118-594: The Caltech Seismological Laboratory is an arm of the Division of Geological and Planetary Sciences of the California Institute of Technology . Known as "the Seismo Lab", it has been a world center for seismology research since the 1920s, and was for many decades a go-to source for rapid (and quotable) commentary to the press on large earthquakes. The Seismo Lab was established under the auspices of
236-480: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that
354-533: A Pass/Fail basis. There is little competition; collaboration on homework is encouraged and the honor system encourages take-home tests and flexible homework schedules. Caltech offers co-operative programs with other schools, such as the Pasadena Art Center College of Design and Occidental College . According to a 2018 PayScale study, Caltech graduates earn a median early career salary of $ 83,400 and $ 143,100 mid-career, placing them in
472-624: A collaboration between the Naval Ordnance Test Station and the Manhattan Project. Caltech scientists worked on a variety of assignments, including B-29 airdrop tests of model atomic bombs and the manufacturing of explosives for use in the atomic bomb’s implosion mechanism. Additionally, the Salt Wells Pilot Plant at Inyokern was developed with Caltech scientists in response to concerns about
590-466: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,
708-594: A division of biology was established under the leadership of Thomas Hunt Morgan , the most distinguished biologist in the United States at the time, and discoverer of the role of genes and the chromosome in heredity. In 1930, Kerckhoff Marine Laboratory was established in Corona del Mar under the care of Professor George MacGinitie . In 1926, a graduate school of aeronautics was created, which eventually attracted Theodore von Kármán . Kármán later helped create
826-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}
944-402: A granitic outcropping. Its first home, at 220 N. San Rafael Avenue, was a modest two-story laboratory building (later known as "Charles Richter Laboratory"). By 1958 it had outgrown the site and expanded to an adjacent property at 295 N. San Rafael, into what had been an 11000 sq ft (1020 sq m) mansion, complete with marble staircases, 9 bedrooms with connecting baths, and gardens. (This property
1062-537: A heavy load, and registrations above 51 units require an overload petition. Approximately 20 percent of students double-major. This is achievable since the humanities and social sciences majors have been designed to be done in conjunction with a science major. Although choosing two options in the same division is discouraged, it is still possible. First-year students are enrolled in first-term classes based upon results of placement exams in math, physics, chemistry, and writing and take all classes in their first two terms on
1180-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement
1298-439: A master plan for the 22 acres (8.9 ha) campus. Goodhue conceived the overall layout of the campus and designed the physics building, Dabney Hall, and several other structures, in which he sought to be consistent with the local climate, the character of the school, and Hale's educational philosophy. Goodhue's designs for Caltech were also influenced by the traditional Spanish mission architecture of Southern California. During
SECTION 10
#17327877082921416-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector
1534-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and
1652-471: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to
1770-601: A sign on the library bearing the simple phrase " Impeach Nixon ". The following week, Ross McCollum, president of the National Oil Company, wrote an open letter to Dabney House stating that in light of their actions he had decided not to donate one million dollars to Caltech. The Dabney family, being Republicans, disowned Dabney House after hearing of the protest. Since 2000, the Einstein Papers Project has been located at Caltech. The project
1888-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of
2006-551: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of
2124-745: A solar astronomer from the University of Chicago , founded the Mount Wilson Observatory in 1904. He joined Throop's board of trustees in 1907, and soon began developing the university, and the whole of Pasadena, into a major scientific and cultural destination. He engineered the appointment of James A. B. Scherer , a literary scholar untutored in science but very capable in administration and fund-raising, to Throop's presidency in 1908. Scherer persuaded retired businessman and trustee Charles W. Gates to donate $ 25,000 in seed money (equivalent to $ 800,000 in 2023) to build Gates Laboratory,
2242-572: A striking illustration of one way in which the Research Council can secure co-operation and advance scientific investigation. This institution, with its able investigators and excellent research laboratories, could be of great service in any broad scheme of cooperation. President Scherer, hearing of the formation of the council, immediately offered to take part in its work, and with this object, he secured within three days an additional research endowment of one hundred thousand dollars. Through
2360-496: A visiting-scholars program soon after joining Caltech. Notable scientists who accepted his invitation include Paul Dirac , Erwin Schrödinger , Werner Heisenberg , Hendrik Lorentz and Niels Bohr . Albert Einstein arrived on the Caltech campus for the first time in 1931 to polish up his Theory of General Relativity , and he returned to Caltech subsequently as a visiting professor in 1932 and 1933. During World War II, Caltech
