Arnold Sommerfeld

How to use this course

Arnold Sommerfeld trained as a pure mathematician, worked for a decade on engineering problems, and then built in Munich the most productive school of theoretical physics of the early twentieth century. Four of his doctoral students won Nobel Prizes; he was nominated 84 times and never did. His own work spans exact solutions of wave equations, lubrication, wireless telegraphy, relativity, the old quantum theory and the electron theory of metals. This course follows that range, lesson by lesson, alongside the story of a teacher who kept his institute open to Einstein’s physics under a regime that called it Jewish.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 to 07 cover the mathematical and engineering physics of 1895–1912; Lessons 08 to 10 the quantum work of 1915–1928. Formulas are in modern notation, with Gaussian units for the atomic lessons; where Sommerfeld’s own notation differs, the text says so.

Timeline

Königsberg, 1868–1893

Arnold Johannes Wilhelm Sommerfeld was born in Königsberg, the capital of East Prussia, on 5 December 1868. His father, Franz Sommerfeld, was a physician with a strong interest in natural history; his mother, Cäcilie Matthias, was the daughter of a builder from Potsdam. He entered the Altstädtisches Gymnasium in 1875 and was a good pupil with wide interests, as fond of literature and history as of mathematics.

He passed the Abitur in 1886 and enrolled at the University of Königsberg. The university had a strong mathematical tradition reaching back to Carl Gustav Jacobi, and Sommerfeld’s teachers included Ferdinand von Lindemann, who had proved the transcendence of \(\pi\) in 1882, Adolf Hurwitz and the young David Hilbert. He also attended physics lectures by Paul Volkmann and joined a student fraternity, a Burschenschaft, which took a good deal of his time.

He received his doctorate on 24 October 1891 with the dissertation Die willkürlichen Functionen in der mathematischen Physik (The arbitrary functions of mathematical physics). The thesis treated the representation of functions by eigenfunction expansions of the kind that arise in heat conduction. Lindemann is usually named as his advisor; the historian David Cassidy has pointed to Volkmann. Sommerfeld then passed the state examination for secondary-school teaching in 1892 and completed a year of military service with a reserve regiment in Königsberg, ending in September 1893.

Göttingen with Klein, 1893–1897

In October 1893 Sommerfeld went to Göttingen as assistant to the mineralogist Theodor Liebisch. The work bored him, and Felix Klein, the most influential mathematician in Germany, noticed the young man in his lectures. In September 1894 Klein made him his own assistant, responsible for writing up Klein’s lecture notes for the students’ reading room. It was the decisive appointment of Sommerfeld’s life. Klein wanted mathematics to serve physics and engineering, and he gave his assistant problems from both.

The first was diffraction. The theory of Gustav Kirchhoff gave good approximate answers for light passing an edge but rested on inconsistent boundary conditions. Sommerfeld looked for exact solutions of the wave equation. His habilitation thesis on the mathematical theory of diffraction was accepted in March 1895, and he became a Privatdozent in mathematics at Göttingen. The next year he published the exact solution for a perfectly reflecting half-plane, the subject of Lesson 04. The key idea was to let the solution live on a two-sheeted Riemann surface, a device borrowed from Bernhard Riemann’s function theory and a long way from the optics laboratory.

The second was the spinning top. Klein lectured on the top in 1895–96 and asked Sommerfeld to work the lectures into a book. It grew into the four-volume Über die Theorie des Kreisels (1897–1910): the first two volumes on the mathematical theory, the last two on applications to astronomy, ship stabilizers, torpedoes and gyrocompasses. Klein also asked him to edit the volume on physics of the Encyklopädie der mathematischen Wissenschaften, a task that brought him into contact with nearly every theoretical physicist in Europe. In Göttingen he met Johanna Höpfner, daughter of the university curator Ernst Höpfner.

Clausthal, Aachen and Munich, 1897–1911

In October 1897 Sommerfeld became professor of mathematics at the Bergakademie in Clausthal, a mining academy in the Harz mountains. The salary allowed him to marry Johanna Höpfner the same year; they had three sons and a daughter. In 1900 he moved to the Technische Hochschule in Aachen as professor of technical mechanics. Engineers expected little of a mathematician, and Sommerfeld set out to earn their respect. He worked on the vibrations of machine foundations, the strength of materials, and, in 1904, the theory of lubricated bearings (Lesson 05). He took part in the debates of engineering societies and learned to write for engineers. Peter Debye, a student of electrical engineering at Aachen, became his assistant.

