One hundred years ago, Wolfgang Pauli, Werner Heisenberg, Erwin Schrödinger, and others laid down cornerstones of quantum mechanics. UNESCO has designated 2025 the International Year of Quantum Science and Technology in recognition of those breakthroughs and of the revolutionary impact that quantum science and technology have had and promise to have. To celebrate this centennial, the Editors of the Physical Review journals have collected the papers below that established the foundations for the field.

The collection opens with an Editorial by Dagmar Bruß of Heinrich Heine University Düsseldorf, offering an introduction to this exciting history and to the blossoming applications that, as UNESCO puts it, “are driving some of the most exciting breakthroughs of the 21st century.”

—The Physical Review Editors

1900  1910  1920  1930  1940  1950  1960  1970  1980  1990  2000  2010

Celebrating the first century of quantum physics and preparing for the next one

Max Planck and Albert Einstein (Hebrew University of Jerusalem)

Max Planck introduced his namesake constant, which relates quanta of energy to frequency. At the time, he attributed no physical significance to this quantization, considering it a mathematical trick for explaining blackbody radiation. 

M. Planck, Ueber das Gesetz der Energieverteilung im Normalspectrum, Ann. Phys. (Berlin) 309, 553 (1901).

Albert Einstein showed that Planck’s 1901 mathematical “trick” of assuming that light comes in energy packets, or particles (now known as photons), elegantly explains the photoelectric effect, whereby light ejects electrons from metal surfaces.

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt, Ann. Phys. (Berlin) 322, 132 (1905).

designua/stock.adobe.com

An early success for quantum physics was Niels Bohr’s model of the atom as a small solar system with discrete orbits for electrons. The model successfully reproduced the absorption spectrum of hydrogen, but it had trouble with more complex atoms.

N. Bohr, On the constitution of atoms and molecules, Philos. Mag. Ser. 5 26, 1 (1913).

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By observing the deflection of atoms in a spatially varying magnetic field, Walther Gerlach and Otto Stern showed experimentally that the spin of an atom is quantized and exists in two discrete states.

W. Gerlach and O. Stern, Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld, Z. Phys. 9, 349 (1922).

Arthur Compton discovered the phenomenon now known as the Compton effect, by which a high-energy photon transfers some of its momentum to a charged particle. The result confirmed that light can act both as a wave and as a stream of particles. 

A. Compton, A quantum theory of the scattering of x-rays by light elements, Phys. Rev. 21, 483 (1923).

Wolfgang Pauli posited that two electrons in an atom cannot simultaneously inhabit the same quantum state. For the case of electrons in atoms, the principle implies that they can never have the same values of their four quantum numbers, including the spin quantum number. Pauli’s principle explained how Bohr’s model of the atom can be connected to the structure of the periodic table and was later extended to all fermions.

W. Pauli, Über den Einfluß der Geschwindigkeitsabhängigkeit der Elektronenmasse auf den Zeemaneffekt, Z. Phys. 31, 373 (1925).

After his famous trip to Helgoland, Werner Heisenberg derived the quantized energy levels of a one-dimensional oscillator in terms of physical quantities, such as transition probabilities, that could in principle be measured. Three fundamental papers by Heisenberg, Max Born and Pascual Jordan developed the first consistent formulation of quantum mechanics, based on matrices.

W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik 33, 879 (1925); M. Born and P. Jordan, Zur Quantenmechanik, Z. Phys. 34, 858 (1925); M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik. II., Z. Phys. 35, 557 (1925).

In his doctoral thesis, Louis de Broglie theorized that any moving particle had an associated wave. His work cemented the concept of wave-particle duality in quantum mechanics.

L. De Broglie, Recherches sur la théorie des Quanta, Ann. Phys. (Berlin) 10, 22 (1925).

George Uhlenbeck and Samuel Goudsmit theorized that electrons possess an intrinsic angular momentum, now known as spin. Their idea was introduced to explain the splitting of spectral lines in atoms. 

G. E. Uhlenbeck and S. Goudsmit, Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons, Zuschriften Und Vorläufige Mitteilungen 13, 953 (1925).

