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{{Quantum book backlink|Timeline}}

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[[File:10 Quantum Modern Color.jpg|320px|Key figures in contemporary quantum physics and quantum information]]
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Key figures in contemporary quantum physics and quantum information (post-1950), including pioneers of quantum computing, entanglement, and modern experiments.
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'''Modern quantum mechanics''' is the form of quantum theory developed from about 1925 onward, when the older, partly phenomenological quantum rules were replaced by a systematic mathematical theory of states, observables, operators, and probabilities. Its creation was one of the central events in the history of modern physics.

The older quantum theory had introduced quantized energy, photons, atomic spectra, and the Bohr model, but it lacked a complete mathematical foundation. The transition to modern quantum mechanics began with [[Louis de Broglie]]'s proposal that matter has wave-like properties and was completed through the nearly simultaneous development of [[matrix mechanics]] by [[Werner Heisenberg]], [[Max Born]], and [[Pascual Jordan]], and [[wave mechanics]] by [[Erwin Schrödinger]].<ref name="Whittaker">{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900–1926 |date=1989 |publisher=Dover Publications |isbn=978-0-486-26126-3 |edition=Reprint |location=New York}}</ref><ref name="Hanle">{{citation |last=Hanle |first=P. A. |title=Erwin Schrödinger's Reaction to Louis de Broglie's Thesis on the Quantum Theory |journal=Isis |volume=68 |issue=4 |pages=606–609 |date=December 1977 |doi=10.1086/351880 |s2cid=121913205}}</ref>

== Origin of the modern theory ==

In 1924, de Broglie proposed that particles of matter, such as electrons, possess an associated wavelength. This hypothesis connected the quantization of Bohr orbits with wave behavior and suggested that atomic structure could be understood through standing waves rather than classical planetary motion.<ref name="Whittaker" /> De Broglie's idea was soon supported experimentally by electron diffraction, showing that electrons can behave as waves.

In 1925, Heisenberg formulated a new quantum theory based only on observable quantities such as spectral-line frequencies and intensities. Born recognized that Heisenberg's calculations could be expressed using matrices, leading to the formulation known as '''matrix mechanics'''.<ref name="Heisenberg1925">{{cite book |last=Van der Waerden |first=B. L. |title=Sources of Quantum Mechanics |date=1967 |publisher=Dover Publications |location=Mineola, New York |pages=261–276}}</ref>

Almost simultaneously, Schrödinger developed '''wave mechanics''', based on a wave equation for quantum systems. His equation described the allowed stationary states of systems such as the hydrogen atom and gave the correct energy levels. Schrödinger later showed that wave mechanics and matrix mechanics were mathematically equivalent, even though they looked very different.<ref name="SchrBiog">{{cite web |last=Nobel Prize Organization |title=Erwin Schrödinger – Biographical |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1933/schrodinger-bio.html |access-date=28 March 2014}}</ref>

== Matrix mechanics ==

'''Matrix mechanics''' was the first complete formulation of modern quantum mechanics. It replaced classical variables such as position and momentum with mathematical arrays whose order of multiplication mattered. This non-commutativity became one of the essential features of quantum theory.

Heisenberg's approach was motivated by atomic spectra. Instead of trying to picture electron orbits directly, he focused on observable transition frequencies and intensities. Born and Jordan developed the formal matrix structure of the theory, turning Heisenberg's insight into a systematic mechanics of quantum systems.<ref name="Heisenberg1925" />

This approach also led naturally to the uncertainty principle. In 1927, Heisenberg argued that certain pairs of physical quantities, such as position and momentum, cannot both be assigned exact values at the same time. The principle became one of the defining conceptual features of quantum mechanics.<ref>{{Cite journal |last1=Busch |first1=Paul |last2=Lahti |first2=Pekka |last3=Werner |first3=Reinhard F. |date=17 October 2013 |title=Proof of Heisenberg's Error-Disturbance Relation |journal=Physical Review Letters |volume=111 |issue=16 |article-number=160405 |arxiv=1306.1565 |bibcode=2013PhRvL.111p0405B |doi=10.1103/PhysRevLett.111.160405 |pmid=24182239 |s2cid=24507489}}</ref>

== Wave mechanics ==

'''Wave mechanics''' was developed by Schrödinger in 1926. Building on de Broglie's matter-wave hypothesis, Schrödinger introduced a wave equation whose solutions describe the possible quantum states of a system.

The central object in wave mechanics is the [[wave function]], usually denoted by <math>\psi</math>. The wave function does not describe a classical material wave. Instead, through Max Born's interpretation, it provides probability amplitudes: the square of its magnitude gives probabilities for measurement outcomes.