He also kept up with electron theory. In 1904–05 he worked out the fields of an electron moving faster than light in a medium, then a hypothetical question. When Albert Einstein published special relativity in 1905, Sommerfeld was at first sceptical, but he came round within a few years.

In 1906 the University of Munich called him to its chair of theoretical physics, vacant since Ludwig Boltzmann left in 1894, and gave him a small institute with a laboratory. Debye followed him and took his doctorate in 1908. In 1910 he recast relativity in four-dimensional vector notation, building on the work of Hermann Minkowski, whose 1907 lecture on the relativity principle he later edited for publication. His seminar quickly drew able students. In 1909 he published a theory of radio-wave propagation over the Earth (Lesson 06); from 1911 he turned to the quantum, presenting a paper on the role of Planck’s constant in molecular processes at the first Solvay Conference in Brussels.

Diffraction by a half-plane

For the general reader. Light passing the edge of an obstacle bends into the shadow and forms fringes. For a century, physicists computed these fringes with approximations. In 1896 Sommerfeld found the first exact solution: a wave meeting the straight edge of a thin perfectly conducting sheet. He did it by imagining the wave living on two copies of the plane, glued together along a cut.

The problem

Let a time-harmonic wave \(e^{-i\omega t}\) with wavenumber \(k\) fall on the half-plane \(\{y=0,\ x\ge0\}\), the screen. Use polar coordinates \((r,\phi)\) with \(0\le\phi\le2\pi\), so the screen is at \(\phi=0\) and \(\phi=2\pi\). For an electric field polarized parallel to the edge (the \(z\)-axis), \(u=E_z\) satisfies the Helmholtz equation and vanishes on the screen:

The incident wave is \(u_i=e^{-ikr\cos(\phi-\phi_0)}\), arriving from the direction \(\phi_0\), with \(0<\phi_0<\pi\).

The idea

A plane wave on the ordinary plane has period \(2\pi\) in \(\phi\). Sommerfeld built a wave function with period \(4\pi\), a “branched” plane wave on the double cover of the plane with branch point at the edge. The image method then works as for a full mirror, with the image wave on the same two-sheeted surface. The result can be written with the Fresnel-type integral

which tends to 1 as \(a\to+\infty\) and to 0 as \(a\to-\infty\).

Sketch. Each term has the form \(e^{-ikr\cos\psi}F(\sqrt{2kr}\cos\frac\psi2)\). Write \(\xi=\sqrt{2kr}\cos\frac\psi2\). A direct calculation shows that \(v=e^{-ikr\cos\psi}F(\xi)\) satisfies the Helmholtz equation because \(F'(\xi)\propto e^{i\xi^2}\) and \(e^{-ikr\cos\psi}e^{i\xi^2}=e^{ikr}\) is independent of \(\psi\), which makes the extra terms cancel. At \(\phi=0\) the two terms are equal, and at \(\phi=2\pi\) the half-angles change sign together, so \(u=0\) on both faces. Changing \(\phi\) by \(2\pi\) flips the sign of each argument of \(F\): the solution has period \(4\pi\), which is the branching.

Reading the solution

Far from the edge, \(F\approx1\) where its argument is large and positive, and \(F\approx0\) where it is large and negative. The first term thus gives the incident wave in the lit region and nothing in the shadow; the second gives the reflected wave where reflection is geometrically possible. Near the shadow boundaries, where the argument of \(F\) passes through zero, \(F\) varies smoothly and produces the Fresnel fringes. Elsewhere the asymptotic expansion \(F(a)\sim\frac{e^{-3i\pi/4}}{2\sqrt\pi}\frac{e^{ia^2}}{a}\) as \(a\to-\infty\) (and the same form for \(F(a)-1\) as \(a\to+\infty\)) gives a cylindrical wave from the edge:

This is the edge-diffracted wave. Its amplitude falls as \(r^{-1/2}\) and blows up on the shadow and reflection boundaries, where the full Fresnel form must be used. The geometrical theory of diffraction of Joseph Keller (1950s) takes such edge coefficients from Sommerfeld’s solution and uses them for general obstacles.

Tops, bearings and turbulence

For the general reader. Between 1895 and 1908 Sommerfeld worked on the mechanics engineers needed: why a fast top stands upright, why a shaft in an oiled bearing does not touch the metal, and why smooth flow in a pipe turns turbulent. In each case he turned a practical question into a precise equation and solved it, or showed how it should be solved.

