Rydberg Formula

The Rydberg formula organizes atomic spectral lines by writing inverse wavelengths as differences of simple terms. For hydrogen, it generalizes the Balmer Formula from the visible series to a family of spectral series.

Its historical importance is not merely numerical. The formula made the spectrum look like a difference law before energy levels had been formulated. Later quantum theory explained that structure as differences between atomic energy eigenvalues.

Rydberg Formula

For hydrogen at leading nonrelativistic Coulomb order, the modern Rydberg formula is

$$\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right), \qquad n_i>n_f.$$

Here $\lambda$ is the photon wavelength in vacuum, $R_H$ is the hydrogen Rydberg constant, and $n_i,n_f$ are positive integers in modern notation. The formula is normally used with $n_i$ larger than $n_f$ for emission; absorption reverses the physical process while using the same transition frequency.

It is often useful to introduce the spectroscopic wavenumber

$$\tilde{\nu} \equiv \frac{1}{\lambda}.$$

Then the same formula becomes

$$\tilde{\nu} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right).$$

Historical caution: $\tilde{\nu}$ is a wavenumber, not the ordinary frequency $\nu$. In vacuum they are related by

$$\nu=c\tilde{\nu}.$$

Spectral Series

Holding $n_f$ fixed and varying $n_i$ gives a spectral series. The main hydrogen series are:

• Lyman series: $n_f=1$, ultraviolet;
• Balmer series: $n_f=2$, visible and near ultraviolet;
• Paschen series: $n_f=3$, infrared;
• Brackett series: $n_f=4$, infrared;
• Pfund series: $n_f=5$, infrared.

The series limit occurs when $n_i$ becomes arbitrarily large. Taking $n_i\to\infty$ gives

$$\frac{1}{\lambda_{\mathrm{limit}}} = \frac{R_H}{n_f^2}, \qquad \lambda_{\mathrm{limit}} = \frac{n_f^2}{R_H}.$$

For the Balmer series, this gives a limiting wavelength near $365\,\mathrm{nm}$. That is the constant $B$ in Balmer’s wavelength formula.

Rydberg Constant

The Rydberg constant is a wavenumber scale. For an infinitely heavy nucleus, the modern constant is

$$R_\infty \simeq 1.0973731568\times10^7\,\mathrm{m}^{-1}.$$

The leading hydrogen spectrum uses a slightly different constant because both the electron and proton participate in the relative motion. Separating center-of-mass and relative coordinates introduces the reduced mass; literal electron orbits are not required. In modern notation,

$$R_H = R_\infty \frac{\mu}{m_e}, \qquad \mu= \frac{m_em_p}{m_e+m_p}.$$

Here $\mu$ is the electron-proton reduced mass. This refinement is a modern interpretation of the constant, not what Rydberg needed in order to recognize the spectral pattern.

The product $hcR_\infty$ is the infinite-nuclear-mass Rydberg energy scale. Numerically it is close to $13.6\,\mathrm{eV}$, the familiar leading hydrogen ionization energy scale.

Difference of Terms

A powerful way to read the formula is to define term values

$$T_n=\frac{R_H}{n^2}.$$

Then a hydrogen spectral wavenumber is a difference:

$$\tilde{\nu}_{i\to f} = T_{n_f}-T_{n_i}.$$

This difference structure matters. It says that the spectrum is not merely a list of unrelated lines. Lines are related by a common set of terms. Ritz later formulated a combination principle for spectral lines: observed wavenumbers can often be understood as differences between term values.

Modern quantum mechanics translates this into energy language. If photon energy is $h\nu=hc\tilde{\nu}$, then the Rydberg formula suggests energies of the form

$$E_n=-hc\frac{R_H}{n^2},$$

with spectral lines coming from

$$hc\tilde{\nu} = E_{n_i}-E_{n_f}.$$

The minus sign places the bound states below the ionization threshold.

Connection to Energy-Level Differences

The Rydberg formula prepared a central quantum idea: transition frequencies depend on energy differences, not on the mechanical frequency of a classical orbit. Bohr’s model used this idea explicitly through the frequency condition

$$h\nu=E_i-E_f.$$

For hydrogen, Bohr’s model reproduced the Rydberg pattern because it produced energies proportional to $-1/n^2$. Wave mechanics later explained the same energy pattern by solving the Schrödinger equation for the Coulomb potential.

