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CARL HENRY ECKART
May 4, 1902-October 23,1973
BY WALTER H. MUNK AND
RUDOLPH W. PREISENDORFER
~ .
CARL ECKART was a major participant in the development of
quantum mechanics and atomic physics. At the age of forty,
with the advent of World War II, he turned his attention to
underwater acoustics and related problems in geophysical hydro-
dynamics; this was to remain Eckart's primary interest for the
rest of his life. His contributions to physics and oceanography
are about equally divided. For ten years he directed the Uni-
versity of California Division of War Research and its successor,
the Marine Physical Laboratory. For two years he was Director
of the Scripps Institution of Oceanography. A shy man, he dis-
charged these responsibilities with precision, elegance, and
gentle care.
Carl Eckart, an only child and the son of conservative people
of German heritage, was born in St. Louis, Missouri. During his
high school years in St. Louis, Eckart's interests were in science
and mathematics. These interests, along with an innate ability
in fine draftsmanship, left little time for social pursuits. Upon
graduation he was awarded a full scholarship to Washington
University, in St. Louis, where he received B.S. and M.S. degrees
with a major in engineering. Eckart's intention was to turn his
interest to mathematics, but this changed to physics, evidently
under the influence of Arthur Holly Compton, a physics faculty
member (later Chancellor). Compton influenced Eckart to con-
195

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196
BIOGRAPHICAL MEMOIRS
tinue graduate work at Princeton University, where he went on
an Edison Lamp Works Research Fellowship and received a
Ph.D. in 1925. It was during this period that Eckart produced
his first recorded research paper (with G. E. M. Jauncey) sug-
gesting an extension of Compton's classic photon-scattering ex-
periment to X rays in crystal lattices. Other papers in this period
followed (jointly with Arthur's older brother Karl): a study of
low-voltage arcs, particularly the oscillatory phenomena arising
in the diffusion of electrons against low-voltage fields. He con-
tinued this work as National Research Council Fellow at the
California Institute of Technology from 1925 to 1927.
During the winter of 1925, Max Born came to Pasadena and
gave a lecture on quantum mechanics. This lecture aroused
Eckart's interest in a possible general operator formalism for
quantum mechanics. Working through the winter of 1925-1926,
Eckart developed the formalism and completely familiarized
himself with what is now known as the Schrodinger energy
operator. In January 1926, when Schrodinger's first paper (of
the famous set of four) on wave mechanics appeared in the
Annalen der Physik, Eckart immediately recognized its revolu-
tionary content. There was, in particular, the puzzling presence
of another formulation alternative to Schrodinger's wave me-
chanics: the matrix mechanics of Heisenberg, which used not
the partial differential equation for matter waves, but rather
infinite-ordered matrices. Despite their outwardly different struc-
ture, the theories yielded identical predictions of atomic spectra
and identical relations between atomic constants. Evidently,
they were equivalent ways of viewing the same physical phe-
nomena, and Eckart felt that there should be a general mathe-
matical framework that would encompass both formalisms as
alternative representations. Working in relative isolation in
California, far from the exciting German scientific centers,
Eckart soon found the connecting link between the Hilbert
space of eigenfunctions of Schrodinger's equation and the

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CARL HENRY ECKART
197
matrices of the Jordan-Born matrix algebra (which lay at the
base of Heisenberg's mechanics). Eckart's solution was submitted
to the Proceedings of the National Academy of Sciences on May
31, 1926. But the credit generally went to Schrodinger, whose
note to the Annalen der Physik containing essentially the same
solution was dated March 18. Later that year, in June 1926,
Eckart completed his general study of the operator calculus (see
the "note in proof" appended to his paper in Physical Review).
This near miss was a source of disappointment for Carl; on the
few occasions when a friend could approach him on the subject,
he would comment on his isolation in 1926 from the main-
stream of quantum physical activity.
But this was soon to change. In 1927, Eckart received a
Guggenheim Fellowship to study with Arnold SommerfelcT in
Munich. Here he worked on the quantum mechanical behavior
of simple oscillators using Schrodinger's equation, developing
further the operator calculus that would allow rapid and almost
mechanical manipulations of the newly discovered matrix me-
chanics and gaining new insights into the correspondence prin-
ciple. Applications were made to the electron theory of metallic
conduction using Fermi statistics, with particular attention to
the Volta effect.
The German fellowship coincided with the culmination of
the twer~ty-year search by European physicists for the key in-
sights that would consolidate the long series of experimental
and theoretical advances in the "old" quantum mechanics begun
in 1905 by Planck. The search came to an end in the period
1925-1928 with the advent of Heisenberg's matrix mechanics
and Schrodinger's wave mechanics. As we saw, Eckart was an
integral part of these exciting developments. During his Ger-
~ For further discussion of this period of time, see M. Jammer, The Concep-
tual Development of Quantum Mechanics (New York: McGraw-Hill Book Co.,
1966), p. 275. (We have used the above-cited communication dates as they appear
in the original papers.)

