Section 38–5 Energy levels

(Ritz’s Combination Principle / Energy states / Energy quantization)

 

In this section, Feynman discusses Ritz’s Combination Principle of spectroscopy (or spectral lines), energy states, and energy quantization instead of simply energy levels. The section could be titled Ritz’s Combination Principle because it is about how the principle can be explained using energy states (or energy levels) and quantization of energy.

 

1. Ritz’s Combination Principle

“… if we find two spectral lines, we shall expect to find another line at the sum of the frequencies (or the difference in the frequencies), and that all the lines can be understood by finding a series of levels such that every line corresponds to the difference in energy of some pair of levels...... it is called the Ritz combination principle (Feynman et al., 1963, p. 38-7).”

 

According to Sommerfeld, Ritz states: “[b]y additive or subtractive combination, whether of the series formulae themselves, or of the constants that occur in them, formulae are formed that allow us calculate certain newly discovered lines from those known earlier (Sommerfeld, 1923, p. 205).” To acknowledge Ritz’s generalization of Rydberg’s empirical findings, the principle is often referred to as the Rydberg–Ritz combination principle. For example, if two spectral lines are observed with frequencies n₁₂ and n₂₃, one may expect a third line at n₁₃ = n₁₂ + n₂₃ (by addition). Conversely, if two spectral lines are known with frequencies n₁₂ and n₁₃, a third line may appear at the frequency n₂₃ = n₁₃ - n₁₂ (by subtraction). This principle was helpful in organizing spectral data, but it lacked a theoretical foundation until the development of quantum theory.

“This remarkable coincidence in spectral frequencies was noted before quantum mechanics was discovered, and it is called the Ritz combination principle (Feynman et al., 1963, p. 38-8).”

It may not be misleading to describe the spectral frequencies as remarkable coincidence because the observation reflects a systematic pattern encapsulated by Ritz’s combination principle, which is not completely accurate. Feynman might have offered greater clarity by specifying the conditions under which the principle holds. As Ritz noted in 1908, “The new principle of combination also finds application to other spectra, particularly to helium and the earth alkalies.” However, the principle is more accurate for hydrogen-like atoms, where electron-electron interactions are negligible. In multi-electron atoms, these interactions shift energy levels, leading to deviations from the simple additive or subtractive relationship between spectral frequencies. Thus, the predictive power (or “remarkable” coincidence) of the combination principle depends on the nature of the atomic system and the complexity of electron interactions.

 

Note: Feynman’s reference text Introduction to Modern Physics (Richtmyer et al., 1956) outlines four features of the Ritz’s combination principle:

1. Spectral Lines as Differences of Terms: The wave number of each line is conveniently represented as the difference between two numbers. These numbers have come to be called terms.

2. Ordered Term Sequences: The terms group themselves naturally into ordered sequences, the terms of each sequence converging toward zero.

3. Combinability of Terms: The terms can be combined in various ways to give the wave numbers of spectral lines.

4. Spectral Series and Convergence: A series of lines, all having similar character, results from the combination of all terms of one sequence in succession with a fixed term of another sequence. Series formed in this manner have wave numbers which, when arranged in order of increasing magnitude, converge to an upper limit.

In essence, the Ritz’s combination principle means that the wavenumber (inverse wavelength, ν) of any spectral line can be expressed as the difference between two spectral terms as follows: n = T₁ – T₂ = R/(2 + S)² – R/(m + P)² where m = 3, 4, 5…..

(Each term represents a quantized energy level divided by hc.)

2. Energy states

“Let us observe how it comes about from the point of view of amplitudes that the atom has definite energy states (Feynman et al., 1963, p. 38-8).”

It should be worthwhile to clarify the distinction between energy level and energy state, as the two terms—while related—have distinct meanings. This distinction is also noted in Feynman’s reference text, Introduction to Modern Physics. For instance, Richtmyer et al. (1956) write “[i]t should be emphasized that the energy levels represented in Fig. 199 do not necessarily represent the energy states of any single nucleon (p. 515).” In short, an energy level refers to a quantized energy values (eigenvalues), which can be experimentally measured, typically through spectroscopy. On the other hand, an energy state refers to the quantum state (eigenstate) associated with a given energy level, defined by a specific set of quantum numbers. However, the term energy state is less commonly used in quantum mechanics as compared to energy level nowadays.

The observation of spectral lines arises from the interaction between a quantum system (such as an atom) and discrete units of light energy, or light quanta. Einstein (1905) introduced the concept of light quanta to explain the photoelectric effect, but it is from a heuristic viewpoint. The term photon was later coined by American chemist Gilbert N. Lewis in a 1926 letter, originally to describe a unit of radiant energy; it was subsequently adopted to refer specifically to Einstein’s light quanta. In 1913, Bohr proposed that when an electron transitions between two stationary states, it emits a quantum of light whose energy is equal to the energy difference between the two states. This relationship is expressed by the Planck-Einstein-Bohr radiation condition w_nm = E_n – E_m in which w_nm is the frequency of emitted or absorbed radiation, whereas E_n and E_m are the energy of the initial and final states respectively.

