Carl Eckart
Encyclopedia
Carl Henry Eckart was an American
physicist, physical oceanographer, geophysicist, and administrator. He co-developed the Wigner-Eckart theorem
and is also known for the Eckart conditions
in quantum mechanics.
where he received his B.S. and M.S. degrees with a major in engineering. However, due to Arthur Holly Compton, a physics faculty member and later Chancellor, Eckart was influenced to continue his education in physics at Princeton, where he went in 1923 on an Edison Lamp Works Research Fellowship. Eckart was awarded his Ph.D. in 1925.
During his graduate studies, Eckart co-authored a paper with Karl Compton, brother of Arthur Compton on low-voltage arcs, particularly the oscillatory phenomena arising in the diffusion of electrons against low-voltage fields. He continued this line of work after receipt of his Ph.D. on a National Research Council Fellowship at the California Institute of Technology
(Caltech) during the period 1925 to 1927.
Max Born
, Director of the Institute for Theoretical Physics at the Georg-August University of Göttingen
and co-developer of the matrix mechanics
formulation of quantum mechanics
with Werner Heisenberg
, came to Caltech in the winter of 1925 and gave a lecture on his work. Born’s lecture gave Eckart the impetus to investigate the possible general operator formalism for quantum mechanics. Working into early 1926, Eckart developed the formalism. When Erwin Schrödinger’s
first paper in the series of four on the wave mechanics formulation of quantum mechanics was published in January, Eckart soon realized that the matrix formulation and wave formulation
of quantum mechanics were equivalent; he submitted his paper to the Proceedings of the National Academy of Sciences of the United States of America for publication. However, it was communicated on May 31, 1926, and Schrödinger’s
paper on the equivalence was received on March 18, 1926, thus giving him credit for the realization.
In 1927 Eckart received a Guggenheim Fellowship
to do postdoctoral study and research with Arnold Sommerfeld
at the Ludwig Maximilians University of Munich
, one of the three main centers for the development of quantum mechanics, the others being Göttingen under Born and the University of Copenhagen
under Niels Bohr
. Also at Munich simultaneous with Eckart were Rudolf Peierls
, and two other Guggenheim Fellows, Edwin C. Kemble
and William V. Houston. In Munich, Eckart worked on the quantum mechanical behavior of simple oscillators using the Schrödinger equation
and on operator calculus related to the matrix formulation of quantum mechanics. He also applied his work to the theory of electrons and the conductivity of metals using Fermi statistics, and he co-authored a paper on the subject with Sommerfeld
and William V. Houston.
, where he continued his work on quantum mechanics for another 14 years. Noteworthy was a paper co-authored with Helmut Hönl
, who received his doctorate under Sommerfeld in 1926; the paper, on the foundations of quantum mechanics, dealt with the role of group theory in quantum dynamics in monatomic systems and comparisons of the nuclear theories of Werner Heisenberg
and Eugene Wigner. During this period, Eckart developed his formulation of the Wigner-Eckart theorem
– a link between symmetry transformation groups applied to the Schrödinger equation and the laws of conservation of energy, momentum, and angular momentum. The theorem is particularly useful in spectroscopy
. With F. C. Hoyt, Eckart translated Heisenberg’s book on the physical principles of quantum mechanics. During the 1934-1935 academic year, Eckart took a sabbatical at the Institute for Advanced Study
in New Jersey
, as he also did in the academic years 1952-1953 and 1960-1961.
In December 1938 in Germany, Otto Hahn
and Fritz Strassmann
conducted an experiment which pointed towards the fission
of uranium
. They communicated their results to their former colleague Lisa Meitner, who had fled Germany
earlier in the year. In January 1939, Meitner and her nephew Otto Frisch correctly interpreted the experimental results as the fission of uranium. News of the discovery spread very rapidly. With the potential of making a fission-based atomic weapon and the threat of war in Europe, this caused anxiety in many, Leó Szilárd
for example, that Germany would develop an atomic weapon. As a result of two meetings with Albert Einstein
, the first with Szilárd and Eugene Wigner and the second with Szilárd and Edward Teller
, Einstein signed the Einstein–Szilárd letter to President Franklin Delano Roosevelt (FDR) in August. World War II
broke out in Europe in September. The letter was delivered to FDR in October by economist and banker Alexander Sachs
. In response to the letter, the Uranium Committee was formed that month. The Committee was organized into subsections by topic. The Theoretical Aspects Subsection, chaired by Enrico Fermi
, was located at the University of Chicago, and Eckart was a member of it. However, in 1941, Eckart withdrew from the Committee because of his anti-atomic bomb sentiments. Also notable during this period are his papers on the thermodynamics of irreversible processes.