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BIOGRAPHICAL MEMOIRS
man fellowship, Eckart became ever more deeply absorbed in
"the mathematics of the period, which was miraculously made
available and compiled for quantum physicists in almost fully
developed forms by the applied and pure mathematics schools
at Gottin~en headed by Felix Klein and David Hilbert. It was
this mathematics that would guide Carl Eckart's approaches to
all his subsequent theoretical investigations.
On his return to the United States in 1928, Eckart was ap-
pointed to an Assistant Professorship in the Physics Department
at the University of Chicago. Although once again removed
from the physics centers of Munich and Gottingen, Eckart con-
tinued his quantum mechanical studies over the subsequent
fourteen-year period. Particularly noteworthy is the paper (with
H. Honl) on the foundations of wave mechanics, an exposition
of the role of group theory in the quantum dynamics of mon-
atomic systems, and the comparisons of the nuclear theories of
Heisenberg and Wigner. It was in this period that Eckart built
on his formulations of the so-called Wigner-Eckart theorem, a
link between the symmetry transformation groups of space
(applied to the Schrodinger equations) and the laws of conserva-
tion of energy, momentum, and angular momentum.! It is of
practical use in atomic spectroscopy. These researches went
hand in hand with teaching activities and with a translation
(together with F. C. Hoyt) of Heisenberg's tract on the Physical
Principles of Quantum Theory. In all, the decade of the 1930s
saw twenty important papers by Eckart in quantum physics.
Eckart's paper on the electrodynamics of material media (in
1938) suggests a transition in his interests. By that time he had
begun to lose interest in the submicroscopic world of matter
~ H. Weyl, "David Hilbert and His Mathematical Work," Bulletin of the
American Mathematical Society 50(1944):612 (the section on integral equations).
~ The basic idea occurs in E. P. Wigner, "Some Consequences for Term Struc-
ture from Schrodinger's Theory," Zeitschrift fur Physik 43(1927):624. The idea
was elaborated in Eckart's 1930 group theory paper. Our description covers only
the simpler cases. For a fuller description, see P. Roman, Advanced Quantum
Theory (Reading, Mass.: Addison-Wesley Publishing Co., Inc., 1965>, p. 583.

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C AR L H E N R Y E C K ART
199
waves. Perhaps he felt that the trend of quantum mechanical
research into atomic systems was toward less-rigorous and only
partial analyses of solutions of the associated Schrodinger equa-
tions. Physicists, facing the complicated multiple interactions
of electors systems in the heavier atoms, were adopting simpli-
fied models (such as the shell and liquid drop model) that could
only partially describe the physical facts. On the other hand,
such venerable subjects as electrodynamics and thermodynamics,
worked over as they were by several generations of physicists,
still contained obscurities and curious gaps between the pure
and applied levels. For example, the thermodynamic basis of
heat transport and the mechanism of mixing of fluids needed
attention.
The title of the 1938 paper is somewhat misleading, for it
implies a reworking of the Minkowski or Lorentz formulations
of fiche subject. In fact, Eckart achieved a unified theory of
Maxwellian and quantum electromagnetics, leading to a gauge-
invariant formulation of electrodynamics (previously attempted
notably by Mie and Weyl, but without success). He was not
successful in extending the formulation to contain as special
cases Schrodinger's, Heisenberg's, and Dirac's equations of elec-
trodynamics, but his approach did yield equations closely re-
sembling these famous equations and also portions of gas theory
for irrotational motion. This latter feature may seem somewhat
incongruous, but it falls out quite naturally from a general
approach that postulates a set of moving particles of matter
characterized by "states" that can be electric, magnetic, or of
other forms. (The resultant variational formulation of the kinetic
theory of such particles- is subject to the constraint of Ampere's
law.) By shutting off the electric states, a portion of gas theory is
recovered. This paper is one of Eckart's major contributions
towards a physical synthesis.
The 1938 paper prepared the way for "The Thermodynam-
ics of Irreversible Processes," parts I, II, and III of which were
published in 1940. The first paper showed how the entropy