Erwin Schrödinger derived the wave equation that is central to modern quantum mechanics. It was soon proven that his formulation was equivalent to Heisenberg's matrix mechanics formulation. Physicists’ familiarity with wave mechanics allowed Schrödinger’s approach to rapidly gain popularity. 

E. Schrödinger, Quantisierung als Eigenwertproblem, Ann. Phys. (Berlin) 384, 361 (1926).

Three years after Louis de Broglie proposed that matter, like light, has a dual nature. Clinton Davisson and Lester Germer provided an experimental confirmation of his prediction, showing that electrons scatter from a crystal the same way x rays do.

C. Davisson and L. H. Germer, Diffraction of electrons by a crystal of nickel, Phys. Rev. 30, 705 (1927).

Max Born provided a probabilistic interpretation of Schrödinger's wave function—stating that the squared amplitude of the wave function gives the probability that a particle will be observed in a particular position or that a measurement will yield a particular result. The Born rule became a postulate of quantum mechanics.

M. Born, Das Adiabatenprinzip in der Quantenmechanik, Z. Phys. 40, 167 (1927). 

By combining quantum mechanics and relativity, Paul Dirac developed a relativistic equation for the electron. The theory explained the emergence of the electron spin and implied the existence of an antielectron, or positron.

P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. A 117, 610 (1928).

Howard P. Robertson extended the uncertainty principle beyond the position-momentum relationship. His general formula established uncertainty relations between observables corresponding to non-commuting operators.

H. P. Robertson, The uncertainty principle, Phys. Rev. 34, 163 (1929).

Felix Bloch proved a theorem that allowed the description of the behavior of electrons in the periodic structure of crystalline solids. His theory offered the basis for the concept of electronic band structures.

F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52, 555 (1929).

Niels Bohr and Albert Einstein (P. Ehrenfest, courtesy AIP Emilio Segrè Visual Archives, Gamow Collection)

Albert Einstein, Boris Podolsky, and Nathan Rosen introduced the concept of entanglement (a term coined later that year by Schrödinger) and discussed its implications. Using a thought experiment involving two distant entangled particles, the trio concluded that measurement of either the position or the momentum of one particle can reveal the value of position or momentum of the other. The fact that the second particle seemingly possesses predetermined values for both position and momentum would contradict the uncertainty principle, which led them to conclude that quantum mechanics cannot be a complete description of physical reality. They suggested that a complete, classical-like theory might be constructed, possibly by including inaccessible, or “hidden,” variables. 

A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered

complete?, Phys. Rev. 47, 777 (1935).

Niels Bohr responded to the EPR paper through an article with the same title and an opposing answer. He contended that the EPR paradox was based on an erroneous argument: since one cannot physically perform a simultaneous measurement of position and momentum—two complementary variables—there is no way to verify experimentally that they coexist as definite properties of any given particle. The two papers were part of a long-running debate about the nature of quantum mechanics that was ultimately resolved in favor of Bohr by Bell’s theorem and its experimental validations.

N. Bohr, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 48, 696 (1935).

In a review of the field of quantum mechanics, Erwin Schrödinger introduced his famous cat-in-a-box thought experiment to illustrate paradoxes deriving from quantum superpositions and their relation to the macroscopic world. The concept of a Schrödinger's cat became a familiar tool for foundational discussions and has been part of popular culture since the 1970s. 

E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften 23, 807 (1935).

APS/Alan Stonebraker

Richard Feynman presented the path integral formulation of quantum mechanics, in which a particle's evolution is calculated by summing all quantum mechanically possible trajectories. His path-integral formulation has proven crucial to many theoretical developments in fields ranging from quantum field theory to cosmology.

R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20, 367 (1948).

 Illustration of the Aharonov-Bohm effect (Physics Today 62, 38 (2009)/AIP)

Julian Schwinger laid one of the foundation stones of quantum field theory by developing a rigorous theoretical framework for treating the interaction of electron-positron pairs with an electromagnetic field. His namesake effect involves the production of particle-antiparticle pairs in the presence of an electric field, which is associated with a nonlinear polarization of the vacuum.

J. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951). 