2478-473: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of
SECTION 20
#17327877082922596-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of
2714-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes
2832-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in
2950-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate
3068-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes
3186-491: Is governed by a permanent trustee committee and administered by an investment office. The institute is organized into six primary academic divisions: Biology and Biological Engineering (founded 1927), Chemistry and Chemical Engineering (founded 1926), Engineering and Applied Science (founded 1926), Geological and Planetary Sciences (founded 1927), Humanities and Social Sciences (founded 1926), Physics, Mathematics, and Astronomy (founded 1926). Given Caltech's historical prestige and
3304-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In
3422-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,
3540-646: Is located approximately 11 mi (18 km) northeast of downtown Los Angeles , in Pasadena. First-year students are required to live on campus, and 95% of undergraduates remain in the on-campus housing system at Caltech . Students agree to abide by an honor code which allows faculty to assign take-home examinations . The Caltech Beavers compete in 13 intercollegiate sports in the NCAA Division III 's Southern California Intercollegiate Athletic Conference (SCIAC). Scientists and engineers at or from
3658-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making
Caltech Seismological Laboratory - Misplaced Pages Continue
3776-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then
3894-540: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent
4012-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,
4130-605: Is the largest ever for environmental sustainability research and the second-largest private donation to a US academic institution (after Bloomberg's gift of $ 1.8 billion to Johns Hopkins University in 2018). On account of President Robert A. Millikan's affiliation with the Human Betterment Foundation , in January 2021, the Caltech Board of Trustees authorized the removal of Millikan's name (and
4248-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation
4366-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because
4484-505: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}
4602-509: The California Legislature calling for the establishment of a publicly funded "California Institute of Technology," with an initial budget of a million dollars, ten times the budget of Throop at the time. The board of trustees offered to turn Throop over to the state, but the presidents of Stanford University and the University of California, Berkeley successfully lobbied to defeat the bill, which allowed Throop to develop as
4720-517: The Carnegie Institution of Washington in 1921 under leadership of Harry O. Wood. By 1926 it had become a cooperative venture between Carnegie and Caltech. In 1937 it was formally transferred in full to Caltech. Requiring accessible bedrock on which to place seismometers, the Seismo Lab could not originally be located on the Caltech campus, which is on alluvium. Instead, its first two homes were in Pasadena's Linda Vista neighborhood,
4838-714: The Fat Man implosion bomb, crucial to the Trinity Test and the subsequent bombing of Nagasaki . Lauritsen’s team at Caltech developed detonators that would later be used in atomic bombs. In November 1943, Caltech and the U.S. Navy established the Naval Ordnance Test Station (NOTS) in Inyokern, California , near the Mojave Desert to work on aircraft ordnance and rocket development. One of
Caltech Seismological Laboratory - Misplaced Pages Continue
4956-491: The Jet Propulsion Laboratory (JPL) is a federally funded research and development center (FFRDC) owned by NASA and operated as a division of Caltech through a contract between NASA and Caltech. In 2008, JPL spent over $ 1.6 billion on research and development and employed over 5,000 project-related and support employees. The JPL Director also serves as a Caltech Vice President and is responsible to
5074-713: The National Research Council to coordinate and support scientific work on military problems. While he supported the idea of federal appropriations for science, he took exception to a federal bill that would have funded engineering research at land-grant colleges, and instead sought to raise a $ 1 million national research fund entirely from private sources. To that end, as Hale wrote in The New York Times : Throop College of Technology, in Pasadena California has recently afforded
5192-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators
5310-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called
5428-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to
5546-516: The 1960s, Caltech underwent considerable expansion, in part due to the philanthropy of alumnus Arnold O. Beckman . In 1953, Beckman was asked to join the Caltech Board of Trustees. In 1964, he became its chairman. Over the next few years, as Caltech's president emeritus David Baltimore describes it, Arnold Beckman and his wife Mabel "shaped the destiny of Caltech". In 1971 a magnitude-6.6 earthquake in San Fernando caused some damage to