The heavy symmetric top

A symmetric top with moments of inertia \(I_1=I_2\), \(I_3\), mass \(M\), and centre of mass at distance \(l\) from the fixed point has Lagrangian, in Euler angles,

Klein and Sommerfeld reduced its motion to elliptic integrals, using Klein’s complex parameters for rotations (now the Cayley–Klein parameters, the same objects that later describe electron spin). The angles \(\phi\) and \(\psi\) are cyclic, so \(p_\psi=I_3\omega_3\) and \(p_\phi\) are conserved, and \(\theta(t)\) moves in a one-dimensional effective potential.

Lubrication (1904)

In 1886 Osborne Reynolds showed that a thin oil film between a shaft and its bearing obeys, for a long bearing,

where \(R\) is the shaft radius, \(U\) its surface speed, \(\mu\) the viscosity, and \(h(\theta)=c(1+\varepsilon\cos\theta)\) the film thickness for clearance \(c\) and eccentricity ratio \(\varepsilon\). Reynolds solved it by series. Sommerfeld found a closed form in 1904.

Derivation. Integrate once: \(h^3p'=6\mu UR(h-h_0)\) for a constant \(h_0\). The integrals \(\int d\theta/(1+\varepsilon\cos\theta)^n\) are evaluated with the substitution \(1+\varepsilon\cos\theta=(1-\varepsilon^2)/(1-\varepsilon\cos\gamma)\), now called the Sommerfeld substitution. Periodicity of \(p\) fixes \(h_0=2c(1-\varepsilon^2)/(2+\varepsilon^2)\), and integrating again gives the formula. The engineering design parameter \(S=\big(\frac Rc\big)^2\frac{\mu N}{P}\), with \(N\) the shaft speed in revolutions per second and \(P\) the load per projected area, is the Sommerfeld number. In this full-film solution it depends only on \(\varepsilon\): \(S=\frac{(2+\varepsilon^2)\sqrt{1-\varepsilon^2}}{12\pi^2\varepsilon}\).

Stability of shear flow (1908)

At the International Congress of Mathematicians in Rome in 1908, Sommerfeld proposed to explain turbulence as an instability of laminar flow. Linearize the Navier–Stokes equations about a parallel flow \(U(y)\) and take a disturbance stream function \(\varphi(y)e^{i\alpha(x-ct)}\). Independently of William Orr (1907), he arrived at

the Orr–Sommerfeld equation. Instability means an eigenvalue \(c\) with positive imaginary part. In the same lecture he gave the dimensionless parameter \(\mathrm{Re}\) the name Reynolds number. The equation is hard: for plane Couette flow it has no unstable modes at any Reynolds number, and for plane Poiseuille flow his student Werner Heisenberg gave an asymptotic estimate of the critical Reynolds number in his 1923 thesis; the accurate value, \(\mathrm{Re}\approx5772\), was computed only in 1971.

Radio waves and the radiation condition

For the general reader. In 1901 Guglielmo Marconi sent radio signals across the Atlantic, and nobody could say precisely how waves travel over the curved, imperfectly conducting Earth. In 1909 Sommerfeld solved the model problem of an antenna above a flat conducting ground. In working out what makes a wave “outgoing”, he gave mathematics the condition that singles out the physical solution of a wave problem in unbounded space.

Outgoing waves

The Helmholtz equation \(\Delta u+k^2u=f\) in all of \(\mathbb R^3\) has many solutions: one can add any free wave, including incoming ones. With time dependence \(e^{-i\omega t}\), the physical solution radiates energy outward.

Why uniqueness. If \(u\) is radiating and solves the homogeneous equation everywhere, Green’s identity on a large sphere gives \(\operatorname{Im}\oint\bar u\,\partial_ru\,dS=0\). The radiation condition turns this into \(\oint|u|^2dS\to0\), and Rellich’s lemma then forces \(u\equiv0\). Without the condition, \(\sin(kr)/r\) would be a nonzero solution.