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BIOGRAPHICAL MEMOIRS
increase in a simple viscous fluid can be calculated and the
Kelvin—Fourier hypothesis of heat conduction can be rigorously
deduced from the laws of thermodynamics. Thus, this ostensibly
empirical law was placed into the fold of classical thermodynam-
ics. In the second paper, in a similar vein, Ohm's law and Fick's
law of diffusion were shown to be special cases of more general
thermodynamic laws. This paper also discussed the general
theory of entropy increase in fluid mixtures. (Eckart's subse-
quent work in hydrodynamics of the oceans and the atmosphere
had its beginnings in this work.) In a third paper, the concepts
of relativity and of fluid dynamics were related. The influence
of these three papers in the field of irreversible processes was
immediate and long-lasting, as a study of the subsequent work
by Tolman and Prigogine shows. Twice more, Eckart would
reach for a thermodynamic formulation of physical laws, start-
ing from the particle level. Paper IV (1948) deals with elasticity
and Inelasticity; paper V, with shock waves and phase bound-
aries (written in 1965 but never submitted for publication).
In December 1941, the United States entered World War II.
It was already clear to responsible scientists in mid-1941 that
preparations for the national defense must be made realistically
and without further delay. Axis submarines were taking their
toll of shipping. University scientists were being approached by
the U.S. Navy concerning the problems of optical and acoustical
detection of enemy submarines. V. O. Knudsen, director of the
embryonic University of California Division of War Research,
and his close associate L. P. Delsasso asked Eckart for advice and
help. He spent a week in June of that year in San Diego review-
ing reports by the British Naval Laboratories and the Naval
Research Laboratory in the United States. Impressed with the
need for understanding the fundamentals underlying submarine
detection, he took leave of absence from the University of
Chicago (still an Associate Professor), a momentous decision not
only in his own life, but also in the lives of many others. When

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CARL HENRY ECKART
201
Eckart began his thirty-one-year stay in California, during those
hectic days of preparation for the national defense, he was forty
years old. He had at his beck all of classical mathematics and
physics of that time. Now, before him, there were new problems
to solve and new concepts to clarify.
Eckart found himself in the unaccustomed role of working
with technicians and engineers on a day-to-day basis. He was
greatly stimulated by this contact, and many of his later contri
buttons had their roots in this period. In a series of classified
reports (some of which are now available), we can trace his grow-
ing interest in the problems of sound attenuation in the sea, in
the effect of randomly moving sea surfaces on the reflection of
sound waves and electromagnetic waves, and in the analysis of
time series of acoustical signals. Following the war, Eckart col-
lected his work and the work of others into Principles and A ppli-
cations of Underwater Sound, first issued in 1946, declassified
in 1954, and reprinted in 1968. Principles serves as standard
reference even today. By 1946 many of Eckart's wartime col-
leagues had returned home, but Eckart decided to remain in
California. He terminated his appointment at the University of
Chicago and became Professor at the University of California
and the first Director of its Marine Physical Laboratory (MPL),
established by Eckart, Roper Revelle and Admiral Rawson
Bennett to continue geophysical research of common interest to
the academic and navy communities. MPL became an integral
part of the Scripps Institution of Oceanography in 1948, and
Eckart served as Director until 1952. Under the present leader-
ship of Fred Spiess, MPL continues its vital function.
One of the major puzzles unsolved in the war research was
the anomalously high attenuation of sound in seawater. Eckart
encouraged experimentalists to work on this problem, particu-
larly Leonard Liebermann, whom Eckart had brought from
Woods Hole Oceanographic Institution to join the staff of
MPL. As a result of the work by Liebermann and the late