5664-626: The 2021–2022 school year was $ 56,394 and total annual costs were estimated to be $ 79,947 excluding the Caltech Student Health Insurance Plan. In 2012–2013, Caltech awarded $ 17.1 million in need-based aid, $ 438k in non-need-based aid, and $ 2.51 million in self-help support to enrolled undergraduate students. The average financial aid package of all students eligible for aid was $ 38,756 and students graduated with an average debt of $ 15,090. The full-time, four-year undergraduate program emphasizes instruction in
5782-543: The Caltech Division in which it is administratively situated. California Institute of Technology The California Institute of Technology (branded as Caltech ) is a private research university in Pasadena, California , United States. The university is responsible for many modern scientific advancements and is among a small group of institutes of technology in the United States that are devoted to
5900-671: The Caltech campus. Engineers who evaluated the damage found that two historic buildings dating from the early days of the Institute—Throop Hall and the Goodhue-designed Culbertson Auditorium—had cracked. New additions to the campus include the Cahill Center for Astronomy and Astrophysics and the Walter and Leonore Annenberg Center for Information Science and Technology, which opened in 2009, and
6018-654: The Class of 2027 (enrolled Fall 2023), Caltech received 13,136 applications and accepted 412 applicants for a 3.14% admit rate; 270 enrolled. The subsequent year, for the Class of 2028, Caltech reduced the number of seats by almost one hundred, accepting 315 applicants out of approximately 13,000 total applications. For the Class of 2025, 32% were of underrepresented ancestry (which includes students who self-identify as American Indian/Alaska Native, Hispanic/Latino, Black/African American, and/or Native Hawaiian/Pacific Islander), and 6% were foreign students. Undergraduate tuition for
SECTION 50
#17327877082926136-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on
6254-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system
6372-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by
6490-637: The Jet Propulsion Laboratory, and played an integral part in establishing Caltech as one of the world's centers for rocket science . In 1928, construction of the Palomar Observatory began. Millikan served as "Chairman of the Executive Council" (effectively Caltech's president) from 1921 to 1945, and his influence was such that the institute was occasionally referred to as "Millikan's School". Millikan initiated
6608-562: The National Research Council, Hale simultaneously lobbied for science to play a larger role in national affairs, and for Throop to play a national role in science. The new funds were designated for physics research, and ultimately led to the establishment of the Norman Bridge Laboratory, which attracted experimental physicist Robert Andrews Millikan from the University of Chicago in 1917. During
6726-598: The President of the Institute for the management of the laboratory. Caltech is a small four-year, highly residential research university with slightly more students in graduate programs than undergraduate. The institute has been accredited by the Western Association of Schools and Colleges since 1949. Caltech is on the quarter system : the fall term starts in late September and ends before Christmas,
6844-512: The SAT Physics Subject Test was 760–800; SAT Chemistry Subject Test was 760–800; SAT Biology Subject Tests was 760–800. In June 2020, Caltech announced a test-blind policy where they would not require nor consider test scores for the next two years. The moratorium was extended twice, starting July 2021, but was subsequently cancelled starting with the Class of 2029. The institute is need-blind for domestic applicants. For
6962-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses
7080-435: The United States among national universities overall, 11th for most innovative, and 15th for best value. U.S. News & World Report also ranked the graduate programs in chemistry and earth sciences first among national universities. Admission to Caltech is extremely rigorous. Prior to going test blind, Caltech students had the highest test scores in the nation. In admissions for the Class of 2028 (entering 2024), Caltech
7198-524: The University Librarian. Faculty are responsible for establishing admission requirements, academic standards, and curricula. The Faculty Board is the faculty's representative body and consists of 18 elected faculty representatives as well as other senior administration officials. Full-time professors are expected to teach classes, conduct research, advise students, and perform administrative work such as serving on committees. Founded in 1930s,
SECTION 60
#17327877082927316-630: The Warren and Katherine Schlinger Laboratory for Chemistry and Chemical Engineering followed in March 2010. The institute also concluded an upgrading of the South Houses in 2006. In late 2010, Caltech completed a 1.3 MW solar array projected to produce approximately 1.6 GWh in 2011. Caltech is incorporated as a non-profit corporation and is governed by a privately appointed 46-member board of trustees who serve five-year terms of office and retire at
7434-411: The age of 72. The trustees elect a president to serve as the chief executive officer of the institute and administer the affairs on the institute on behalf of the board, a provost who serves as the chief academic officer of the institute below the president, and ten other vice presidential and other senior positions. Thomas F. Rosenbaum became the ninth president of Caltech in 2014. Caltech's endowment
