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.