Building on a hidden-variable theory first proposed by Louis de DeBroglie, David Bohm developed a deterministic interpretation of quantum mechanics. In Bohmian mechanics, also known as pilot-wave theory, an ensemble of particles is described by a wave that evolves according to the Schrödinger equation, with each particle following a deterministic trajectory.

David Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables. I, Phys. Rev. 85, 166 (1952).

Hugh Everett proposed his “relative-state” formulation of quantum mechanics to solve the quantum measurement problem. He asserted that the wave function of the Universe is a real entity that evolves according to Schrödinger's equation, and that decoherence explains what we perceive as a wavefunction collapse in a measurement. His approach was popularized as the “many-worlds interpretation” based on the underlying idea that each observation splits the Universe into multiple copies of itself, branching into countless parallel worlds.  

H. Everett, Relative state formulation of quantum mechanics, Rev. Mod. Phys. 29, 454 (1957).

Through a thought experiment, Yakir Aharonov and David Bohm showed that an electromagnetic potential can affect the phase of quantum particles moving in that potential even when the field vanishes along the particles’ trajectories.

Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115, 485 (1959).

John Stewart Bell in 1982 (CERN)

Turning to the unresolved debate raised by the EPR paradox, John Stewart Bell shifted the discussion to the realm of experimental physics. He showed that a local, hidden-variable theory implies mathematical constraints, now known as Bell inequalities, on measurements performed on two separated particles of an entangled pair. Quantum physics, on the other hand, would predict measurement correlations that violate such inequalities. First observed in 1982, violations of Bell inequalities have been observed in increasingly sophisticated experiments removing several “loopholes” that could still allow the possibility of a local, hidden-variable theory.

J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).

Complementing the work of Bell, Simon Kochen and Ernst Specker established stronger constraints on hidden-variable theories that reproduce quantum-mechanical predictions. Specifically, they concluded that such theories must feature not only nonlocality, but also contextuality—the property by which measurement outcomes depend on the measurement context (the choice of measured observables).

S. Kochen and E. Specker, The problem of hidden variables in quantum mechanics, J. Math. Mech. 17, 59 (1967).

Artistic rendition of quantum entanglement (V. de Schwanberg/sciencesource.com)

Using photons from excited calcium atoms, Stuart Freedman and John Clauser provided the first experimental observation of a violation of a Bell inequality, demonstrating the quantum entanglement of two widely separated particles. 

S. J. Freedman and J. F. John F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972).

John Wheeler proposed a thought experiment in which the decision to measure wavelike or particlelike features of a quantum system, such as a photon, is made after the particle has passed through parts of an experimental apparatus. The surprising implication is that, if the measurement decision determines whether the photon behaves as a wave or a particle, it retroactively affects the photon after its journey has been completed. This conclusion rules out the idea that particle-propagation models could explain quantum phenomena.

J. A. Wheeler, The “past” and the “delayed-choice” double-slit experiment, Mathematical Foundations of Quantum Theory, edited by A. R. Marlow (Academic Press, New York, 1978), pp. 9–48, 10.1016/B978-0-12-473250-6.X5001-8.

Artistic rendition of quantum key cryptography (Yao Zheng/Micius Salon)

Using the example of a polarized photon, William Wooters and Wojciech Zurek proved that the linearity of quantum mechanics forbids the replication of an arbitrary unknown quantum state. This “no-cloning theorem” has profound implications for quantum computing and cryptography.

W. K. Wootters and Wojciech H. Zurek, A single quantum cannot be cloned, Nature (London) 299, 802 (1982).

Alain Aspect, Jean Dalibard, and Gérard Roger launched pairs of entangled photons and rapidly switched the settings of the measurement device while the photons were in flight. The setup’s shuffling eliminated the possibility that observed correlations were the result of classical effects, closing the “nonlocality loophole” (the possibility that hidden signals exchanged between two measurement sites could produce correlations violating Bell inequalities).

A. Aspect, J. Dalibard, and C. Roger, Experimental test of Bell’s inequalities using time-varying analyzers, Phys. Rev. Lett. 49, 1804 (1982).