7552-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of
7670-641: The arts and sciences and has high graduate coexistence. Caltech offers 28 majors (called "options") and 12 minors across all six academic divisions. Caltech also offers interdisciplinary programs in Applied Physics, Biochemistry, Bioengineering, Computation and Neural Systems, Control and Dynamical Systems, Environmental Science and Engineering, Geobiology and Astrobiology, Geochemistry, and Planetary Astronomy. The most popular options are Chemical Engineering, Computer Science, Electrical Engineering, Mechanical Engineering and Physics. The most popular majors of
7788-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy
7906-439: The campus tripled in size. DuBridge, unlike his predecessors, welcomed federal funding of science. New research fields flourished, including chemical biology , planetary science , nuclear astrophysics , and geochemistry . A 200-inch telescope was dedicated on nearby Palomar Mountain in 1948 and remained the world's most powerful optical telescope for over forty years. Caltech opened its doors to female undergraduates during
8024-519: The class of 2023 were Computer Science, Mechanical Engineering, Physics, and Electrical Engineering. Prior to the entering class of 2013, Caltech required students to take a core curriculum of five terms of mathematics, five terms of physics, two terms of chemistry, one term of biology, two terms of lab courses, one term of scientific communication, three terms of physical education, and 12 terms of humanities and social science. Since 2013, only three terms each of mathematics and physics have been required by
8142-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for
8260-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in
8378-522: The course of the war, Hale, Noyes and Millikan worked together in Washington on the NRC. Subsequently, they continued their partnership in developing Caltech. Under the leadership of Hale , Noyes , and Millikan (aided by the booming economy of Southern California ), Caltech grew to national prominence in the 1920s and concentrated on the development of Roosevelt's "Hundredth Man". On November 29, 1921,
8496-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}
8614-498: The eighth president, announced on February 19, 2013, that he would be stepping down to accept the presidency at King Abdullah University of Science and Technology . Thomas F. Rosenbaum was announced to be the ninth president of Caltech on October 24, 2013, and his term began on July 1, 2014. In 2019, Caltech received a gift of $ 750 million for sustainability research from the Resnick family of The Wonderful Company . The gift
8732-426: The end of the war, Caltech had essentially become an extension of the U.S. Navy’s Bureau of Ordnance , with its rocket research providing important technology to U.S. combat capabilities. From April to December 1951, Caltech was the host of a federal classified study, Project Vista . The selection of Caltech as host for the project was based on the university's expertise in rocketry and nuclear physics. In response to
8850-458: The first science building on campus. In 1910, Throop moved to its current site. Arthur Fleming donated the land for the permanent campus site. Theodore Roosevelt delivered an address at Throop Institute on March 21, 1911, and he declared: I want to see institutions like Throop turn out perhaps ninety-nine of every hundred students as men who are to do given pieces of industrial work better than any one else can do them; I want to see those men do
8968-672: The funds went into the support of Caltech programs and projects. In 2010, Caltech, in partnership with Lawrence Berkeley National Laboratory and headed by Professor Nathan Lewis , established a DOE Energy Innovation Hub aimed at developing revolutionary methods to generate fuels directly from sunlight. This hub, the Joint Center for Artificial Photosynthesis , will receive up to $ 122 million in federal funding over five years. Since 2012, Caltech began to offer classes through massive open online courses (MOOCs) under Coursera , from 2013, edX , and bootcamps. Jean-Lou Chameau,
9086-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,
9204-493: The hotel, Vista del Arroyo Hotel, which housed the study. The study operated under a committee with the supervision of President Lee A. DuBridge . William A. Fowler , a professor at Caltech, was selected as research director. More than a fourth of Caltech's faculty and a group of outside scientists staffed the project. Moreover, the number increases if one takes into account visiting scientists, military liaisons, secretarial, and security staff. In compensation for its participation,
9322-403: The institute, with the remaining two terms each required by certain options. A typical class is worth 9 academic units and given the extensive core curriculum requirements in addition to individual options' degree requirements, students need to take an average of 40.5 units per term (more than four classes) to graduate in four years. 36 units is the minimum full-time load, 48 units is considered