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202
BIOGRAPHICAL MEMOIRS
Robert Leonard at UCLA, the attenuation could be ascribed
to molecular resonances of certain trace constituents. Eckart's
wartime acoustical studies had called attention to the fact that
the usual linear formulation was incomplete and that additional
effects could be predicted if nonlinear terms were included in
the equations. He showed that irradiation of a fluid by sound
led to streaming and that the streaming could be used to mea-
sure the "second viscosity" of fluids, as distinct from the classical
notion of dilational viscosity. This theoretical work was directly
responsible for a series of experiments by Liebermann.
During this period there was time to consolidate some pre-
war studies into two major review papers. Eckart's exposition of
the one-dimensional Schrodinger equation for the Reviews of
Modern. Physics was enriched by his wartime experiences with
sound and light waves and internal waves in the sea.
Eckart's close attention to the logical development of ideas
with regard to fundamentals is perhaps brought out in its most
explicit form in his Encyclopacdia Britannica article on the
ether in physics. Eckart clearly put an immense amount of work
into this, and it should remain a classic. It is unfortunate that
with the passing of the 1948 edition this article will no longer
be generally available. The article traces the evolution of ether
from its initial concept as a passive backdrop of space-time
events for matter to that of an active participant In these events.
In its passive role, Eckart likened the ether to a movie screen
that is unaffected by the events cast on it by the movie projector.
The Einstein equations changed the passive role of space-time
to an active one by equating the Einstein curvature tensor of
space-time to the energy momentum tensor of matter. In this
way, seemingly empty space between atoms, planets, and stars
responded to their presence by accommodating its curvature;
and the matter in turn evolved and moved in response to space's
curvature. Eckart concluded his 1948 article on a tentative note,
anticipating future changes in the ether concept when quantum
. ~ . . . .

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C AR L H E N R Y E C K A R T
203
effects would direct attention to the structure of the ether
(space-time) in "the small." Such concepts were in fact devel-
oped in the following decade, mainly by the Princeton research
group headed by Eckart's friend and colleague John Archibald
Wheeler. In 1957, Wheeler and one of his students, Charles
Misner, showed that Einstein's field equations could be so inter-
preted that matter itself (their example used electromagnetic
fields) was a property of empty space. In conversations with
Eckart about these advances, he expressed a neutral attitude,
preferring to defer judgment until more empirical evidence was
available.
In 1948, Harald Sverdrup, Director of the Scripps Institu-
tion, decided that he would return to his native Norway and
that Roger Revelle (a Scripps Ph.D. then on duty with the Office
of Naval Research in Washington, D.C.) should succeed him.
There was some determined opposition, which was resolved by
Eckart's appointment to the Scripps directorship, with Revelle
as Associate Director. After two years, Eckart resigned, and
Revelle succeeded him. Concerning this period, Revelle has
written (personal communication):
"After I assumed the job, I rapidly gained a reputation as a
poor administrator. But in some ways, compared to Carl, I was
an administrative genius. The difficulty was that he took the
job too seriously. The rigor in definition and precision of
thought, and the inability to leave any loose strings untied,
which were his great strengths as a scientist, were just what was
not needed as an administrator. I remember he spent a good
deal of time trying to tidy the Scripps Institution up; it was
quite a messy place in those days and this was a completely
frustrating job for him.
~ C. W. Misner and J. A. Wheeler, "Classical Physics as Geometry. Gravita-
tion, Electromagnetism, Unquantized Change, and Mass as Properties of Curved
Empty Space," Annals of Physics 2(1957):525. C. W. Misner, "Feynman Quantiza-
tion of General Relativity," Reviews of Modern.Physics 29(1957):497.

