirs so good? The reason is simple. Lenard and I began with mathematical tricks and hacked our way through a forest of inequalities without any physical understanding. Lieb and Thirring began with physical understanding and went on to find the appropriate mathematical language to make their understanding rigorous. Our proof was a dead end. Theirs was a gateway to the new world of ideas (Lieb, 2005, p. xi)”.

Review Questions:

1. Should the Bohr radius be derived using Planck constant or the reduced Planck constant?

2. Should the Rydberg energy be derived using the uncertainty principle and Bohr radius?

3. Would you explain the “why we do not fall through the floor” using the term Pauli exclusion principle or/and Coulomb force?

 

The moral of the lesson: While Bohr radius, Rydberg energy, and stability of matter could be explained using the uncertainty relation by working backward, this is not a rigorous method for establishing stability of hydrogen atom and stability of matter.

 

References:

1. 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.

2. Lieb, E. H. (2005). The stability of matter: from atoms to stars. Heidelberg, Berlin: Springer.

Section 38–3 Crystal diffraction

Bragg diffraction / Bragg condition / Bragg cutoff

 

In this section, Feynman discusses Bragg diffraction, Bragg condition, and Bragg cutoff. Interestingly, the section is titled as “crystal diffraction,” but he explains the phenomenon as the reflection of particle waves from a crystal. However, the term Bragg diffraction is more appropriate to acknowledge the contributions of W. H. Bragg and his son W. L. Bragg in x-ray diffraction, a discovery for which they received the 1915 Nobel Prize in Physics.

 

1. Bragg diffraction

“Next let us consider the reflection of particle waves from a crystal. A crystal is a thick thing which has a whole lot of similar atoms—we will include some complications later—in a nice array. The question is how to set the array so that we get a strong reflected maximum in a given direction for a given beam of, say, light (x-rays), electrons, neutrons, or anything else. In order to obtain a strong reflection, the scattering from all of the atoms must be in phase. There cannot be equal numbers in phase and out of phase, or the waves will cancel out. The way to arrange things is to find the regions of constant phase, as we have already explained; they are planes which make equal angles with the initial and final directions (Feynman et al., 1963).”

 

Feynman could have continued using the term wave packets or wave trains instead of particle waves to model x-rays, electrons, and neutrons. More importantly, the term reflection is a misnomer, as the underlying process is diffraction, not simple specular reflection. A more precise term is Bragg diffraction, which accurately describes the phenomenon as wave interference arising from periodic layers of atoms rather than mere bouncing off a surface. The process involves the scattering of incoming waves that interact with parallel atomic planes, leading to constructive interference among outgoing waves. The scattering phenomenon is also known as Bragg scattering or elastic scattering, as the interaction between the incoming waves and the crystal lattice does not result in an observable change in energy—only a change in direction.

 

Historically, in 1912, Max von Laue proposed that crystals act as three-dimensional diffraction gratings for x-rays. To simplify analysis, the x-ray source and detector are idealized as being far from the crystal, allowing both the incident and outgoing waves to be treated as plane waves.  Specifically, x-rays induce oscillations in the electrons within the crystal, causing them to emit secondary x-rays. These scattered waves interfere and give rise to diffraction patterns at certain angles. This process is a form of elastic scattering, meaning that while the x-rays interact with the crystal lattice, their wavelength remains constant. The experiments showed that x-rays have wavelike properties and provided insight into the periodic arrangement of atoms in crystals.

 

2. Bragg conditions:

“… the waves scattered from the two planes will be in phase provided the difference in distance travelled by a wavefront is an integral number of wavelengths. This difference can be seen to be 2dsinθ, where d is the perpendicular distance between the planes. Thus the condition for coherent reflection is 2dsinθ = nλ (n=1,2,…) (Feynman et al., 1963).”

 

Feynman states the condition for coherent reflection as 2d sin θ = nλ (n = 1, 2,…), where d is the interplanar spacing. However, instead of single condition, we may emphasize three key Bragg conditions:

(1) Bragg’s equation: For diffraction to occur, the scattered waves must interfere constructively satisfying Bragg’s equation, nλ = 2dsin θ.

(2) Angle of diffraction: The incident and diffracted waves must obey the relation: Angle of Incidence = Angle of Diffraction.

(3) Interplanar spacing: The crystal must have a regular, periodic arrangement of atoms with a well-defined interplanar spacing d.

