Arnold Sommerfeld

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Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist and renowned mentor of fellow physicists (he has supervised the greatest number of Nobel prize winners of any other supervisor). His notable achievements include the introduction of the 2nd and 4th quantum number, as well as x-ray wave theory.

Arnold's Life

Early Life

Arnold was born to Cacille Matthis and Franz Sommerfeld. The latter was a medical doctor from a prominent Königsberg family. They both fostered an intellectually curious environment for young Arnold, to which he attributes his similar zeal for the sciences.

He excelled in school, attending the Altstädtisches Gymnasium in Königsberg (notable peers include Hermann Minkowski, who would go on to introduce "Minkowski space-time", a geometrical representation of his former student Albert Einstein's theory of relativity, and Wilhelm Wien, a Nobel prize winning physicist). Arnold wasn't drawn to the sciences initially, in fact, preferring the humanities. Even so, he went on to study mathematics at the Albertina University in Königsberg.

Notable Teachers

While in university, Arnold was inspired and influenced by the esteemed staff of the mathematics department: David Hilbert, Adolf Hurwitz, and Carl Louis Ferdinand von Lindemann. Hilbert proved to be especially influential, as it was his course on the theory of ideal numbers that convinced Arnold to pursue abstract mathematics. In fact, despite there being a renowned school of Theoretical Physics, Arnold did not study physics. The quality of the university's teaching staff and his involvement with the Burschenschaft resulted in Arnold staying in Königsberg, while his peers moved on to other universities.

Burschenschaft

The Burschenschaft was a student organization/fraternity that Arnold joined in his college days. The resulting drunk antics proved to be detrimental to Arnold’s diligence with his schoolwork, often foregoing studying in favor of social events with the Burschenschaft. He also received a notable scar from fencing one of his fraternity brothers.

Doctorate/Post-University

Arnold received his pHD in 1891 after writing a dissertation about arbitrary functions in mathematical physics, under the advise of Lindemann. Afterwards, he decided to pursue a teaching diploma, in hopes of teaching high school mathematics. The following year, he passed the examinations, but embarked on a year-long stint with the military instead of teaching right away. He continued to engage in voluntary military exercises for years after.

Professional Career

Göttingen

At the time, Göttingen was regarded as the hub for advanced mathematics studies, drawing Arnold in. Upon arriving, he spent a year as an assistant for his family friend and Königsberg professor, Theodor Liebisch, at the Mineralogical Institute. He moved on to be Felix Klein’s assistant, becoming intimately familiar with his lectures due to his duties making copies of them for students to review. The direction of Arnold’s work was strongly influenced by Klein, including the theory of partial differential equations, the behavior of electromagnetic waves in wires, and studying fields created by electrons in motion.

In 1895, Arnold’s achievement under Klein’s supervision (particularly his habilitation thesis) granted him the position of Privatdozent. Those who hold this honor are deemed capable of independently advising pHD students, and are highly esteemed professors. His lectures at Göttingen were eventually compiled into a textbook series. Klein and Arnold also collaborated, producing a four-volume textbook called Die Theorie des Kreisels, which dealt with theory about gyroscopes.

It was during this time in his life that Arnold met his future wife, Johanna Höpfner, daughter of the university’s curator. They did not wed until after Arnold was appointed Chair of Mathematics at Clausthal, which he accepted for its salary. He remained in contact and collaboration with Klein and his other Gottingen peers, resulting in his becoming involved in the editing of Volume V of the mathematical encyclopedia. He continued to do so even after leaving Clausthal.

Encyklopädie der mathematischen Wissenschaften

Arnold was responsible for standardizing notation used in the encyclopedia, particularly for vectors and electromagnetic magnitudes, going as far as forming a vector commission to created a consistent, official symbolism.

Aachen

Thanks to recommendations from Klein, Arnold was appointed a professor of mechanics at the Technische Hochschule in Aachen, in an attempt to root engineering in a solid mathematical basis. His newfound colleagues proved tough to convince at first, regarding him with suspicion initially for his devotion to pure mathematics. He was eventually accepted by his peers, producing the theory of hydrodynamics.

Munich

Arnold’s reputation begins paying off for him as he becomes a theoretical physics professor at Munich. They provide him with facilities for seminars, assistants, and laboratory work, allowing him to field his theories with concrete experiments. He supervised almost 30 students in Munich’s doctorate program, primarily in theoretical physics. He was well-liked, treating students casually and appearing approachable while fostering skills to allow them to complete fulfilling research. He invited them to his home and his ski hut to break down the barrier between student and teacher.

The notable topics of study for Arnold while at Munich include the atomic spectra, quantum theory, and statistical mechanics.

Notable Work

Atomic Spectra

Arnold hypothesized and proved that x-rays are waves in an experiment using crystals and three-dimensional diffraction.

Quantum Theory

Arnold is responsible for introducing the second (magnetic/azimuthal) quantum number, and the fourth (inner/spin) quantum number, which proved valuable to the discovery of electron spin. He also modified the Bohr model of an atom to have elliptical orbits.

Second Quantum Number

The second quantum number, or the azimuthal quantum number, determines the shape and angular momentum of the atomic orbital it describes.

Fourth Quantum Number

The fourth quantum number represents the normalized spin of a particle.

Properties of Metals

Arnold propose a model of metals that regards electrons as degenerate electron gas, replacing Lorentz’s theory.

See also

• Quantum Theory – Arnold was very influential on this topic.
• Albert Einstein – Arnold’s work on the theory of relativity helped it to become accepted by critics.

Further reading

• M Eckert, Mathematics, experiments, and theoretical physics : the early days of the Sommerfeld school, Phys. Perspect. 1 (3) (1999), 238-252.

References

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Sommerfeld.html http://www.famousscientists.org/arnold-sommerfeld/ http://www.encyclopedia.com/doc/1G2-2830904080.html