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BIOGRAPHICAL MEMOIRS
"In other ways, however, he was a great leader. He had good
taste in people, in choosing staff, and even more, he had the
ability to see what was good about them, what was original
about them, and to help them fulfill their promise."
Following the two-year directorship, Eckart returned with
vigor to research and teaching. Several generations of students
were to benefit from his outstanding teaching abilities. Many
of Eckart's studies are in the form of lecture notes and Scripps
reports that have never been published. This is particularly the
case for his work on stochastic processes and geophysical time
series. These studies originated in wartime and were conducted
in parallel with, but quite independently of, a large effort at the
Massachusetts Institute of Technology's radiation laboratory.
The MIT work was published promptly after the war, but
Eckart's contributions have not been generally recognized.
Eckart's blackboard work was a reflection of the working of
his mind: he would start in the upper left-hand part of the left-
most board and then work slowly, with his elegant handwriting,
down the board, and then onto the next, developing as he went
a set of ideas woven through with a logical thread. Eckart's
"rough" notes, written in ink as they occurred, usually without
corrections, are so well-worded, annotated, and spaced that they
could not be improved. (They lose some of their clarity when
subsequently set in print.) In the solitude of his study, there
would be held lightly in the fingers of his left hand the ever-
present smoldering cigarette; its fumes randomly swirling
about his head as he strove, oblivious to the thickening haze,
toward the end of some syllogistic trail.
Eckart depended on his colleagues for ideas and stimulation;
yet he was impatient and driven to distraction by the lack of
rigor with which some of his scientific associates presented their
problems. Turbulence, described with an appropriate wave of
the hands, was the all-encompassin~ sink of oceanographic igno-
rance. This drove Eckart to his 1948 paper on stirring and mix-

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CARL HENRY ECKART
209
developed earlier for shock waves, Eckart specifically considered
phase changes and added to the classical boundary conditions
the generation of entropy across the phase discontinuity bound-
ary. This led to a theory of evaporation and condensation, pre-
dicting that energy is added or removed via radiation rather
than convection. In evaporation, a thermal boundary layer is
formed in the liquid, and a high-pressure gradient opposes the
flow in a thin layer in the vapor. In condensation, the thermal
layer occurs in the vapor, the pressure gradient in the liquid.
The mathematical and physical similarities between shock
waves and phase boundaries in chemically homogeneous sub-
stances are thus established.
This study, as his earlier ones, was marked by Eckart's abid-
~ng preoccupation with the powerful methods of classical mathe-
matical physics. The mathematical themes were those of the
Sturm-Liouville differential systems, of integral operators, of
classical groups of motions in Euclidean spaces, of useful analyt-
ical transformations, of variational principles; in short, of all the
notions arising in prerelativistic, prequantum physics. Revelle
remarked to van Neumann that he never really understood
Eckart's mathematics, and that they were difficult to follow.
Von Neumann replied that Eckart's mathematics were quite
simple and easy to follow, and that Eckart had great ability as a
mathematician; but that he was first, last, and always a physicist.
The single-minded and intense devotion to his work took
their toll, first of Carl Eckart's private life, and later of his
health. He was a lonely man, who had little supportive home
life in his first marriage. This finally ended in divorce after some
eighteen years of his constant, but unsuccessful, attempts to help
his wife through severe psychological problems. Carl Eckart,
always shy, remained somewhat aloof from the social activities
of La Jolla, then a maturing seaside village. After the death of
Eckart's great and good friend John van Neumann, Klara van
Neumann turned to Carl Eckart for solace and companionship.