9440-440: The instruction of pure and applied sciences. The institution was founded as a preparatory and vocational school by Amos G. Throop in 1891 and began attracting influential scientists such as George Ellery Hale , Arthur Amos Noyes , and Robert Andrews Millikan in the early 20th century. The vocational and preparatory schools were disbanded and spun off in 1910, and the college assumed its present name in 1920. In 1934, Caltech
9558-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior
9676-460: The kind of work that is now being done on the Panama Canal and on the great irrigation projects in the interior of this country—and the one-hundredth man I want to see with the kind of cultural scientific training that will make him and his fellows the matrix out of which you can occasionally develop a man like your great astronomer, George Ellery Hale. Also in 1911, a bill was introduced in
9794-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;
9912-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It
10030-493: The most successful innovations was the development of the 5-inch High-Velocity Aircraft Rocket , commonly known as the " Holy Moses ," which was used in combat against enemy fortifications and ships. The partnership between the Navy and Caltech continued to deepen throughout the war, leading to the creation of several military technologies, and by 1945, the focus of Caltech’s war contributions expanded further with Project Camel ,
10148-625: The names of five other historical figures affiliated with the Foundation), from campus buildings. Caltech's 124-acre (50 ha) primary campus is located in Pasadena, California , approximately 11 miles (18 km) northeast of downtown Los Angeles . It is within walking distance of Old Town Pasadena and the Pasadena Playhouse District and therefore the two locations are frequent getaways for Caltech students. In 1917 Hale hired architect Bertram Goodhue to produce
10266-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude
10384-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of
10502-552: The only individual in history to win two unshared prizes). In addition, 66 National Medal of Science Recipients, 43 MacArthur Fellows , 14 National Medal of Technology and Innovation recipients, 11 astronauts , 5 Science Advisors to the President , 4 Fields Medalists , and 6 Turing Award winners have been affiliated with Caltech. Caltech started as a vocational school founded in present-day Old Pasadena on Fair Oaks Avenue and Chestnut Street on September 23, 1891, by local businessman and politician Amos G. Throop . The school
10620-438: The only scientific research-oriented educational institute in southern California, public or private, until the onset of World War II necessitated the broader development of research-based science education. The promise of Throop attracted physical chemist Arthur Amos Noyes from MIT to develop the institution and assist in establishing it as a center for science and technology. With the onset of World War I , Hale organized
10738-455: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }
10856-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For
10974-462: The presidency of Harold Brown in 1970, and they made up 14% of the entering class. The portion of female undergraduates has been increasing since then. Protests by Caltech students are rare. The earliest was a 1968 protest outside the NBC Burbank studios, in response to rumors that NBC was to cancel Star Trek . In 1973, the students from Dabney House protested a presidential visit with
11092-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that
11210-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and
11328-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present
11446-641: The safety of explosive production at Los Alamos and began producing high explosives just days before the Trinity Test in July 1945. Early in the war, Caltech scientists, including Lauritsen’s son, Thomas Lauritsen , worked on various rocket designs at the Kellogg Radiation Laboratory. These rockets, including the " Tiny Tim " and the " Mighty Mouse ," were used in critical military operations, from naval engagements to land assaults. By
11564-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while
11682-1019: The scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately
11800-543: The second term starts after New Year's Day and ends in mid-March, and the third term starts in late March or early April and ends in early June. Caltech was ranked within the top ten universities in the world by the Times Higher Education World University Rankings , QS World University Rankings , and Academic Ranking of World Universities . For 2022, U.S. News & World Report ranked Caltech as tied for 9th in
11918-517: The small size of its faculty in many major fields, the institution is exceptionally careful in selecting candidates. This rigorous process can result in some positions remaining unfilled for several years until the right candidate is found. Caltech dedicates significant resources to attract top-tier faculty and provides them with substantial financial support to foster their research and academic endeavors. The voting faculty of Caltech include all professors, instructors, research associates and fellows, and
12036-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then
12154-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space
12272-406: The top 5 among graduates of US colleges and universities. The average net return on investment over a period of 20 years is $ 887,000, the tenth-highest among US colleges. Caltech offers Army and Air Force ROTC in cooperation with the University of Southern California . Quantum mechanics Quantum mechanics is a fundamental theory that describes the behavior of nature at and below
