This is a modification/extension of a paper I wrote for a January 2000 UMAIE course in Europe, studying the history of quantum theory. For photos of that trip, please see my Trip to Europe page (still and perpetually under construction). The paper for that class was mostly biographical, and I have chosen not to remove any of the original material, so it may be somewhat heavy on biographical information compared to scientific work.
The Life and Work of Arnold Sommerfeld
Karin Shafer
May, 2002
Photos obtained from http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Sommerfeld.html
Additional photos are available at http://www.lrz-muenchen.de/~Sommerfeld/WWW/AS_Bilder.html . The page is in German, but a www.google.com for Sommerfeld or Johanna Höpfner (it is the first result under a search for "Johanna Höpfner") will give this search result and will translate the page, though somewhat poorly.
In the story of quantum theory, one does not often
hear the name Arnold Sommerfeld. His contributions to quantum mechanics
were indirect. Sommerfeld is best known as a teacher. He had
a rapport with his students, many of whom went on to win the Nobel prize,
something their teacher never received, despite support from colleagues.
Sommerfeld is also notable in his support of Jews and his clashes with
the establishment in Nazi Germany, especially concerning the issue of his
own successor upon forced retirement from the University of Munich.
Arnold Johannes Wilhelm Sommerfeld was born in Königsberg,
East Prussia on the December 5, 1868. His father was a doctor, Franz
Wilhelm Sommerfeld, and both Franz and Arnold’s mother, Cäcile Matthias,
challenged their son intellectually. Franz’s interest in “natural
objects (amber, shells, minerals, beetles, etc.)” (Dictionary of Scientific
Biography) and in the natural sciences would lay the foundations for his
son’s studies. Arnold attended the Altstädtisches Gymnasium.
His interests while there included literature, languages, and history as
much as the natural sciences. He then attended the University of
Königsberg. After some deliberation, he decided to study mathematics,
but also attended lectures on philosophy and political economy. He
was also involved in fraternity life, which somewhat hampered his studies
(Dictionary of Scientific Biography). His doctoral thesis focused
on the application of the “theory of functions of a complex variable to
boundary-value problems” (Notable Twentieth-Century Scientists).
After graduation, Sommerfeld served a year (1892-1893)
in the military, then held various teaching posts. The first of these
was at the University of Göttingen. During his time there, Sommerfeld
worked as an assistant for Felix Klein at the Mineralogical Institute.
He wrote out lectures for Klein’s students and supervised the mathematics
reading room. As a result of his work, Sommerfeld’s mastery of mathematics
increased, as well as his understanding of mathematical physics and mechanics
(Dictionary of Scientific Biography). In 1895, Sommerfeld gained
the post of Privatdozent in mathematics. To gain this teaching position
he presented for his Habilitationsschrift an “exact solution of a diffraction
problem, which he gave as a complex integral in closed form suitable for
numerical evaluation,” (Dictionary of Scientific Biography) the first exact
solution for such a problem. After five years, Sommerfeld left Göttingen
for the Bergakademie, located in Clausthal. At the Bergakademie,
Sommerfeld was given a full professorship of mathematics. Due to
the increase in salary, Sommerfeld was able to marry Johanna Höpfner,
whose father was the Kurator at Göttingen (Dictionary of Scientific
Biography). While at Clausthal, Sommerfeld studied the propagation
of electromagnetic waves and X-ray diffraction and, together with Klein,
wrote and published in a series of volumes the Theorie des Kreisels, or
Theory of Crystals. They published this work in several volumes between
1897 and 1910. Klein was able to pull enough strings that, in 1900,
Sommerfeld became a full professor of technical mechanics at the Technische
Hochschule at Aachen. Sommerfeld, seen as a pure mathematician, had
to overcome much skepticism from his colleagues in applied engineering.