The hydrogen atom derives the spectrum from the Coulomb Hamiltonian; the hydrogen spectrum formula collects the conventions.

Modern Interpretation

In the modern view, the Rydberg formula is the leading nonrelativistic spectrum of a one-electron Coulomb bound state. For a hydrogenic ion of nuclear charge $+Ze$ and nuclear mass $M$, the leading formula is $1/\lambda=R_\infty(\mu/m_e)Z^2(1/n_f^2-1/n_i^2)$, with $\mu=m_eM/(m_e+M)$. The $Z^2$ factor must not be omitted when using the hydrogen pattern for ions. Real spectra require corrections and selection rules:

• reduced-mass effects shift the constant;
• fine structure splits levels that are degenerate in the simplest formula;
• Lamb shifts and hyperfine structure further split hydrogen levels;
• finite nuclear size matters at high precision;
• transition intensities require matrix elements, not just energy differences;
• multi-electron atoms require electron-electron interactions and quantum statistics.

The formula is therefore both historically profound and physically bounded. It is not the whole theory of spectra, but it is one of the cleanest clues that atomic radiation is organized by quantized energy differences.

Common Mistakes

• Confusing wavenumber $\tilde{\nu}=1/\lambda$ with frequency $\nu$.
• Treating $R_\infty$ and $R_H$ as the same constant when reduced-mass precision matters.
• Reading the Rydberg formula as a complete theory of line intensities and selection rules.
• Assuming that all atoms obey the exact hydrogenic inverse-square formula.
• Forgetting that the formula was empirical before it was explained by Bohr theory or wave mechanics.

Exercises

1. If recoil is neglected, how does a corresponding transition wavelength in He⁺ compare with hydrogen?

Solution

He⁺ has one electron and $Z=2$. Its wavenumber is four times larger for the same pair of principal quantum numbers, so its wavelength is one quarter as large. For example, the transition corresponding to hydrogen Hα near $656\,\mathrm{nm}$ lies near $164\,\mathrm{nm}$ in He⁺. Reduced-mass differences give a smaller correction.

2. Estimate the wavelength of the Balmer Hα line using $R_H\simeq1.0968\times10^7\,\mathrm{m}^{-1}$.

Solution

For Hα, $n_i=3$ and $n_f=2$, so

$$\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R_H\frac{5}{36}.$$

Thus

$$\lambda \simeq \frac{36}{5R_H} \simeq 6.56\times10^{-7}\,\mathrm{m},$$

or about $656\,\mathrm{nm}$.

3. Derive the series limit for fixed $n_f$.

Solution

The Rydberg formula is

$$\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right).$$

As $n_i\to\infty$, the second term goes to zero. Therefore

$$\frac{1}{\lambda_{\mathrm{limit}}} = \frac{R_H}{n_f^2}, \qquad \lambda_{\mathrm{limit}} = \frac{n_f^2}{R_H}.$$

4. Why does the difference-of-terms form point naturally toward energy levels?

Solution

Photon energy is proportional to frequency and therefore to wavenumber in vacuum: $E_\gamma=hc\tilde{\nu}$. If spectral wavenumbers are differences $T_{n_f}-T_{n_i}$, then photon energies are differences of quantities proportional to those terms. Quantum theory identifies those quantities with atomic energy eigenvalues, up to the sign convention that bound states lie below the ionization threshold.

References

• J. J. Balmer, “Notiz über die Spectrallinien des Wasserstoffs,” Annalen der Physik und Chemie 25, 80–87 (1885), DOI: 10.1002/andp.18852610506.
• N. Bohr, “On the Constitution of Atoms and Molecules,” Philosophical Magazine 26, 1-25 (1913), DOI: 10.1080/14786441308634955.
• B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed., Pearson, 2003.
• H. Haken and H. C. Wolf, The Physics of Atoms and Quanta, 7th ed., Springer, 2005.
• NIST, CODATA Fundamental Physical Constants.
• W. Ritz, “Ueber ein neues Gesetz der Serienspektren,” Physikalische Zeitschrift 9, 521-529 (1908).
• J. R. Rydberg, “Recherches sur la constitution des spectres d’emission des elements chimiques,” Kungliga Svenska Vetenskapsakademiens Handlingar 23, 1-177 (1890).