The Sommerfeld integral

To satisfy boundary conditions on a flat ground \(z=0\), Sommerfeld expanded the spherical wave of a dipole in cylindrical waves:

Each cylindrical component meets the plane boundary separately, and the reflected field is the same integral weighted by a reflection coefficient. For a vertical dipole at height \(h\) over ground of complex wavenumber \(k_E\), the field above ground is

for the Hertz potential \(\Pi\). Sommerfeld evaluated this integral by deforming the contour in the complex \(\lambda\)-plane around its branch points and a pole. The branch-cut contributions give the space wave and the ground wave; the pole gives a surface wave. Whether the surface-wave term describes a physically separate wave was argued over for decades; Hermann Weyl gave an alternative evaluation in 1919, and the question was settled only by later asymptotic analysis. The approach remains the standard method for antennas over layered media.

Relativity and fast electrons

For the general reader. Sommerfeld was not an early convert to Einstein’s relativity, but once convinced he became one of its best expositors. He showed that relativistic velocities add like angles on a sphere of imaginary radius, wrote the theory in the four-dimensional vector notation that physicists still use, and computed how a fast electron that is suddenly stopped sprays X-rays forward.

Velocities as angles (1909)

A boost along \(x\) is a rotation in the \((x,ict)\) plane through an imaginary angle. Sommerfeld wrote \(v/c=\tanh\eta\), with rapidity \(\eta\); in his notation, \(v=ic\tan\varphi\) with imaginary \(\varphi\). Successive collinear boosts add their rapidities, just as rotations about one axis add their angles.

Four-vectors and six-vectors (1910)

In two long papers of 1910 in the Annalen der Physik, Sommerfeld translated Minkowski’s matrix calculus into vector language. The potentials form a four-vector \(A^\mu=(\phi,\mathbf A)\), the fields a “six-vector”, the antisymmetric tensor \(F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu\), and Maxwell’s equations become

(Gaussian units). Laue’s textbook of 1911 and most later expositions adopted this language.

Bremsstrahlung (1909)

X-rays from a tube are produced when cathode-ray electrons are stopped in the anode. Sommerfeld computed the angular distribution of radiation from an electron decelerated along its line of flight. From the Liénard–Wiechert fields, for acceleration parallel to velocity:

Sommerfeld used this to explain why the X-rays from a tube are partly polarized and more intense in the forward direction. In 1931 he returned to bremsstrahlung with wave mechanics, giving the quantum cross section for the continuous X-ray spectrum.

Quantum conditions and elliptic orbits

For the general reader. In 1913 Niels Bohr explained the hydrogen spectrum with electrons on circular orbits whose angular momentum comes in whole multiples of \(h/2\pi\). Real orbits in a Coulomb field are ellipses, and an atom moves in three dimensions. In 1915 Sommerfeld gave the rule for quantizing every periodic coordinate, which brought in new quantum numbers for the shape and orientation of the orbit.

Wilson in London and Jun Ishiwara in Tokyo stated similar rules independently; Sommerfeld applied them systematically to spectra. Paul Ehrenfest’s principle that the \(J_i\) are adiabatic invariants, and Karl Schwarzschild’s and Paul Epstein’s use of action–angle variables in 1916, gave the conditions their theoretical footing.

The Kepler problem

For an electron in the field of a nucleus of charge \(Ze\), in the plane of the orbit with polar coordinates \((r,\varphi)\), the angular momentum \(L=p_\varphi\) is constant and

The azimuthal condition gives \(2\pi L=kh\), so \(L=k\hbar\), \(k=1,2,\dots\) (\(k=0\) would be an orbit through the nucleus). The radial condition uses the integral of Exercise 8.1.

The energy depends only on the sum \(n=n_r+k\): the \(n\) ellipses \(k=1,\dots,n\) all have the same energy. This degeneracy is special to the \(1/r\) potential. Any perturbation, such as relativity (Lesson 09) or the screening by inner electrons in alkali atoms, lifts it. Sommerfeld also quantized the orientation of the orbit relative to a magnetic field: the projection \(L_z=m\hbar\) with \(|m|\le k\). This “space quantization” explained the normal Zeeman effect, and in 1922 Otto Stern and Walther Gerlach observed it directly.

The fine structure of hydrogen

For the general reader. Seen through a good spectroscope, each hydrogen line is a close group of lines. Sommerfeld traced this to relativity: on an elongated ellipse the electron speeds up near the nucleus, its mass grows, and the orbit slowly rotates. Quantizing the rotating orbit gave a formula that matched the measurements, and introduced a pure number near 1/137 that measures the strength of electromagnetism.