He was able to apply his mastery of mathematics to various engineering
problems. These included resonance in bridges and related effects
in ships and also locomotive construction. Some of his most important
work concerned the hydrodynamics of viscous liquids, an attempt to explain
the onset of turbulence and also develop a theory of machine lubrication
(Dictionary of Scientific Biography). Sommerfeld spoke at the 1903
Kassel congress at the request of the council of the Gesellschaft Deutscher
Naturforscher und Ärtze. He also “declined a highly complimentary
offer of the chair of mathematics and technical mechanics at the Berlin
mining academy.” (Dictionary of Scientific Biography)
As his reputation grew, especially in light of his work
on electron dynamics (focusing on movement faster than the speed of light,
remarkable despite the fact that it was soon negated by Einstein’s theory
of relativity), Sommerfeld caught the eye of the University of Munich.
In 1906 Sommerfeld became the chair of theoretical physics. He directed
a vigorous experimental research program and dealt with the newest and
most important issues facing the physics of the time in his classes.
Sommerfeld would remain at the University of Munich until 1940 (Dictionary
of Scientific Biography). Arnold Sommerfeld also contributed to the
development of physics in his role in the German Physical Society.
He was elected presiding officer of that organization in 1918 (Notable
Twentieth-Century Scientists). The German Physical Society was riddled
with internal strife and was deeply divided. It fell to Sommerfeld
to clean up the mess, which he did admirably (Notable Twentieth-Century
Scientists). The furor surrounding his successor is a story in and
of itself, and is addressed below. Arnold Sommerfeld was killed in
a 1951 automobile accident while walking with his grandchildren.
Sommerfeld worked in many varied areas of physics and
engineering. His work on diffraction has already been mentioned,
as has the Theorie des Kreisels, written in collaboration with Klein.
He published a textbook of atomic physics, Atombau and Spektrallinien,
considered definitive until the rise of quantum mechanics. Sommerfeld
was one of the earliest supporters of Einstein’s theory of relativity,
along with Planck. He presented a vector form of the theory and used
the it in his work on electron deceleration (Notable Twentieth-Century
Scientists). Sommerfeld met Einstein and developed a friendly relationship
with him. Sommerfeld also discussed quantum theory with Einstein,
and Einstein’s theory of light as a particle (Dictionary of Scientific
Biography). At another time, Sommerfeld proposed that action, the
time integral of energy, is primarily quantized, and energy is quantized
secondarily to this, and action is also equal to h/2p.
He suggested that “the ubiquity of an h in the interactions of atoms and
radiation is not to be regarded as a secondary expression of the size,
structure, and internal energy of atoms, ‘but rather the existence of molecules
[atoms] is to be regarded as a function and consequence of the existence
of an elementary quantum of action’” (Dictionary of Scientific Biography).
Sommerfeld also worked with specific heat and electron theory in metals,
assisted by his students, especially Bethe (Dictionary of Scientific Biography).
Other work aside, Arnold Sommerfeld’s best known work
is with the Bohr atomic theory. While initially doubtful, Sommerfeld
soon became one of the theory’s best defenders. Sommerfeld took Bohr’s
theory and expanded it to include elliptical electron orbits, with the
nucleus of the atom at one focus. He also noted the “relativistic
increase in the mass of the electron as it sped about the nucleus, which
led him to introduce a second quantum number, the azimuthal quantum number
(l), in addition to that introduced by Bohr (n)” (Notable Twentieth-Century
Scientists). For additional information, see http://scienceworld.wolfram.com/physics/SommerfeldModel.html
.Sommerfeld was able to apply this explanation to the Zeeman effect, the
splitting of spectral lines in a magnetic field. Because of Sommerfeld’s
support and refinement, the Bohr model spread quickly and widely throughout
the scientific community (and still surfaces today). The final result
of the collaboration between Bohr and Sommerfeld was the Bohr-Sommerfeld
atomic theory, much more complete than the first theory (Notable Twentieth-Century
Scientists). As a result of this new theory, he was able to establish
a “quantitative theory of the fine structure of the spectral lines of hydrogen
and of the X-ray spectra of the heavy elements” (Dictionary of Scientific
Biography). He pioneered the field of theoretical spectroscopy, working
backwards from spectra to deduce atomic energy levels. Data from
this work was then set in the frame of quantum rules (Dictionary of Scientific
Biography). While not a founder of the theory, when quantum mechanics
came to the fore, Sommerfeld was quick to begin using it (Dictionary of
Scientific Biography).