Additionally, while Bragg’s equation provides a simplified scalar description of diffraction, the Laue condition offers a more general vector-based formulation that relates the incident and diffracted wave vectors. Bragg’s equation can be derived as a special case of the Laue condition, particularly when considering diffraction from parallel atomic planes.

 

“If, on the other hand, there are other atoms of the same nature (equal in density) halfway between, then the intermediate planes will also scatter equally strongly and will interfere with the others and produce no effect. So d in (38.9) must refer to adjacent planes; we cannot take a plane five layers farther back and use this formula! (Feynman et al., 1963).”

 

Perhaps Feynman could have clarified the distinction between intermediate planes and adjacent planes. In a crystal, multiple sets of parallel planes exist, each with its own interplanar spacing, leading to different pairs of incidence and diffraction angles (as shown below) that satisfy the conditions for constructive interference. Bragg’s law applies not only to regular lattice structures but also to specific lattice planes, such as hexagonal planes in certain crystals. Furthermore, if the diffraction conditions hold for a particular atomic layer and its neighboring layers, they can be assumed to apply consistently across all layers with identical spacing. Experimentally, x-rays penetrate deeply into the crystal, allowing diffraction to arise from thousands or even millions of layers, collectively contributing to the observed diffraction pattern.

 

Source: (Mansfield & O'sullivan, 2020)

3. Bragg cutoff:

“Incidentally, an interesting thing happens if the spacings of the nearest planes are less than λ/2. In this case (38.9) has no solution for n. Thus if λ is bigger than twice the distance between adjacent planes then there is no side diffraction pattern, and the light—or whatever it is—will go right through the material without bouncing off or getting lost. So in the case of light, where λ is much bigger than the spacing, of course it does go through and there is no pattern of reflection from the planes of the crystal (Feynman et al., 1963).”

 

Bragg cutoff refers to the wavelength λ_b beyond which Bragg diffraction cannot occur. This wavelength can be determined by substituting two extreme values, θ = 90° (maximum angle) and n =1 (minimum order) into Bragg’s equation. Mathematically, if λ > 2d​, no real angle θ satisfies Bragg’s law, making diffraction impossible. However, Feynman’s claim that “light will go right through the material without bouncing off or getting lost” oversimplifies the situation. Even if Bragg diffraction does not occur, incident waves can still interact with the crystal through scattering, absorption, or transmission. In materials with sufficient electron density, electromagnetic radiation can be significantly absorbed rather than simply passing through unaffected. Bragg cutoff represents a fundamental limit in crystal diffraction, defining the range of wavelengths that can undergo diffraction.

 

“If we take these neutrons and let them into a long block of graphite, the neutrons diffuse and work their way along (Fig. 38–7). They diffuse because they are bounced by the atoms, but strictly, in the wave theory, they are bounced by the atoms because of diffraction from the crystal planes. It turns out that if we take a very long piece of graphite, the neutrons that come out the far end are all of long wavelength!... In other words, we can get very slow neutrons that way. Only the slowest neutrons come through; they are not diffracted or scattered by the crystal planes of the graphite, but keep going right through like light through glass, and are not scattered out the sides (Feynman et al., 1963).”

 

Feynman’s statement—“if we take a very long piece of graphite, the neutrons that come out the far end are all of long wavelength!”— oversimplifies the underlying physics. As neutrons diffuse through graphite, they undergo multiple collisions with carbon atoms, losing kinetic energy in a process known as neutron moderation. This slowing-down effect is why graphite serves as a moderator in nuclear reactors, reducing neutron energy to facilitate optimal fission reactions. In a sufficiently long piece of graphite, the neutrons that emerge at the far end are mainly slower neutrons with longer de Broglie wavelengths. This occurs partly because high-energy (short-wavelength) neutrons satisfy Bragg's diffraction condition for scattering from the crystal planes and are thus deflected. In contrast, slow neutrons, which do not satisfy Bragg’s condition, pass through the lattice with minimal scattering, similar to light passing through glass.

 

Review questions:

1. What is Bragg diffraction? Is it due to the reflection of particle waves from a crystal?

2. How would you explain the Bragg condition(s)? How many are there?

3. How would you explain the Bragg cutoff? Does light simply pass through the material without bouncing off or getting lost?

 

The moral of the lesson: The wave properties of x-rays, electrons, and neutrons are revealed through Bragg diffraction, which occurs when the path difference between the adjacent waves scattered from different planes is an integer multiple of the wavelength.