The relativistic Kepler problem

Sommerfeld presented the theory to the Bavarian Academy in December 1915 and January 1916 and published the full account, Zur Quantentheorie der Spektrallinien, in the Annalen der Physik in 1916. For an electron of rest mass \(m\) in the field \(-Ze^2/r\), the total energy \(E\) (including \(mc^2\)) satisfies

So \(p_r^2=A+\frac{2B}{r}-\frac{C}{r^2}\) with \(A=\frac{E^2-m^2c^4}{c^2}\), \(B=\frac{EZe^2}{c^2}\), \(C=L^2-\frac{Z^2e^4}{c^2}\). The reduced \(C\) is why the orbit does not close: the perihelion advances on each revolution, producing the rosette of the figure at the top of this page.

Derivation. Exercise 8.1 gives \(n_rh=2\pi\big(\frac{B}{\sqrt{-A}}-\sqrt C\big)\), i.e. \(n_r+\sqrt{k^2-\alpha^2Z^2}=\frac{\alpha ZE}{\sqrt{m^2c^4-E^2}}\). Call the left side \(D\). Then \(D^2(m^2c^4-E^2)=\alpha^2Z^2E^2\), so \(E^2=m^2c^4D^2/(D^2+\alpha^2Z^2)\), which is the formula.

Test and coincidence

The splitting of the level \(n=2\) of hydrogen between \(k=1\) and \(k=2\) is \(\frac{mc^2\alpha^4}{32}\approx4.53\times10^{-5}\) eV, or \(0.365\ \mathrm{cm^{-1}}\). The lines of ionized helium, with \(Z=2\), are split sixteen times more. Friedrich Paschen measured the helium line at 4686 Å in 1916 and found the pattern Sommerfeld predicted. The exact formula reappeared in 1928 in Paul Dirac’s relativistic theory of the electron, with \(k\) replaced by \(j+\tfrac12\). Sommerfeld’s model ignored spin and Dirac’s ignored orbits, and the two agree because spin–orbit coupling and the relativistic correction combine to depend on \(j\) alone in a way that mimics the orbit model. The levels with equal \(j\) but different orbital angular momentum, degenerate in both theories, were split by the Lamb shift measured in 1947.

Electrons in metals

For the general reader. Around 1900 Paul Drude and Hendrik Lorentz explained the electrical and thermal conductivity of metals by a gas of free electrons. The theory had a fatal flaw: such a gas should add a large amount to the heat capacity of a metal, and it does not. In 1927 Sommerfeld replaced the classical gas by one obeying the new statistics of Enrico Fermi and Paul Dirac. The flaw disappeared and the modern theory of metals began.

The degenerate electron gas

Treat the conduction electrons as free particles of spin \(\tfrac12\) in a box of volume \(V\), with occupation given by the Fermi–Dirac distribution \(f(\varepsilon)=1/(e^{(\varepsilon-\mu)/kT}+1)\). At \(T=0\) the states fill a sphere in momentum space up to the Fermi momentum \(\hbar k_F\), with

For copper, \(n\approx8.5\times10^{28}\ \mathrm m^{-3}\) gives \(E_F\approx7.0\) eV, a Fermi temperature \(E_F/k\approx8\times10^4\) K. At room temperature the gas is deeply degenerate.

Sketch. Integrate by parts with \(K(\varepsilon)=\int^\varepsilon H\): the integral becomes \(\int K(\varepsilon)(-f'(\varepsilon))d\varepsilon\). The function \(-f'\) is a narrow even peak at \(\mu\) of width \(kT\). Expand \(K\) in a Taylor series about \(\mu\); odd terms vanish, and \(\int x^2(-f')dx=\frac{\pi^2}{3}(kT)^2\) gives the second term.

Only electrons within about \(kT\) of the Fermi energy can be excited, a fraction \(\sim kT/E_F\) of the total; this is the whole content of (i). The Lorenz number in (ii) agrees with measurements on many metals at room temperature to within ten percent. The classical theory had got a similar value from compensating errors. Sommerfeld presented the theory at the Volta Congress in Como in September 1927, and published it in the Zeitschrift für Physik in 1928. The free-electron model left open why electrons pass through a lattice of ions so easily; Felix Bloch’s theory of electrons in periodic potentials, written in Leipzig under Heisenberg in 1928, answered that. Sommerfeld and Hans Bethe combined the two in their long Handbuch der Physik article of 1933.