Sommerfeld has been well published. A search in
SciFinder under his name gives 91 references between 1910 and 1951, as
well as several papers entries published after his death in 1951 that still
bear his name. Approximately 31 of these references address spectroscopy
in some shape or form; approximately 21 address quantum theory or atomic
structure in general; about 11 address the electronic structure or heat
capacity of metals. Among his work is the paper "A model of
the neutral Helium Atom." It was published in the Journal of
the Optical Society of America in 1923, and is the only one of Sommerfeld's
paper's I have to date received that is in English. The paper is
available upon request, as well as two others which are in German.
Following is a summary of the paper:
A model of the neutral Helium Atom. A. Sommerfeld. Journal of the Optical Society of America, 7, 509 (1923).
Previous models of the helium atom had been unsuccessful. The Bohr model (1913) had two electrons moving in the same direction on a single circle (at opposite sides of the circle). It predicts an a paramagnetic atom with an ionization potential of 28.75 V. A 1921 Kemble model proposed crossed circular orbits at 60° to each other and predicted the ionization potential at 20 V. Experimentally, the ionization potential was known to be 24.5 V. Because the neutral helium atom is diamagnetic and has no net angular momentum, Sommerfeld concludes that the orbits are coplanar (but not necessarily the same) and that the electrons move oppositely.
1. Quantum Condictions for the Hydrogen Atom, Considering the Motion of the Nucleus.
At the time, the hydrogen atom is the only system that could be assigned quantum conditions with certainty. Through a mathematical derivation (using polar coordinates), he concludes that "the single phase integrals in the problem of two bodies moving about a fixed center of force are multiples of a traction of h; only the sums of corresponding phase integrals for the two bodies are equal to integral multiples of h, according to equations (3)." Equations 3 are given by
ò prdr + òpRdR=n'h,
òpjdj + òpFdF=kh,
where n', k=1, 2, 3...
2. Quantum Conditions for the Unexcited State of the Neutral Helium Atom
Here the conditions for the hydrogen atom are extended to the helium atom. The first approximation is to consider the nucleus of infinite mass and neglect its motion. Extending equations 3, he gives
ò pr1dr1 + òpr2dr2=n'h,
òpj1dj1 + òpj2dj2=kh,
Neither quantum numbers (n' or k) can be zero, but the lowest state will be characterized by the lowest possible values of these numbers (n'=1 and k=1). The equations above are split into ò pr1dr1=òpr2dr2=òpj1dj1 + òpj2dj2=½h. The eight integration constants can be determined through various conditions, including ther fact that the orbitals are complex and that the period of r1 must be the same as the period of r2. The equations are further extended to take account of the motion of the nucleus:
pr1dr1 + òpr2dr2 + òpRdR=n'h,
òpj1dj1 + òpj2dj2 + òpFdF=kh.
In a similar treatment, the first two terms in each exprpession can be equated, but they are no longer equal to h/2. Again, the constants can be determined. The choices of coordinates are discussed and justified.
3. First Approximation, Elliptical Orbits
When electron repulsion is neglected, the orbits are approximated by a Keplerian ellipse. The axes depend on the quantum numbers n and k, the hydrogen radius, a1 (a0 in newer literature), and the atomic number.
a=n2a1/Z, b=nka1/Z (n=n' + k=principal quantum number)
The quantum numbers represent the entire system, not a single electron, but partial qunatum numbers are used:
n1'=k1=½;
n2'=k2=½.
The equations for the elliptical orbits are given:
a=(½)a1, b=(¼)a1, b/a=(1-Î2)½=½, Î=(3½)/2
This approximation allow no conclusions to be drawn about the orientation of the orbits or the relative positions of the elctrons on their orbits. Despite this, it is assumed that the elpses have their aphelia at the nucleus but in opposite directions; also, it is assumed that one electron is at the aphelion while the other is at the perihelion, to avoid collisions and preserve symmetry and periodicity. Sommerfeld is "certain that the average position of the two orbits will not be changed sytematically, without limit, but the perturbations" because the influences on one electron during one half of its orbit are the same as the influences on the other electron during the opposite half of its orbit. Here is a scan of the image in the article:
Millikan's work on ionization by a-particles is applied to this system. Helium atoms are the only ones that can be doubly ionized by a-particles, and approximately 1/6 of the atoms are doubly ionized, compared to singly ionized particles. This is intepreted to mean that the electrons are in conjunction 1/6 of the time compared to the time where they are in opposition. This theory seems to fit the theorem of areas, that the electron at the perihelion will move faster than the one at the aphelion.