In 1946, Eckart officially resigned his position at the University of Chicago to become a professor of geophysics at the Scripps Institution of Oceanography
of the University of California at San Diego (UCSD), a position he held until 1971. In 1946 he also became the first director of the Marine Physical Laboratory (MPL) of the University of California. The MPL was founded by Eckart, Roger Revelle
, and Admiral Rawson Bennett to conduct geophysical research of common interest to the academic and naval communities. In 1948, the MPL became an integral part of the Scripps Institution of Oceanography, with Eckart serving as director until 1952. Eckart contributed to geophysics by linking theoretical hydrodynamic exercises to actual physical properties of water. In the following decades he did research on thermal layering in the ocean and atmospheres on which he wrote a book, the transmission of sound in the sea, turbulence, air-sea interactions, the generation and structure of surface, and internal ocean waves.
After WW II, Eckart collected his work and the work of others on underwater detection and published it a classified volume entitled Principles and Applications of Underwater Sound, which was first published in 1946. It was declassified in 1954 and reprinted in 1968. It is a standard reference.
During the period 1957 to 1959, Eckart was a member of the Editorial Advisory Board for the Johns Hopkins University
Applied Physics Laboratory’s series on applied mathematics and mechanics. From 1959 through 1970, he was also a consultant for commercial enterprises such as General Dynamics Corporation and the Rand Corporation.
From 1965 to 1967, Eckart was vice-chancellor for academic affairs at the UCSD. He then served the University of California, from 1967 to 1968, as alternate representative to the Institute for Defense Analyses
, which was made up of 12 member universities and functioned as an independent source for studies and advice for the Department of Defense
.
Eckart contributed to the posthumous publication of some works by the mathematician John von Neumann
.
; Klara died in 1963 in a drowning accident.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
physicist, physical oceanographer, geophysicist, and administrator. He co-developed the Wigner-Eckart theorem
Wigner-Eckart theorem
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum...
and is also known for the Eckart conditions
Eckart conditions
The Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
in quantum mechanics.
Education
Eckart began college in 1919 at Washington University in St. LouisWashington University in St. Louis
Washington University in St. Louis is a private research university located in suburban St. Louis, Missouri. Founded in 1853, and named for George Washington, the university has students and faculty from all fifty U.S. states and more than 110 nations...
where he received his B.S. and M.S. degrees with a major in engineering. However, due to Arthur Holly Compton, a physics faculty member and later Chancellor, Eckart was influenced to continue his education in physics at Princeton, where he went in 1923 on an Edison Lamp Works Research Fellowship. Eckart was awarded his Ph.D. in 1925.
During his graduate studies, Eckart co-authored a paper with Karl Compton, brother of Arthur Compton on low-voltage arcs, particularly the oscillatory phenomena arising in the diffusion of electrons against low-voltage fields. He continued this line of work after receipt of his Ph.D. on a National Research Council Fellowship at the California Institute of Technology
California Institute of Technology
The California Institute of Technology is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering...
(Caltech) during the period 1925 to 1927.
Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
, Director of the Institute for Theoretical Physics at the Georg-August University of Göttingen
Georg-August University of Göttingen
The University of Göttingen , known informally as Georgia Augusta, is a university in the city of Göttingen, Germany.Founded in 1734 by King George II of Great Britain and the Elector of Hanover, it opened for classes in 1737. The University of Göttingen soon grew in size and popularity...
and co-developer of the matrix mechanics
Matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...
formulation of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
with Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, came to Caltech in the winter of 1925 and gave a lecture on his work. Born’s lecture gave Eckart the impetus to investigate the possible general operator formalism for quantum mechanics. Working into early 1926, Eckart developed the formalism. When Erwin Schrödinger’s
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
first paper in the series of four on the wave mechanics formulation of quantum mechanics was published in January, Eckart soon realized that the matrix formulation and wave formulation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
of quantum mechanics were equivalent; he submitted his paper to the Proceedings of the National Academy of Sciences of the United States of America for publication. However, it was communicated on May 31, 1926, and Schrödinger’s
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
paper on the equivalence was received on March 18, 1926, thus giving him credit for the realization.