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BIOGRAPHICAL MEMOIRS
Their marriage led to a brief period of happiness and active
participation in the sprawling life of the postwar Scripps Insti-
tution and beginnings of a general university campus, until
Klara's tragic drowning in 1963.
During the two remaining years of his life after retirement
in 1971, Eckart's thoughts turned to a task which he had been
considering for years: the summing up in book form of his
beliefs concerning the role of mathematical science in further-
ing human society. This task became urgent as his eyesight
began to fail. With the help of friends, notebooks and reference
materials were supplied as needed. The evolving manuscript
was called, "Our Modern Idol: Mathematical Science." Eckart
found that the promise of mathematical science in furthering
man's social progress was hollow. This realization, though un-
pleasant for him at the time, was in the end salutary. For he
realized that great scientists such as Ernst Abbe and Bertrand
Russell could use their clear insights into social problems with-
out recourse to mathematical analysis. These men were for
Eckart fine examples of concerned scientists who could use their
knowledge to educate fellow humans so that the latter could in
turn further the social progress of mankind. In Essay 9, Part IV,
of the manuscript, Eckart writes:
"When men of proven mathematical creativity become seri-
ously concerned with the problems of people and society, they
abandon the mathematical methods of which they are masters.
Their actions show that they do not consider that problems of
society are amenable to mathematical theories and calculations.
No matter how pessimistic they may be about the future, or how
ungratefully their efforts to improve it are received, they do not
become fatalistic. Their hope for improving, the future of Man
rests not on inexorable mathematical calculations but on the
ability of people to make decisions, to make plans, and to
implement them."
Another principal concern of Eckart in this, his final study,

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CARL HE N RY E CKART
211
was the often fallacious use of language by the ancient western
philosophers in cataloging their perceptions and conceptions of
the real world, and the malevolent persistence of their con-
fusions down to the present. In particular, these errors arose in
the improper separation of the kind of thinking that scientists
do about the real world from that kind of thinking all of us do
about the thinking of humans. These two kinds of thinking are
designated by A. N. Whitehead as homogeneous and hetero-
geneous thinking, respectively. Eckart made three applications
of this classification: to the long-standing problems of the faulty
development of knowledge and its faulty communication from
one generation to another; to the needless mental confusion of
ethical and scientific matters; and last and most painful for him,
to the seeming impotence of mathematical reasoning to show
society the way clear of its political, economic, and social prob-
lems. These applications were developed by Eckart in a series
of several dozen essays over a span of 2,000 handwritten pages,
showing his concern for social progress and the responsibility of
scientists to assure the proper use of their discoveries.
Death overtook Eckart before this work was finished. Plans
are being made by friends to prepare the incomplete manuscript
for publication as a book.
When Eckart first accepted the challenge of the oceans and
the Earth as a test of his mathematical and physical insight, he
had available the most powerful tools of his generation. He felt
that if only he could spend ten concentrated years on the prob-
lems of oceanography and geophysics, he would "solve" them on
some level of satisfaction. As he made progress, the complexity
and difficulty of the problems grew at approximately the same
rate as the evolving solutions, perhaps at a slightly greater rate.
After having worked in this oceanographic setting during his
~ A. N. Whitehead, The Concept of Nature (Cambridge: Cambridge University
Press, 1964), chap. 1.

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212
BIOGRAPHICAL MEMOIRS
mature lifetime, Eckart probably overreacted and was left with
the impression that the problems of the oceans (like the social
problems) are unsolvable. Nevertheless, in the thirty or so years
between the time when he thought he could solve the problems
and the times when he thought that they were unsolvable, he
provided inspiration to a generation of oceanographers.
Eckart was elected to the National Academy of Sciences in
1953 and received the Academy's Alexander Agassiz Medal in
1966.

