Johann Jakob Balmer

Swiss mathematician

• Born: May 1, 1825; Lausanne, Switzerland
• Died: March 12, 1898; Basel, Switzerland

Nineteenth-century Swiss mathematician Johann Balmer deduced a formula that predicted the wavelengths of the Balmer series, which are spectral lines emitted by hydrogen atoms. The formula, now known as the Balmer formula, proved a key contribution to the later development of atomic theory.

Also known as: Johann Jacob Balmer

Primary field: Mathematics

Specialty: Atomic and molecular physics

Early Life

Johann Jakob Balmer was born on May 1, 1825, in the town of Lausanne, Switzerland, outside the city of Basel. He was the eldest son of Johann Jakob Balmer and Elisabeth Rolle Balmer. Balmer’s father was a prominent judge in Switzerland, at one point serving as the country’s chief justice.

After completing his primary education in the town of Liestal, Balmer attended high school in Basel. Throughout his schooling, Balmer displayed a strong mathematical bent and went on to pursue the discipline in college and beyond. He studied for a time at the University of Karlsruhe in the Grand Duchy of Baden, now part of Germany, and at the University of Berlin, located in the city that was then the capital of the Prussian Empire.

Returning to Switzerland, Balmer continued his education at the University of Basel, completing his PhD in mathematics in 1849. Balmer wrote his doctoral thesis on the cycloid—the curve that is traced when a point on the outside of a circle rotates along a straight line.

For the next ten years, Balmer taught mathematics in Basel. In 1859, he accepted a position at a women’s high school in Basel, teaching mathematics and calligraphy there until his death. In 1865, Balmer completed a postdoctoral dissertation bearing the unusual title “The Prophet Ezekiel’s Vision of the Temple, Clearly Portrayed and Architecturally Explained,” in which he relied on passages from the biblical Old Testament book of Ezekiel to describe the geometric contours of the Temple. After completing this thesis, Balmer served as a lecturer in mathematics at the University of Basel for the next thirty-five years.

Though he dedicated his career to mathematics, with an emphasis on geometry, Balmer did not build any enduring legacy in the field. As his thesis demonstrates, he explored some unconventional territory in his scholarly pursuits. One professed interest was numerology, a pseudoscience that holds that numbers, formulas, and the like can have mystical, divinatory qualities.

In 1868, at the age of forty-three, Balmer married Christine Pauline Rinck. Together they had six children.

Life’s Work

In the early 1880s, nearing the age of sixty, Balmer was at a loss as to what to study next. One of his friends and colleagues at the University of Basel, Professor Jakob Edward Hagenbach-Bischoff, suggested he bring his mathematical skills to bear on a problem in the field of physics.

In the previous decades, the science of spectroscopy had made vast strides and offered important insights into the realm of atomic physics, among other areas of scientific inquiry. By examining the light emitted and absorbed by the elements, the German physicist Gustav Kirchhoff had demonstrated that each element had its own unique spectrum that was expressed in distinct lines. What these lines indicated was difficult to decipher, but scientists believed that once properly interpreted, the various phenomena they were observing would shed light on some of the fundamental mysteries of physics. As the renowned physicist Arthur Schuster remarked, “It is the ambitious object of spectroscopy to study the vibrations of atoms and molecules in order to obtain what information we can about the nature of forces which bind them together.”

In 1862, the Swedish scientist Anders Jonas Ångström examined the emission spectrum of the element hydrogen. He observed three lines in the visible spectrum, a red line, a blue-green line, and a violet line. He later determined that the violet line was, in fact, two violet lines spaced closely together. These hydrogen lines would become known in time as the Balmer lines or the Balmer series.

Over the next decade or so, Ångström continued his investigations and perfected his techniques, even developing a unit to measure the wavelengths of the lines emitted by the various elements. Now known as the Ångström (Å), this unit measures about one-ten-billionth of a meter and is still in use today. In 1871, Ångström measured the wavelengths of the four visible lines of the hydrogen spectrum. He determined that the two violet lines were 4340.47 Å and 4101.74 Å, while the red and blue-green lines came out to 6562.852 Å and 4861.33 Å, respectively. These measurements stumped the scientific community for the next fourteen years. Scientists knew the lines were related, but no one could come up with the formula that tied them all together.