The Munich school, 1906–1935

Sommerfeld’s institute in Munich became a nursery of theoretical physics. His method was to take students into his seminar early, give them open problems that he was working on himself, and treat them as colleagues. Nearly thirty took doctorates with him. Among them were Peter Debye (1908), Paul Peter Ewald (1912), Wilhelm Lenz (1911), Paul Epstein (1914), Alfred Landé (1914), Wolfgang Pauli (1921), Gregor Wentzel (1921), Werner Heisenberg (1923), Otto Laporte (1924), Albrecht Unsöld (1927), Hans Bethe (1928) and Herbert Fröhlich (1930). Debye, Pauli, Heisenberg and Bethe won Nobel Prizes. Visitors included Max von Laue, Linus Pauling and Isidor Rabi. Einstein wrote to Sommerfeld in 1922 praising his gift for bringing out the talents of young people.

In 1912 Max von Laue, then a Privatdozent at the institute, suggested that a crystal might diffract X-rays. Sommerfeld’s assistant Walter Friedrich and the doctoral student Paul Knipping did the experiment in April 1912, using the institute’s X-ray tube. The photographs showed that X-rays are waves and crystals are lattices; Laue received the Nobel Prize in 1914. Ewald’s thesis, on optics of crystal lattices, led to the dynamical theory of X-ray diffraction.

Sommerfeld’s Atombau und Spektrallinien (1919) gathered the old quantum theory into a book that physicists called the bible of atomic physics. It went through four editions by 1924; a wave-mechanical supplement followed in 1929. He succeeded Einstein as chairman of the German Physical Society in 1918. He spent 1922–23 as Carl Schurz professor at the University of Wisconsin, and toured India, China, Japan and the United States in 1928–29, lecturing on the new quantum mechanics. In 1927 he was offered Planck’s chair in Berlin, the most prestigious in Germany, and after two months of negotiations declined it to stay with his school. Among his many honours were the Matteucci Medal (1924), foreign membership of the Royal Society (1926), the Max Planck Medal (1931) and the Lorentz Medal (1939).

Dictatorship, last years and legacy, 1933–1951

After 1933 Sommerfeld lost colleagues and students to emigration, Bethe and Fröhlich among them, and he helped several find positions abroad. He reached the retirement age and became emeritus on 1 April 1935. The faculty named Heisenberg as his preferred successor, with Debye and Richard Becker. The proponents of “Deutsche Physik”, led by Philipp Lenard and Johannes Stark, opposed Heisenberg and attacked Sommerfeld as a teacher of “Jewish” theoretical physics. The fight lasted more than four years, during which Sommerfeld served as his own temporary replacement. On 1 December 1939 the chair went to Wilhelm Müller, an aerodynamicist with no record in theoretical physics and a supporter of the Deutsche Physik movement. The appointment ended the Munich school as Sommerfeld had built it.

He spent the war years turning his lecture courses into the six-volume Vorlesungen über theoretische Physik: mechanics, mechanics of deformable media, electrodynamics, optics, thermodynamics and statistics, and partial differential equations of physics. The first volume appeared in 1943; the last was edited after his death. After the war he taught again for a time. He received the Oersted Medal of the American Association of Physics Teachers, awarded for 1948, in January 1949. In early 1951, walking in Munich with his grandchildren, he was struck by a car; he was hard of hearing and did not hear the warning. He died of his injuries on 26 April 1951, aged 82, and is buried in the Nordfriedhof in Munich.

Legacy

• Mathematical physics: exact diffraction theory, the radiation condition, the Sommerfeld integral for layered media, and the Sommerfeld–Malyuzhinets methods for wedges still used in antenna and scattering theory.
• Engineering: the Sommerfeld number and substitution in bearing design; the Orr–Sommerfeld equation in hydrodynamic stability.
• Atomic physics: the azimuthal and magnetic quantum numbers, the fine-structure constant \(\alpha\), and the exact fine-structure formula.
• Solid-state physics: the Sommerfeld free-electron model and the Sommerfeld expansion, the starting point of every course on metals.
• Teaching: a school whose members founded theoretical physics in Leipzig, Zürich, Hamburg, Cornell and elsewhere, and a textbook series still in print. The lunar crater Sommerfeld and asteroid 32809 Sommerfeld bear his name.