Extending concepts from optics, Michael Berry showed that a quantum system evolving through an “adiabatic” cycle accumulates a geometric phase that depends on the path, but not the speed, of the evolution. The Berry phase plays an important role in quantum phenomena such as the Aharonov-Bohm effect.

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A 392, 45 (1984).

Charles Bennett and Gilles Brassard proposed the first quantum cryptography protocol, now known as BB84. This quantum-key-distribution protocol allows two parties to share a random string of bits (the encryption key) in a way that is inherently secure from eavesdroppers. 

C. H. Bennett and Gilles Brassard, Quantum cryptography: Public key distribution and coin tossing, Theoretical Computer Science, Vol. 560 (Part 1), pp. 7—11.

David Deutsch showed the possibility of a universal quantum computer that’s able to simulate any physical system, pinpointing tasks on which such a machine could vastly outperform a classical computer.

D. Deutsch, Quantum theory, the Church–Turing principle and the universal quantum computer, Proc. R. Soc. A 400, 97 (1985).

Rendering of a quantum computer (BPawesome/stock.adobe.com)

Daniel Greenberger, Michael Horne, Abner Shimony, and Anton Zeilinger demonstrated that entangled states of three or more particles can produce measurement results that are incompatible with any classical-like theory. These “GHZ” states (which Greenberger, Horne, and Zeilinger previously introduced for states of four or more particles) have been widely used in quantum metrology, cryptography, and computing. 

D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Bell’s theorem without inequalities, Am. J. Phys. 58, 1131 (1990).

Leonard Mandel and collaborators demonstrated that the indistinguishability of two particles is a requirement for the particles to produce quantum interference effects. The researchers generated two independent pairs of entangled photons and observed interference patterns only if the interfering photons’ paths were arranged such that they could not be distinguished. 

X. Y. Zou, L. J. Wang, and L. Mandel, Induced coherence and indistinguishability in optical interference, Phys. Rev. Lett. 67, 318 (1991).

Lucien Hardy discussed another thought experiment in which a particle may interact with its antiparticle without annihilating—a result that no classical-like theory could explain. Hardy’s idea was experimentally confirmed in 2009. 

L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states, Phys. Rev. Lett. 71, 1665 (1993).

Sandu Popescu and Daniel Rohrlich explored the implications of assuming that nonlocality is an axiom of quantum mechanics, rather than a theorem. They conjectured that quantum theory could be derived from the combination of two axioms: relativistic causality and nonlocality. Their work indicated the possibility of quantum correlations stronger than any previously identified ones.

S. Popescu and D. Rohrlich, Quantum nonlocality as an axiom, Found. Phys. 24, 379 (1994).

In 1994, Peter Shor developed a quantum algorithm that can determine the prime factors of a large number with exponentially fewer operations than those needed in classical algorithms—an approach that could allow a quantum computer to break standard cryptography schemes. In 1997, Lov Grover provided a quantum algorithm that offers a speedup for the task of searching unsorted databases. 

P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, Proceedings 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM (1994), pp. 124–134; L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79, 325 (1997).

A team led by Alain Aspect and Jean-François Roch experimentally realized Wheeler’s delayed-choice thought experiment, feeding single photons into a 48-m-long interferometer.

V. Jacques et al., Experimental realization of Wheeler’s delayed-choice gedanken experiment, Science 315, 966 (2007).

Sketch of an apparatus for performing a Bell test (APS/Alan Stonebraker)

Since Bell demonstrated his inequalities, experimentalists have tested them with increasingly sophisticated tests. These tests, however, typically had at least one “loophole” that could allow a local, realistic interpretation of the experiments. In 2015, three independent teams, led by Ronald Hanson, Anton Zeilinger, and Lynden Shalm, were able to carry out Bell-test experiments that closed simultaneously the two main loopholes of such tests, known as “detection” and “locality” loopholes. 

B. Hensen et al., Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature (London) 526, 682 (2015); M. Giustina et al., Significant-loophole-free test of Bell’s theorem with entangled photons, Phys. Rev. Lett. 115, 250401 (2015); Lynden K. Shalm et al., Strong loophole-free test of local realism, Phys. Rev. Lett. 115, 250402 (2015).

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