Schrödinger's equation successfully reproduced the hydrogen spectrum and gave a deeper explanation of atomic orbitals. In this picture, electrons do not move in definite planetary orbits around the nucleus; instead, they occupy quantum states represented by probability distributions.

[[File:Atomic orbitals spdf m-eigenstates.png|thumb|400px|right|Atomic orbitals as described by modern quantum mechanics. Electrons are represented by wave functions and probability distributions rather than classical orbits.]]

== Equivalence of the formulations ==

Although matrix mechanics and wave mechanics appeared very different, Schrödinger showed in 1926 that they were mathematically equivalent. Matrix mechanics emphasized observable transitions and algebraic structure, while wave mechanics gave a more visually intuitive representation in terms of waves and differential equations.

The equivalence of these approaches showed that quantum mechanics was not merely a collection of special rules for atoms, but a general physical theory with multiple mathematical representations.

== Dirac and the formal structure ==

[[File:Paul Dirac, 1933.jpg|thumb|250px|right|Paul Dirac helped unify quantum mechanics with special relativity and introduced powerful formal methods.]]

Paul Dirac played a major role in the formal development of quantum mechanics. Around 1927, he began to connect quantum theory with [[special relativity]], eventually producing the [[Dirac equation]] for the electron. The equation predicted electron spin and led to the prediction of the [[positron]].

Dirac also introduced influential mathematical notation and operator methods, including [[bra–ket notation]], which became standard in quantum theory. His 1930 textbook helped establish the abstract formulation of quantum mechanics used in modern physics.

During the same period, [[John von Neumann]] developed a rigorous mathematical foundation for quantum mechanics using linear operators on Hilbert spaces. This abstract framework remains central to the modern formulation of the theory.

== Copenhagen interpretation ==

The rapid development of quantum mechanics raised difficult questions about measurement, probability, and physical reality. Niels Bohr, Werner Heisenberg, and others developed ideas later grouped under the name '''Copenhagen interpretation'''.<ref name="Faye">{{Cite book |last=Faye |first=Jan |title=Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |year=2019 |editor-last=Zalta |editor-first=Edward N. |chapter=Copenhagen Interpretation of Quantum Mechanics |chapter-url=https://plato.stanford.edu/entries/qm-copenhagen/}}</ref>

The Copenhagen view emphasized the probabilistic character of quantum theory, the role of measurement, the uncertainty principle, and Bohr's complementarity principle. It held that experiments may reveal particle-like or wave-like aspects of matter, but not both in the same experimental arrangement.

Although there was never a single, perfectly unified Copenhagen doctrine, the interpretation strongly shaped the way quantum mechanics was taught and discussed throughout the twentieth century.<ref>{{cite journal |first1=K. |last1=Camilleri |first2=M. |last2=Schlosshauer |title=Niels Bohr as Philosopher of Experiment: Does Decoherence Theory Challenge Bohr's Doctrine of Classical Concepts? |arxiv=1502.06547 |journal=Studies in History and Philosophy of Modern Physics |volume=49 |pages=73–83 |year=2015 |doi=10.1016/j.shpsb.2015.01.005 |bibcode=2015SHPMP..49...73C |s2cid=27697360}}</ref>

== Toward quantum field theory ==

Modern quantum mechanics was first developed for particles and atoms, but physicists soon attempted to apply quantum principles to fields. Beginning in the late 1920s, work by Dirac, Jordan, Pauli, and others led to early forms of [[quantum field theory]].

The most successful early quantum field theory was [[quantum electrodynamics]], which describes the interaction of charged particles with the electromagnetic field. It was later reformulated and refined by [[Richard Feynman]], [[Julian Schwinger]], [[Shin'ichirō Tomonaga]], and [[Freeman Dyson]]. Quantum field theory eventually became the language of particle physics and the foundation of the [[Standard Model]].

[[File:Feynman Diagram Gluon Radiation.svg|thumb|300px|right|A Feynman diagram. Diagrammatic methods became central in later quantum field theory.]]

== Significance ==

The development of modern quantum mechanics transformed physics. It explained atomic spectra, chemical bonding, the structure of the periodic table, electron diffraction, spin, and many properties of matter that classical physics could not account for.

It also introduced a new conceptual framework in which physical systems are described by states in an abstract mathematical space, observables are represented by operators, and measurement outcomes are generally probabilistic. This framework remains the basis of quantum physics, quantum chemistry, condensed matter physics, particle physics, and quantum information science.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
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{{Author|Harold Foppele}}

{{Sourceattribution|History of quantum mechanics|1}}

Physics:Quantum mechanics/Timeline/Modern quantum mechanics - Revision history

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