In 1927 Eckart received a Guggenheim Fellowship
Guggenheim Fellowship
Guggenheim Fellowships are American grants that have been awarded annually since 1925 by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts." Each year, the foundation makes...
to do postdoctoral study and research with Arnold Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
at the Ludwig Maximilians University of Munich
Ludwig Maximilians University of Munich
The Ludwig Maximilian University of Munich , commonly known as the University of Munich or LMU, is a university in Munich, Germany...
, one of the three main centers for the development of quantum mechanics, the others being Göttingen under Born and the University of Copenhagen
University of Copenhagen
The University of Copenhagen is the oldest and largest university and research institution in Denmark. Founded in 1479, it has more than 37,000 students, the majority of whom are female , and more than 7,000 employees. The university has several campuses located in and around Copenhagen, with the...
under Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...
. Also at Munich simultaneous with Eckart were Rudolf Peierls
Rudolf Peierls
Sir Rudolf Ernst Peierls, CBE was a German-born British physicist. Rudolf Peierls had a major role in Britain's nuclear program, but he also had a role in many modern sciences...
, and two other Guggenheim Fellows, Edwin C. Kemble
Edwin C. Kemble
Edwin Crawford Kemble was an American physicist who made contributions to the theory of quantum mechanics and molecular structure and spectroscopy...
and William V. Houston. In Munich, Eckart worked on the quantum mechanical behavior of simple oscillators using the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
and on operator calculus related to the matrix formulation of quantum mechanics. He also applied his work to the theory of electrons and the conductivity of metals using Fermi statistics, and he co-authored a paper on the subject with Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
and William V. Houston.
University of Chicago
Returning to the United States in 1928, Eckart was appointed Assistant Professor in the Physics Department at the University of ChicagoUniversity of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
, where he continued his work on quantum mechanics for another 14 years. Noteworthy was a paper co-authored with Helmut Hönl
Helmut Hönl
Helmut Hönl was a German theoretical physicist who made contributions to quantum mechanics and the understanding of atomic and molecular structure....
, who received his doctorate under Sommerfeld in 1926; the paper, on the foundations of quantum mechanics, dealt with the role of group theory in quantum dynamics in monatomic systems and comparisons of the nuclear theories of Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
and Eugene Wigner. During this period, Eckart developed his formulation of the Wigner-Eckart theorem
Wigner-Eckart theorem
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum...
– a link between symmetry transformation groups applied to the Schrödinger equation and the laws of conservation of energy, momentum, and angular momentum. The theorem is particularly useful in spectroscopy
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
. With F. C. Hoyt, Eckart translated Heisenberg’s book on the physical principles of quantum mechanics. During the 1934-1935 academic year, Eckart took a sabbatical at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in New Jersey
New Jersey
New Jersey is a state in the Northeastern and Middle Atlantic regions of the United States. , its population was 8,791,894. It is bordered on the north and east by the state of New York, on the southeast and south by the Atlantic Ocean, on the west by Pennsylvania and on the southwest by Delaware...
, as he also did in the academic years 1952-1953 and 1960-1961.
In December 1938 in Germany, Otto Hahn
Otto Hahn
Otto Hahn FRS was a German chemist and Nobel laureate, a pioneer in the fields of radioactivity and radiochemistry. He is regarded as "the father of nuclear chemistry". Hahn was a courageous opposer of Jewish persecution by the Nazis and after World War II he became a passionate campaigner...
and Fritz Strassmann
Fritz Strassmann
Friedrich Wilhelm "Fritz" Strassmann was a German chemist who, with Otto Hahn in 1938, identified barium in the residue after bombarding uranium with neutrons, which led to the interpretation of their results as being from nuclear fission...
conducted an experiment which pointed towards the fission
Nuclear fission
In nuclear physics and nuclear chemistry, nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts , often producing free neutrons and photons , and releasing a tremendous amount of energy...
of uranium
Uranium
Uranium is a silvery-white metallic chemical element in the actinide series of the periodic table, with atomic number 92. It is assigned the chemical symbol U. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons...