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CARL HENRY ECKART
BIBLIOGRAPHY
KEY TO ABBREVIA TIONS
213
Am. I- Sci. = American journal of Science
C. U. W. of C. E. = Collected Unpublished Works of Carl Eckart~
I. Acoust. Soc. Am. = The journal of the Acoustical Society of America
I. Mar. Res. = Journal of Marine Research
Phys. Fluids = The Physics of Fluids
Phys. Rev. = Physical Review
Proc. Natl. Acad. Sci. USA = Proceedings of the National Academy of Sci-
ences of the United States of America
Rev. Mod. Phys. = Reviews of Modern Physics
Scripps Inst. Oceanogr., Ref. No. = Scripps Institution of Oceanography,
Reference Number
Univ. Calif. Div. War Res. Rep. = University of California Division of War
Research Report
Z. Phys. = Zeitschrift fur Physik
19~23
With G. E. M. jounced. Is there a change of wave-length on reflec-
tion clef X-rays from crystals? Nature, 112:325-26.
1924
With K. T. Compton. Oscillations in the low-voltage helium arc.
Science, 59:166-68.
With K. T. Compton. Explanation of abnormal low-voltage arcs.
Nature,114:51.
With K. T. Compton. The abnormal low-voltage arc. Phys. Rev.,
24:97-112.
The wave theory of the Compton effect. Phys. Rev., 24:591.
1925
With K. T. Compton. The diffusion of electrons against an electric
field in the non-oscillatory abnormal low-voltage arc. Phys. Rev.,
26: 139-46.
The life of metastable helium and mercury. Science, 61:517-18.
The conservation of momentum and the width of critical potentials
determined by the method of energy loss. Science, 62:265.
Post-arc conductivity and metastable helium. Phys. Rev., 26:454-64.
# Available at the Library of the Scripps Institution of Oceanography, Uni-
versity of California at San Diego.

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BIOGRAPHICAL MEMOIRS
1926
The solution of the problem of the simple oscillator by a combina-
tion of the Schrodinger and the Lanczos theories. Proc. Natl.
Acad. Sci. USA, 12:473-76.
Operator calculus and the solution of the equations of quantum
dynamics. Phys. Rev., 28:711-26.
The hydrogen spectrum in the new quantum theory. Phys. Rev.,
28:927-35.
Note on the correspondence principle in the new quantum theory.
Proc. Natl. Acad. Sci. USA, 12:684-87.
1928
Ober die Elektronentheorie der Metalle auf Grund der Fermischen
Statistic, insbensondere uber den Volta-Effekt. Z. Phys., 47:38-42.
Die korrespondenzmaszige Beziehung zwischen den Matrizen und
den Fourier-koeffizienten des Wasserstoff-Problems. Z. Phys., 48:
295-301.
1929
The continuous X-ray spectrum. Phys. Rev., 34: 167-75.
1930
With H. Honl. Grundzuge und Ergebnisse der Wellenmechanik.
Physikalische Zeitschrift, 31: 89` 119, 145-65.
Boundary conditions in wave mechanics. Phys. Rev., 35: 1298. (L)
The penetration of a potential barrier by electrons. Phys. Rev., 35:
1303-9.
The application of group theory to the quantum dynamics of mon-
atomic systems. Rev. Mod. Phys., 2:305-80.
Wave mechanics of deflected electrons. Phys. Rev., 36: 1514-15.
With D. S. Hughes. The effect of the motion of the nucleus on the
spectra of Li I and Li II. Phys. Rev., 36:694-98.
The calculation of energy values. Phys. Rev., 36: 149-50.
Theory and calculation of screening constants. Phys. Rev., 36:878-
92.
With F. C. Hoyt. Translation of Physical Principles of the Quantum
Theory, by W. Heisenberg. Chicago: Univ. of Chicago Press.
xii ~ 186 pp.

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CARL HENRY ECKART
1933
215
A general derivation of the formula for the diffraction by a perfect
grating. Phys. Rev., 44: 12-14.
A comparison of the nuclear theories of Heisenberg and Wigner, I.
Phys. Rev., 44: 109-11.
1934
The influence of the ionization chamber on the form of the cosmic-
ray depth-ionization curve. Phys. Rev., 45:451-53.
The analysis of the cosmic-ray absorption curve. Phys. Rev., 45:851-
59.
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Some studies concerning rotating axes and polyatomic molecules.
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The correction of continuous spectra for the finite resolution of the
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Ray-particle analogy. J. Mar. Res., 9: 139-41.
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Surface wake of a submerged sphere. Phys. Fluids, 1:457-61.
The steady flow of a heavy inviscid fluid of constant density. (Notes
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The equations of motion of sea-water. In: The Sea; Ideas and Ob-
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Some transformations of the hydrodynamic equations. Phys. Fluids,
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Extension of Howard's circle theorem to adiabatic jets. Phys. Fluids,
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The N-particle problem and the macroscopic theory of matter.
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The general circulation of the oceans. (Lecture notes, Oceanography
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