 

References:

1. 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.

2. Mansfield, M. M., & O'sullivan, C. (2020). Understanding physics. Hoboken, NJ: John Wiley & Sons.

Section 38–2 Measurement of position and momentum

(Single slit / Diffraction grating / Uncertainty relation)

 

In this section, Feynman discusses the uncertainty of complementary variables involving a slit and diffraction grating, as well as the uncertainty relation in wave theory. A more fitting title for this section might be “Inherent Uncertainty of Position and Momentum.” It emphasizes that while measurement can increase uncertainty, the uncertainty itself is intrinsic to the system rather than solely a result of measurement (as implied by the title, “Measurement of Position and Momentum”).

 

1. Single slit:

“How does the pattern become spread? To say it is spread means that there is some chance for the particle to be moving up or down, that is, to have a component of momentum up or down. We say chance and particle because we can detect this diffraction pattern with a particle counter, and when the counter receives the particle, say at C in Fig. 38–2, it receives the entire particle, so that, in a classical sense, the particle has a vertical momentum, in order to get from the slit up to C (Feynman et al., 1963, p. 38–2).”

 

Feynman explains diffraction through a single slit using a particle-based perspective. While this explanation aligns with quantum mechanics’ probabilistic nature, it reflects Feynman’s preference for a particle-centric view. In wave theory, light can be represented as wave packets, which are superpositions of waves with slightly different wavelengths. The wave packets may provide a better understanding of diffraction, where the single slit acts as a device that spreads the wave’s wavelength components across a range of angles. The relationship between the slit width and the diffraction pattern can be understood via the uncertainty principle: narrowing the slit increases uncertainty in the momentum of wave packets, resulting in a broader angular spread, and vice versa.

 

The spreading of wave packets by the slit can be interpreted as an environmental interaction, but it is equivalent to a measurement. As light passes through a slit, it becomes localized, effectively corresponding to a position measurement. However, this localization in position introduces an uncertainty in the vertical wave number k_y​, which is directly related to the vertical momentum p_y​ (= ℏk_y​). Based on the wave theory, when the wave packet is highly localized, the momentum uncertainty (Δp_y) is large, causing greater spreading. In this framework, the spreading of the wave packet is not a result of the particle "moving up or down" in the classical sense, but rather an inherent property of wave-like behavior arising from the superposition of wave components.

 

“Sometimes people say quantum mechanics is all wrong. When the particle arrived from the left, its vertical momentum was zero. And now that it has gone through the slit, its position is known. Both position and momentum seem to be known with arbitrary accuracy (Feynman et al., 1963, p. 38–3).”

 

It is worth mentioning that Einstein did not say quantum mechanics was all wrong, but critiqued its completeness. His concern was whether quantum mechanics provides a complete description of reality or merely reflects statistical ignorance of some deeper, hidden reality. Instead of assuming a perfect correlation between a wave packet’s position and momentum, it is crucial to recognize the predictive limitations of quantum mechanics. Once a particle or wave packet interacts with the slit, its momentum is irreversibly disturbed, i.e., the act of measurement (position localization) fundamentally alters the system, increasing momentum uncertainty. However, questions about the nature of quantum correlations and their possible connections to deeper underlying mechanisms continue to drive theoretical and experimental investigations.

 

2. Diffraction grating:

“Suppose we have a grating with a large number of lines (Fig. 38–3), and send a beam of particles at the grating…… That is, the waves which form the diffraction pattern are waves which come from different parts of the grating. The first ones that arrive come from the bottom end of the grating, from the beginning of the wave train, and the rest of them come from later parts of the wave train, coming from different parts of the grating, until the last one finally arrives, and that involves a point in the wave train a distance L behind the first point (Feynman et al., 1963, p. 38–3).”

 

It is remarkable that Feynman used Rayleigh’s criterion of resolution to derive an uncertainty relation, but this criterion is not based on a fundamental physical principle—it is a guideline or convention for resolving two overlapping waves. The key factor in determining the diffraction pattern produced by a grating is the spread of wavelengths within the wave trains. The length of a wave train is crucial because it is directly related to its wavelength spread (Δλ). A longer wave train has a narrower spread of wavelengths, leading to sharper and distinct diffraction peaks, and vice versa. The term wave train is appropriate here because it conveys the idea of a longer wave, as compared to the shorter localized wave packet, which is more apt for modeling particles passing through a single slit.