4. The K-Shell of the Heavier Elements
Here Sommerfeld attempts to apply this theory to the inner shell of heavier atoms, using relativity corrections and spectroscopy, but there a lot of holes; and the relativity corrections used may not be applicable. This paper was published after contact with Heisenberg, who had "solved in detail the dynamical problem of our [Sommerfeld's] model for unexcited helium, and has found that the ionization potential is about 24.5 volts, which is correct", and remains incomplete until "quantitative information about the mutual perturbation of the two electrons" is obtained.
Also given are the abstracts as obtained from SciFinder Scholar (with some formatting changes, including correction of "ö" characters) for the two papers in German:
Principles of the quantum theory and the Bohr atomic model. Sommerfeld, A.. Naturwissenschaften (1924), 12 1047-9.
An address in the form of a general review and discussion.The spectrum of hydrogen. Sommerfeld, A.; Unsold, A. Z. Physik (1926), 36 259-75.
The resemblance of Röntgen spectra to H in respect to the relativistic level differences, and the analogy of Röntgen spectra with alkali spectra, suggested that the H spectrum must also be regarded according to the quantum scheme of the alkalies. The spectral laws of the periodic system are in this way extended to its very beginning: neutral He resembling the alk. earths with two outer electrons should have singlet and triplet spectral systems, while ionized He and H with one electron resemble the alkalies and possess a doublet system. The new quantum interpretation is illustrated with energy diagrams showing the fine structure of Ha (6563 A. U.) and He+ (4686 A. U.). The relativistic formula for level differences remains unaltered but the quantum numbering of the levels is somewhat different. It is now possible not only to give the correct comparison with the selection rules but also to compute the relative intensities of the components in a simple manner. Complete agreement with Paschen's measurements on He+ (4686 A. U.) is found. For Ha the long-wave component is theoretically stronger than the shorter one but observations have established the contrary. This intensity anomaly is ascribed to the meta-stability of the 2s term. Furthermore the Paschen-Back effect for Ha as studied by Försterling and Hansen is successfully explained.
Arnold Sommerfeld is perhaps best known as a teacher.
He never received a Nobel Prize, but many of his students did. Sommerfeld
did not go without recognition, nonetheless. He received many other
honors: “prizes, memberships in foreign academies, honorary doctorates…offers
of the chairs of theoretical physics at the University of Vienna…and at
the University of Berlin” (Dictionary of Scientific Biography). According
to the Dictionary of Scientific Biography, “In the twenty-five years following
his arrival in Munich … Sommerfeld had more advanced students and turned
out more doctorates than any other theorist. The near-monopoly that
he held for the first fifteen of these years was seriously challenged only
after Max Born arrived in Göttingen in 1921.” Sommerfeld enjoyed
a close, friendly relationship with his students, breaking many social
barriers. He often took outings with them, to the consternation his
colleagues (Dictionary of Scientific Biography). Sommerfeld’s students
included Hans Bethe, Peter Debye, Paul S. Epstein, P. P. Ewald, Herbert
Fröhlich, Erwin Fues, Hans Grimm, Werner Heisenberg, Walter Heitler,
Helmut Höhl, Ludwig Hopf, Otto Laporte, Alfred Landé, Wilhelm
Lenz, Wolfgang Pauli, Albrecht Unsöld, and Gregor Wentzel (Dictionary
of Scientific Biography). Bethe would go on to help develop the atomic
bomb, though he campaigned for peace (Encarta, “Bethe, Hans Albrecht”).
He also worked with Sommerfeld on electron theory of metals (Dictionary
of Scientific Biography). Debye is was influential in the development
of quantum theory; he also worked with specific heat, dipole moments, molecular
structure, electrolyte ionization, and diffraction of electrons and X-rays
in gases. Encarta, “Debye, Peter Joseph Wilhelm”). Heisenberg, too,
is extremely well known. He developed matrix mechanics and the uncertainty
principle and worked on atomic structure (Encarta “Heisenberg, Werner”).