12390-502: The trustees declared it to be the express policy of the institute to pursue scientific research of the greatest importance and at the same time "to continue to conduct thorough courses in engineering and pure science, basing the work of these courses on exceptionally strong instruction in the fundamental sciences of mathematics, physics, and chemistry; broadening and enriching the curriculum by a liberal amount of instruction in such subjects as English, history, and economics; and vitalizing all
12508-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of
12626-537: The university have played an essential role in many modern scientific breakthroughs and innovations, including advances in space research , sustainability science , quantum physics , and seismology . As of October 2022 , there are 79 Nobel laureates who have been affiliated with Caltech , making it the institution with the highest number of Nobelists per capita in America. This includes 46 alumni and faculty members (47 prizes, with chemist Linus Pauling being
12744-515: The university received about $ 750,000. From the 1950s to 1980s, Caltech was the home of Murray Gell-Mann and Richard Feynman , whose work was central to the establishment of the Standard Model of particle physics. Feynman was also widely known outside the physics community as an exceptional teacher and a colorful, unconventional character. During Lee A. DuBridge 's tenure as Caltech's president (1946–1969), Caltech's faculty doubled and
12862-526: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained
12980-577: The war in Korea and the pressure from the Soviet Union, the project was Caltech's way of assisting the federal government in its effort to increase national security. The project was created to study new ways of improving the relationship between tactical air support and ground troops. The Army, Air Force, and Navy sponsored the project; however, it was under contract with the Army. The study was named after
13098-475: The war, some scientists from Caltech, including J. Robert Oppenheimer , Richard Tolman , and Robert Bacher , were instrumental in the Manhattan Project and contributed to critical aspects of the atomic bomb's development. Caltech was also directly involved in other bomb-related research with a group led by Charles Lauritsen which assisted in the development of the high-explosive lenses used in
13216-529: The work of the Institute by the infusion in generous measure of the spirit of research". In 1923, Millikan was awarded the Nobel Prize in Physics . In 1925, the school established a department of geology and hired William Bennett Munro , then chairman of the division of History, Government, and Economics at Harvard University , to create a division of humanities and social sciences at Caltech. In 1928,
13334-476: Was elected to the Association of American Universities , and the antecedents of NASA 's Jet Propulsion Laboratory , which Caltech continues to manage and operate, were established between 1936 and 1943 under Theodore von Kármán . Caltech has six academic divisions with strong emphasis on science and engineering, managing $ 423 million in sponsored research in 2022. Its 124-acre (50 ha) primary campus
13452-428: Was established in 1986 to assemble, preserve, translate, and publish papers selected from the literary estate of Albert Einstein and from other collections. In fall 2008, the freshman class was 42% female, a record for Caltech's undergraduate enrollment. In the same year, the Institute concluded a six-year-long fund-raising campaign. The campaign raised more than $ 1.4 billion from about 16,000 donors. Nearly half of
13570-425: Was known successively as Throop University , Throop Polytechnic Institute (and Manual Training School) and Throop College of Technology before acquiring its current name in 1920. The vocational school was disbanded and the preparatory program was split off to form the independent Polytechnic School in 1907. At a time when scientific research in the United States was still in its infancy, George Ellery Hale ,
13688-461: Was later seen in the film The Italian Job , as character Ed Norton's home.) The horizontal tunnel in which seismometers were installed was a former passageway (with private elevator) between the mansion and the lower tennis courts. By 1974 it had become possible to transmit seismic signals electronically from remote seismometers, and the Seismo Lab was then relocated to the Caltech campus, becoming over time more integrated into, and less distinct from,
13806-580: Was one of 131 colleges and universities nationally that took part in the V-12 Navy College Training Program which offered students a path to a Navy commission. The United States Navy also maintained a naval training school for aeronautical engineering , resident inspectors of ordinance and naval material, and a liaison officer to the National Defense Research Committee on campus. During
13924-416: Was ranked the hardest college in America to gain acceptance to by admit rate, at an all-time low of 2.7%. For the freshmen who enrolled in 2019 (Class of 2023) the middle 50% range of SAT were 740–780 for evidence-based reading and writing and 790–800 for math, and 1530–1570 total. The middle 50% range ACT Composite score was 35–36. The SAT Math Level 2 middle 50% range was 800–800. The middle 50% range for
#291708