Connections

People linked to Arnold Sommerfeld across this collection. A name without a link has no page here yet. See Arnold Sommerfeld in the network of all 337 lives →

Teachers and advisors

• Ferdinand von Lindemann 1891

Students

• Hans Bethe 1928
• Peter Debye 1908
• Werner Heisenberg 1923
• Wolfgang Pauli 1921
• Albrecht Unsöld 1927
• Alfred Landé 1914
• Gregor Wentzel 1921
• Herbert Fröhlich 1930
• Isidor Isaac Rabi
• Linus Pauling
• Otto Laporte 1924
• Paul Peter Ewald 1912
• Paul Sophus Epstein 1914
• Wilhelm Lenz 1911

Influenced by

• Adolf Hurwitz
• Osborne Reynolds

Influenced

• Paul Dirac
• Subrahmanyan Chandrasekhar 1928

Colleagues and collaborators

• Albert Einstein
• Felix Klein 1894-1910
• Hermann Minkowski
• Max Planck
• Max von Laue 1909-1912
• Niels Bohr
• Paul Ehrenfest
• Tullio Levi-Civita 1920
• Wilhelm Wien 1920-1928
• William McFadden Orr
• William Wilson 1915

Rivals

• Johannes Stark
• Philipp Lenard

Friends

• David Hilbert

Also mentioned in this life

• Friedrich Wilhelm Bessel
• Ludwig Boltzmann
• Arthur Cayley
• Charles-Augustin de Coulomb
• Leonhard Euler
• Enrico Fermi
• Augustin-Jean Fresnel
• William Rowan Hamilton
• Hermann von Helmholtz
• Heinrich Hertz
• Carl Gustav Jacob Jacobi
• Johannes Kepler
• Gustav Kirchhoff
• Pierre-Simon Laplace
• Hendrik Antoon Lorentz
• Guglielmo Marconi
• James Clerk Maxwell
• Claude-Louis Navier
• Hans Christian Ørsted
• Bernhard Riemann
• George Gabriel Stokes
• Alessandro Volta
• Hermann Weyl

Fields

• theoretical physics 11 lives
• mathematical physics 45 lives
• atomic physics 7 lives
• solid-state physics 6 lives
• hydrodynamics 4 lives

On CoursesHub

• Atomic & Optical Physics
• Fluid Mechanics

Sources and further reading

Biography

1. J. J. O’Connor, E. F. Robertson, Arnold Sommerfeld, MacTutor History of Mathematics.
2. Arnold Sommerfeld, Wikipedia; German Wikipedia.
3. M. Eckert, Arnold Sommerfeld: Science, Life and Turbulent Times 1868–1951, Springer, 2013.
4. M. Eckert, Sommerfeld and the quantum theory, lecture slides, Seven Pines Symposium, 2014.
5. M. Eckert, K. Märker (eds.), Arnold Sommerfeld: Wissenschaftlicher Briefwechsel, 2 vols., GNT-Verlag, 2000–2004.
6. D. C. Cassidy, Uncertainty: The Life and Science of Werner Heisenberg, Freeman, 1992.

Physics

1. A. Sommerfeld, Mathematische Theorie der Diffraction, Math. Ann. 47 (1896) 317–374.
2. F. Klein, A. Sommerfeld, Über die Theorie des Kreisels, 4 vols., Teubner, 1897–1910.
3. A. Sommerfeld, Zur hydrodynamischen Theorie der Schmiermittelreibung, Z. Math. Phys. 50 (1904) 97–155.
4. A. Sommerfeld, Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegungen, Atti IV Congr. Int. Mat., Rome, 1908, vol. 3, 116–124.
5. A. Sommerfeld, Über die Ausbreitung der Wellen in der drahtlosen Telegraphie, Ann. Phys. 28 (1909) 665–736.
6. A. Sommerfeld, Die Greensche Funktion der Schwingungsgleichung, Jahresber. DMV 21 (1912) 309–353.
7. A. Sommerfeld, Zur Quantentheorie der Spektrallinien, Ann. Phys. 51 (1916) 1–94, 125–167.
8. A. Sommerfeld, Atombau und Spektrallinien, Vieweg, 1919; English translation Atomic Structure and Spectral Lines, Methuen, 1923.
9. A. Sommerfeld, Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik, Z. Phys. 47 (1928) 1–32, 43–60.
10. A. Sommerfeld, Lectures on Theoretical Physics, 6 vols., Academic Press, 1949–1956.
11. N. W. Ashcroft, N. D. Mermin, Solid State Physics, Saunders, 1976, chapters 1–2.
12. Fine-structure constant, Orr–Sommerfeld equation and Sommerfeld number, Wikipedia.