Pauli is one of the greats of quantum theory. His exclusion principle
is central to modern chemistry and physics. He also predicted the
existence of the neutrino.
Links to biographies of a few of Sommerfeld's students:
Bethe: http://www.nobel.se/physics/laureates/1967/bethe-bio.html
Pauli: http://www.nobel.se/physics/laureates/1945/pauli-bio.html
Debye: http://www.nobel.se/chemistry/laureates/1936/debye-bio.html
Epstein: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Epstein.html
Heisenberg: http://www.nobel.se/physics/laureates/1932/heisenberg-bio.html
Another interesting side of Arnold Sommerfeld is his
interaction with the Nazis and the National Socialism movement. The
Beyerchen text Scientists Under Hitler is an excellent reference, not only
for Sommerfeld, but all scientists in Nazi Germany. There was a movement
to Aryanize physics and remove all Jewish influences. Sommerfeld
was one of the opponents of this movement, though he was not Jewish.
Before the war, Sommerfeld was known to support, among others, Einstein
and his physics (see Beyerchen pp. 88, 90, and 170). Einstein, a
Zionist, was hated by the Nazis. (Encarta: “Einstein, Albert”)
Attempts were made to separate Einstein’s name from the theory of relativity
in order to make it more acceptable to Aryan physicists (Beyerchen 163).
Sommerfeld also supported other physicists considered unacceptable to the
Nazis, including Richard Courant (Beyerchen 25). He was known to
appoint Jews to academic posts if they were qualified (Beyerchen 49).
Sommerfeld’s opposition of the Aryan physics movement
is part of the uproar concerning his successor at the University of Munich.
In 1935, Sommerfeld passed his obligatory retirement age. Until a
successor could be named, he continued to teach on a provisional, semester-to-semester
basis. His colleagues wanted a worthy successor, and hopefully one
of Sommerfeld’s students. They suggested Werner Heisenberg (and also
Peter Debye and Richard Becker), but Aryan physicists objected vehemently
(Beyerchen 153). Unfortunately, Sommerfeld’s assistant, Otto Scherzer,
was also in the process of leaving Munich, and his absence would leave
the university with no theoretical physics at all (Beyerchen 153).
The Reich Education Ministry rejected all three of the faculty’s suggestions
without explanation, despite their eminence as physicists. Opposition
also came from Nazi students, spearheaded by Wilhelm Führer, who had
caused trouble at other times, as well (Beyerchen 154). The Munich
faculty still strongly supported Heisenberg, to the point where he seemed
a sure bet. The Reich Education Ministry had come to support him,
as well; he even received official notice of his appointment as Sommerfeld’s
successor (Beyerchen 156). However, in 1940, disaster struck.
Wilhelm Müller was appointed to Sommerfeld’s chair. Müller
was elected, not because of any skill in physics, but because he was a
firm supporter of Aryan physics (Dictionary of Scientific Biography).
He “had never published in a physics journal, had never attended a physicists’
conference, and did not even belong to the German Physical Society.
His background in aerodynamics involved extensive use of mathematics, but
only in problems related to classical physics” (Beyerchen 166). Sommerfeld
was disgusted. He referred to Müller as “the worst conceivable
successor” (Dictionary of Scientific Biography). After assuming Sommerfeld’s
position, no theoretical physics was taught, only classical mechanics.
Professor Gerlach complained about this to the University’s dean, but his
complaints were not only ignored, but contradicted. According to
the dean’s response, Müller was doing a wonderful job teaching theoretical
physics, despite the obvious fact that he was doing nothing of the sort
(Beyerchen 166).
The five year story of how Müller was appointed
is one of politics and connections. The physicist Johannes Stark
had become one of Sommerfeld’s chief enemies, partly because of a grudge
dating back to late 1920s (Beyerchen 161). He accused Sommerfeld
of being the leader of a Jewish movement in Germany’s physics as early
as 1915. Stark also wormed his way into positions of influence and
tried to purge German science of Jewish influence (Dictionary of Scientific
Biography). Stark probably had a hand in ruining Heisenberg’s chances
at succeeding Sommerfeld. An article in a major SS journal that has
been connected with Stark (if he did not write it himself, he probably
supplied the information for it) identified Heisenberg as a “white Jew”
of science. “White Jew” was a term applied to non-Jews who were supposedly
Jewish in spirit, and so should be purged from science as the Nazis were
eliminating Jews in the rest of society (Dictionary of Scientific Biography).