Equipped only with Ångström’s four measurements of the hydrogen lines, Balmer took up the problem in 1884. Having no background in physics, he approached the dilemma in strictly mathematical terms. The formula he developed for the wavelengths of the observed lines of the hydrogen atom is as follows: h(m²/(m² – n²)). The h in the formula, now known as the Balmer constant, represents what Balmer referred to as “the fundamental number of hydrogen,” which he determined to be 3645.6(mm/10–⁷). The n and the m in the formula were integers. The n equaled the number two, while m represented three, four, five, six, etc. Balmer suggested that altered versions of his formula, in which the value of n was an integer other than two, might predict the wavelengths of other varieties of hydrogen lines. This prediction was borne out in time.

In 1885, Balmer presented his findings in a paper entitled “Notes on the Spectral Lines of Hydrogen.” Among the questions he raised in his concluding remarks was whether the formula could be applied to the lines of other elements or whether each element had its own formula. He also highlighted how difficult it could be to determine an element’s fundamental number. Only the painstakingly precise calculations of hydrogen wavelengths by Ångström and other scholars had enabled him to derive the number. The wavelengths of other elements might not be so easily or accurately measured.

Though he had answered one of the discipline’s fundamental questions, Balmer did not pursue his physics inquiries much further. For the most part, he returned to his mathematical endeavors, writing only one more physics paper after his first in 1885. In this study, published in 1897, he focused on the spectra of several other elements. Balmer passed away in Basel on March 12, 1898.

Impact

Balmer’s formula effectively turned physics on its head. Its mathematical precision and relative simplicity hid an underlying opacity. There was no theory behind it. The formula worked brilliantly, but scientists did not know why.

Still, Balmer’s discoveries opened the door for other researchers. In the years after his 1885 revelations, scientists found other hydrogen series outside the visual spectrum. The wavelengths of these lines could be predicted by a slight alteration to Balmer’s formula.

In 1888, the Swedish physicist Johannes Rydberg rewrote Balmer’s formula to predict the wave number, the number of waves occupying a particular unit of length, on the hydrogen spectrum. Again, as with Balmer’s calculations, the Rydberg formula, with its accompanying Rydberg constant, functioned empirically: The numbers added up, but why they did so remained an elusive question. Researchers were finding valuable pieces to the puzzle; they just did not know what the puzzle was supposed to look like.

In the early 1900s, however, the puzzle became clear. It was at this time that physicists were starting to uncover the structure of the atom. In 1913, the Danish physicist Niels Bohr unveiled his model of the atom. According to Bohr’s theory, an atom’s electrons emit or absorb energy based on their movement from high-energy to low-energy orbits and vice versa. That distinct movement of energy is reflected in the spectral lines emitted and absorbed by particular atoms. With the development of atomic theory, what the lines of the Balmer series of hydrogen demonstrated became clear. They revealed the absorption and emission of energy by electrons in the hydrogen atom. Balmer’s formula, in turn, predicted in mathematical terms how that energy moved.

Given the advancements that followed, Balmer’s formula was a vital step in the development of atomic theory and later of quantum mechanics. Though Balmer had only dabbled briefly in the field of physics, he nevertheless laid the foundation for discoveries that would transform humanity’s understanding of the physical world.

Bibliography

Baggott, Jim. The Quantum Story: A History in 40 Moments. New York: Oxford UP, 2011. Print. Focuses on the development of quantum mechanics in the twentieth century and beyond, including Balmer’s contributions. Illustrations, bibliography, index.

Balmer, Johann Jakob. “Notes on the Spectral Lines of Hydrogen.” The World of the Atom. Trans. and ed. Henry A. Boorse and Lloyd Motz. Vol. 1. New York: Basic, 1966. Print. An English translation of Balmer’s groundbreaking 1885 article from the journal Annalen der Physik und Chemie, detailing the Balmer formula.

Segré, Emilio. From X-rays to Quarks: Modern Physicists and Their Discoveries. Mineola, NY: Dover, 2007. Print. Charts major discoveries in the field of physics from 1895 to 1980, touching on Balmer’s achievements as a prelude to advances that would provide the major components of modern physics. Illustrations, bibliography, index.