 

Note: In a December 18, 1926 letter to Nature, Lewis writes: “… I therefore take the liberty of proposing for this hypothetical new atom, which is not light but plays an essential part in every process of radiation, the name photon.”

The concept of energy levels is a theoretical construct whose existence is inferred from the observation of spectral lines. In 1928, Walter Grotrian introduced the Grotrian diagram (or term diagram) in atomic spectroscopy, a visual representation of allowed electronic transitions between energy levels. Similarly, energy level diagrams can be used to illustrate Ritz’s principle, which showed that spectral frequencies correspond to the differences between two quantities called terms, later recognized as energy levels. Historically, the spectral lines were first explained by Bohr's (1913) atomic model, which introduced the notion of quantum jump between quantized energy levels. Although the concept of energy levels is commonly attributed to Bohr, he originally used the term stationary state to describe what we now understand as an energy state.

 

3. Energy quantization

“When 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).”

According to Kuhn (1997), Planck’s early papers from the 1900s did not explicitly state that the energy of a single oscillator must be restricted to discrete values in accordance with E = nhf, where n is an integer. Instead, Planck treated energy quanta as a mathematical hypothesis (Kragh, 2000). His cautious stance was understandable, given that energy is not inherently quantized in all physical contexts. For instance, in unbound systems—such as free electrons—energy can vary continuously (See below). In such cases, electrons are not confined within a potential well or subject to boundary conditions that would cause energy quantization. The term free electrons can be used to describe electrons that have been ionized or occupy high-energy states where they are no longer bound to atoms or molecules.

“… we are all familiar with the fact that confined waves have definite frequencies. For instance, if sound is confined to an organ pipe, ……. then there is more than one way that the sound can vibrate, but for each such way there is a definite frequency (Feynman et al., 1963, p. 38-8).”

In quantum mechanics, bound particles are described by wavefunctions that must satisfy specific boundary conditions—such as those imposed by the Coulomb potential in a hydrogen atom. These boundary conditions lead to quantized vibrational modes, which correspond to the discrete energy levels. Importantly, this quantization does not arise from any intrinsic discreteness of energy itself, but rather from the continuous wave-like behavior of particles and governed by the Schrödinger equation, combined with the constraints imposed by the system. An analogous situation occurs with standing sound waves in an organ pipe: only certain frequencies are permitted, determined by the pipe’s length and boundary (pressure) conditions. Similarly, in quantum systems, only specific energy states (or levels) are allowed.

The moral of the lesson: 1. Ritz’s combination —which describes spectral lines as sums or differences of spectral terms—can be understood in terms of discrete energy levels. Although Planck is often seen as the pioneer of quantized energy, he was famously reluctant to fully embrace the physical reality of energy quantization (Kragh, 2000).  His caution was not unfounded: in systems involving free electrons, energy can indeed take on a continuous range of values, unconstrained by boundary conditions.

 

2. Bohr remarked that the Rydberg’s formula and Balmer’s line as “the lovely patterns on the wings of butterflies; their beauty can be admired, but they are not supposed to reveal any fundamental biological laws (Cropper, 1970, p. 48).” Bohr's initial remark was not necessarily due to carelessness, but rather a skepticism typical of early quantum theorists faced with mysterious empirical patterns. Yet it was Bohr himself who later revealed the physical meaning behind these patterns with his atomic model—work that earned him the Nobel Prize.

 

Review questions:

1. How would you state Ritz’s combination principle and its limitations?

2. Would you prefer the term “energy state,” “stationary state,” or “energy level” and in which context?

3. How would you explain why energy becomes quantized in bound systems but remains continuous for free particles?

 

References:

Cropper, W. H. (1970). The quantum physicists and an introduction to their physics. Oxford University Press.

Einstein, A. (1905). On a heuristic viewpoint concerning the emission and transformation of light. Annalen der Physik, 17(6), 132-148.

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.

Lewis, G. N. (1926). The conservation of photons. Nature, 118(2981), 874-875.

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

Kuhn, T. S. (1997). The structure of scientific revolutions. Chicago: University of Chicago press.

Richtmyer, F. K., Kennard, E. H., Lauritsen, T., & Stitch, M. L. (1956). Introduction to modern physics (5th ed.). New York: McGraw-Hill.

Ritz, W. (1908). On a new law of series spectra. Astrophysical Journal, 28(10), 237–243.

Sommerfeld, A. (1923) Atomic Structure and Spectral Lines (H. L. Brose, trans.). London: Methuen.