Heisenberg was compared to Carl von Ossietzky, who had been imprisoned
in the prison camps since 1933, and who received the Nobel Peace Prize
in 1936. Identification with someone imprisoned in the concentration
camps was hardly good for Heisenberg’s chances (Beyerchen 158-159).
Heisenberg even attempted to use his family’s friendship with the Himmler
family to clear up the mess. His mother visited the elderly Mrs.
Himmler to ask her to intercede for Heisenberg with her son (Beyerchen
159-160). In addition, the Munich faculty, Sommerfeld included, expressed
outrage at Stark’s article (Beyerchen 161). Himmler’s lack of a stand
on the Heisenberg mess took up a considerable amount of time, but he eventually
exonerated Heisenberg in July of 1938 and gave him the support of the SS
(Beyerchen 163). Sommerfeld and the rest of the Munich faculty fought
all the more for Heisenberg, giving fervent opposition to the candidates
suggested by the Aryan physicists, including Müller (Beyerchen 165).
The Aryan physics movement became stronger through alliance with another
organization, the University Teacher’s League, and both the Reich Education
Ministry and Himmler turned against Heisenberg. There was a plan
to give him a different post later, but there was no immediate response
(Beyerchen 166). Despite losing the battle eventually and having
an incompetent appointed as his successor, Sommerfeld served as an important
protector of true physics despite the repressive influences of Aryan physics.
Even after Müller was appointed, opposition continued,
headed by Sommerfeld and Gerlach, among others. They attempted to
have Müller appointed elsewhere, but were unsuccessful. In 1942,
Fritz Sauter, a former pupil of Sommerfeld, began teaching at a different
school in Munich. His lectures were at the same time as Müller’s,
and many of Müller’s students attended Sauter’s lectures, missing
Müller’s. When Müller complained about this, professor
Gerlach was able to use his complaint to prove that he was incompetent,
though he would not be ousted until 1945 (Beyerchen 181-182).
Arnold Sommerfeld’s contributions to quantum mechanics
came indirectly. The Dictionary of Scientific Biography relays a
quote by Einstein, “What I especially admire about you is the way, at a
stamp of your foot, a great number of talented young theorists spring up
out of the ground.” His students helped develop the theories forming
the modern understanding of the quantum. He also served as a protector
of true physics under the Third Reich.
References
Biographic
“Arnold Sommerfeld.” Notable Twentieth-Century Scientists. V. 4
1995.
“Bethe, Hans Albrecht.” Microsoft Encarta Encyclopedia 99.
CD-ROM. Redmond, WA: Microsoft, 1998.
Beyerchen, Alan D. Scientists under Hitler. London: Yale University
Press, 1977.
“Debye, Peter Joseph Wilhelm.” Microsoft Encarta Encyclopedia 99.
CD-ROM. Redmond, WA: Microsoft, 1998.
“Einstein, Albert.” Microsoft Encarta Encyclopedia 99. CD-ROM. Redmond,
WA: Microsoft, 1998.
“Heisenberg, Werner.” Microsoft Encarta Encyclopedia 99. CD-ROM.
Redmond, WA: Microsoft, 1998.
“Sommerfeld, Arnold (Johannes Wilhelm).” Dictionary of Scientific Biography.
V. 11-12. 1981.
Papers: all obtained using SciFinder Scholar
A model of the neutral Helium Atom. A. Sommerfeld. Journal
of the Optical Society of America, 7, 509 (1923).
Principles of the quantum theory and the Bohr atomic model.
Sommerfeld, A.. Naturwissenschaften (1924), 12
1047-9.
The spectrum of hydrogen. Sommerfeld, A.; Unsold, A.
Z. Physik (1926), 36 259-75.
Not used, but useful links:
http://scienceworld.wolfram.com/physics/SommerfeldModel.html
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sommerfeld.html