. They communicated their results to their former colleague Lisa Meitner, who had fled Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
earlier in the year. In January 1939, Meitner and her nephew Otto Frisch correctly interpreted the experimental results as the fission of uranium. News of the discovery spread very rapidly. With the potential of making a fission-based atomic weapon and the threat of war in Europe, this caused anxiety in many, Leó Szilárd
Leó Szilárd
Leó Szilárd was an Austro-Hungarian physicist and inventor who conceived the nuclear chain reaction in 1933, patented the idea of a nuclear reactor with Enrico Fermi, and in late 1939 wrote the letter for Albert Einstein's signature that resulted in the Manhattan Project that built the atomic bomb...
for example, that Germany would develop an atomic weapon. As a result of two meetings with Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, the first with Szilárd and Eugene Wigner and the second with Szilárd and Edward Teller
Edward Teller
Edward Teller was a Hungarian-American theoretical physicist, known colloquially as "the father of the hydrogen bomb," even though he did not care for the title. Teller made numerous contributions to nuclear and molecular physics, spectroscopy , and surface physics...
, Einstein signed the Einstein–Szilárd letter to President Franklin Delano Roosevelt (FDR) in August. World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
broke out in Europe in September. The letter was delivered to FDR in October by economist and banker Alexander Sachs
Alexander Sachs
Alexander Sachs was an Jewish American economist and banker. In 1939, he delivered the Einstein–Szilárd letter to Franklin D. Roosevelt, in which it was suggested that nuclear research should be fomented....
. In response to the letter, the Uranium Committee was formed that month. The Committee was organized into subsections by topic. The Theoretical Aspects Subsection, chaired by Enrico Fermi
Enrico Fermi
Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
, was located at the University of Chicago, and Eckart was a member of it. However, in 1941, Eckart withdrew from the Committee because of his anti-atomic bomb sentiments. Also notable during this period are his papers on the thermodynamics of irreversible processes.
University of California at San Diego
With the entry of the United States into WW II in December 1941, there was increased incentive for the scientific community to participate in the war effort. Axis submarines were exacting a toll on allied shipping, and university scientists were being approached by the U. S. Navy concerning optical and acoustical detection of submarines. B. O. Knudsen, director of the newly formed University of California Division of War Research, and his associate L. P. Delsasso approached Eckart for help. Eckart (an Associate Professor) took leave from the University of Chicago to work on the problem, thus beginning his 31-year stay in California. From 1942, he was assistant director of the Division of War Research, and eventually he was director, a position he held until 1946.In 1946, Eckart officially resigned his position at the University of Chicago to become a professor of geophysics at the Scripps Institution of Oceanography
Scripps Institution of Oceanography
Scripps Institution of Oceanography in La Jolla, California, is one of the oldest and largest centers for ocean and earth science research, graduate training, and public service in the world...
of the University of California at San Diego (UCSD), a position he held until 1971. In 1946 he also became the first director of the Marine Physical Laboratory (MPL) of the University of California. The MPL was founded by Eckart, Roger Revelle
Roger Revelle
Roger Randall Dougan Revelle was a scientist and scholar who was instrumental in the formative years of the University of California, San Diego and was one of the first scientists to study global warming and the movement of Earth's tectonic plates...
, and Admiral Rawson Bennett to conduct geophysical research of common interest to the academic and naval communities. In 1948, the MPL became an integral part of the Scripps Institution of Oceanography, with Eckart serving as director until 1952. Eckart contributed to geophysics by linking theoretical hydrodynamic exercises to actual physical properties of water. In the following decades he did research on thermal layering in the ocean and atmospheres on which he wrote a book, the transmission of sound in the sea, turbulence, air-sea interactions, the generation and structure of surface, and internal ocean waves.
After WW II, Eckart collected his work and the work of others on underwater detection and published it a classified volume entitled Principles and Applications of Underwater Sound, which was first published in 1946. It was declassified in 1954 and reprinted in 1968. It is a standard reference.