 

The sharpness of diffraction peaks is not fundamentally limited by the grating itself but rather by the spectral composition (distribution of wavelengths and relative strengths) of the wave train.  If the wavelength spread (Δλ) of the wave train exceeds the grating’s resolving capability, the diffraction pattern will remain blurred, regardless of the number of lines on the grating or width of the slits. Interestingly, the distance L, corresponding to the wave train’s coherence length, can also be interpreted as the minimum length required for using the entire grating effectively. More important, the grating functions like a Fourier transform, decomposing the wave train into its constituent wavelengths. This is analogous to how the human ear—specifically, the cochlea, a fluid-filled spiral structure—distinguishes sounds of different frequencies (See below).

(Source: Parker, 2018)

 

3. Uncertainty relation:

“Now this property of waves, that the length of the wave train times the uncertainty of the wave number associated with it is at least 2π, is a property that is known to everyone who studies them. It has nothing to do with quantum mechanics. It is simply that if we have a finite train, we cannot count the waves in it very precisely (Feynman et al., 1963, p. 38–4).”

 

Feynman’s remark underscores a fundamental fact: a finite wave train does not have a precisely defined wavelength (or wave number). More generally, the product of uncertainties in wave number (Δk) and position (Δx, the length of a wave packet or wave train) can be normalized to a constant such as 2π or 1 for simplicity (see below), depending on the chosen units. This reflects the nature of waves rather than being tied to any specific measurement process. The spectral composition of a wave train depends on its length: a long wave train consists of a narrow spread of wavelengths, akin to playing a single note on a flute for an extended time—an almost pure tone. Conversely, a short wave train has a broad range of frequencies, similar to pressing all the keys on a piano simultaneously, thereby produces a short burst of sound. In short, the length of the wave train tells us about how “pure” or “mixed” the wave is (in terms of its wavelength).

Source: Wave Equation, Wave Packet Solution

The uncertainty relation in wave theory is closely related to Fourier series, where any wave can be represented as a sum of sinusoidal components. Though Feynman does not explicitly specify Fourier transforms, his discussion aligns with Fourier’s principles. Essentially, a shorter wave train needs more wave components of differing wavelengths to form its sharp edges and transient nature, whereas a longer wave train is dominated by a narrower set of frequencies, resulting in a smoother, more uniform waveform. This relation underscores the fundamental trade-off between localization in space and precision in wavelength, mirroring the uncertainty principle in quantum mechanics. This insight, deeply rooted in Fourier analysis, provides a mathematical foundation for understanding the intrinsic limits of measurement in both classical wave theory and quantum mechanics.

 

Note: The section title “Measurement of position and momentum” could be misleading, as the uncertainty relation applies universally to wave phenomena, not just quantum measurement. In a paper on the quantum postulate, Bohr (1928) expressed this clearly: “[r]igorously speaking, a limited wave-field can only be obtained by the superposition of a manifold of elementary waves corresponding to all values of ν and σ_x, σ_y, σ_z. But the order of magnitude of the mean difference between these values for two elementary waves in the group is given in the most favourable case by the condition Δt Δν = Δx Δσ_x = Δy Δσ_y = Δz Δσ_z = 1, where Δt, Δx, Δy, Δz denote the extension of the wave-field in time and in the directions of space corresponding to the co-ordinate axes. These relations — well known from the theory of optical instruments, especially from Rayleigh's investigation of the resolving power of spectral apparatus — express the condition that the wave-trains extinguish each other by interference at the space-time boundary of the wave-field.”

 

Review Questions:

1. How would you explain the spreading of a particle (or wave packet) through a slit?

2. Is it legitimate to derive an uncertainty relation using the Rayleigh’s criterion of resolution?

3. How would you explain inherent uncertainties using the Fourier Transform?

 

The moral of the lesson: The uncertainty of complementary variables in quantum mechanics—whether in a slit or diffraction grating—can be changed by environmental conditions or measurement, however, the uncertainty relation is already known as an inherent property of waves before the quantum revolution.

 

References:

1. Bohr, N. (1928). The Quantum Postulate and the Recent Development of Atomic Theory. Nature, 121, 580-590.

2. 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.

3. Parker, J. A. (2018). Image reconstruction in radiology. Boca Raton, FL: CRC press.