During the period 1957 to 1959, Eckart was a member of the Editorial Advisory Board for the Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
Applied Physics Laboratory’s series on applied mathematics and mechanics. From 1959 through 1970, he was also a consultant for commercial enterprises such as General Dynamics Corporation and the Rand Corporation.
From 1965 to 1967, Eckart was vice-chancellor for academic affairs at the UCSD. He then served the University of California, from 1967 to 1968, as alternate representative to the Institute for Defense Analyses
Institute for Defense Analyses
The Institute for Defense Analyses is a non-profit corporation that administers three federally funded research and development centers to assist the United States government in addressing important national security issues, particularly those requiring scientific and technical expertise...
, which was made up of 12 member universities and functioned as an independent source for studies and advice for the Department of Defense
United States Department of Defense
The United States Department of Defense is the U.S...
.
Eckart contributed to the posthumous publication of some works by the mathematician John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
.
Personal
Eckart married Edith Louise née Frazee in 1926; they were divorced in 1948. In 1958, he married Klara Dan von Neumann, the widow of the mathematician John von NeumannJohn von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
; Klara died in 1963 in a drowning accident.
Honors
- 1948 – Certificate of Merit signed by President Harry Truman
- 1952 – Elected to the National Academy of SciencesUnited States National Academy of SciencesThe National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
- 1966 – Awarded the Alexander Agassiz MedalAlexander Agassiz MedalThe Alexander Agassiz Medal is awarded by the U.S. National Academy of Sciences for an original contribution in the science of oceanography. It was established by Sir John Murray in honor of his friend Alexander Agassiz.-Recipients:-References:NotesA...
by the National Academy of SciencesUnited States National Academy of SciencesThe National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
for contributions to oceanography.
- 1972 – Awarded the William Bowie MedalWilliam Bowie MedalThe William Bowie Medal is awarded annually by the American Geophysical Union for 'outstanding contributions to fundamental geophysics and for unselfish cooperation in research"...
by the American Geophysical UnionAmerican Geophysical UnionThe American Geophysical Union is a nonprofit organization of geophysicists, consisting of over 50,000 members from over 135 countries. AGU's activities are focused on the organization and dissemination of scientific information in the interdisciplinary and international field of geophysics...
for outstanding contributions to fundamental geophysics.
Books
- Werner Heisenberg, Translated by Carl Eckart and F. C. Hoyt The Physical Principles of the Quantum Theory (Dover, 1930)
- Carl Eckart and others. Principles and Applications of Underwater Sound (NRDC, 1946). Originally a classified document and published as a Summary Technical Report of Division 6, NDRC Volume 7, Washington, D.C., 1946. Declassified and distributed September 7, 1954. Reprinted and redistributed by Department of the Navy Headquarters Naval Material Command, Washington, D.C., 1968.
- Carl Eckart Hydrodynamics of Oceans and Atmospheres (Pergamon Press, 1960)
Selected Literature
- Carl Eckart The Solution of the Problem of the Simple Oscillator by a Combination of the Schrödinger and the LanczosCornelius LanczosCornelius Lanczos Löwy Kornél was a Hungarian-Jewish mathematician and physicist, who was born on February 2, 1893, and died on June 25, 1974....
Theories, Proceedings of the National Academy of Sciences of the United States of America 12 473-476 (1926). Submitted 31 May 1926.
- Carl Eckart (National Research Fellow) Operator Calculus and the Solution of the Equations of Quantum Dynamics, Phys. Rev. 28 (4) 711 - 726 (1926). California Institute of Technology. Received 7 June 1926.
- A. Sommerfeld, W. V. Houston, and C. Eckart, Zeits. f. Physik 47, 1 (1928)
- Carl Eckart The Application of Group theory to the Quantum Dynamics of Monatomic Systems, Rev. Mod. Phys. 2 (3) 305 - 380 (1930). University of Chicago.
- Carl Eckart, Some Studies Concerning Rotating Axes and Polyatomic Molecules, Physical Review 47 552-558 (1935).
- Carl Eckart The Approximate Solution of One-Dimensional Wave Equations, Rev. Mod. Phys. 20 (2) 399 - 417 (1948). University of California, Marine Physical Laboratory, San Diego, California.