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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Wave function collapse</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">For the constraint-solving algorithm, see <a href="Wave_function_collapse_(algorithm)" class="mw-redirect" title="Wave function collapse (algorithm)">Wave function collapse (algorithm)</a>.</div>
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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }">
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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation>
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</math></span><img src="./1799e4a910c7d26396922a20ef5ceec25ca1871c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }" loading="lazy"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;">
<ul><li><a href="Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li>
<li><a href="Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li>
<li><a href="History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Background</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">
<ul><li><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li>
<li><a href="Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li>
<li><a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul>
<div class="hlist">
<ul><li><a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li>
<li><a href="Wave_interference" title="Wave interference">Interference</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Fundamentals</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">
<ul><li><a href="Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li>
<li><a href="Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li>
<li><a href="Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li>
<li><a href="Energy_level" title="Energy level">Energy level</a></li>
<li><a href="Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li>
<li><a href="Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li>
<li><a href="Quantum_number" title="Quantum number">Quantum number</a></li>
<li><a href="Quantum_state" title="Quantum state">State</a></li>
<li><a href="Quantum_superposition" title="Quantum superposition">Superposition</a></li>
<li><a href="Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li>
<li><a href="Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li>
<li><a href="Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li>
<li><a href="Wave_function" title="Wave function">Wave function</a>
<ul></ul></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Experiments</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">
<ul><li><a href="Bell_test" title="Bell test">Bell's inequality</a></li>
<li><a href="CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li>
<li><a href="Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li>
<li><a href="Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li>
<li><a href="Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li>
<li><a href="Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li>
<li><a href="Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li>
<li><a href="Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li>
<li><a href="Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li>
<li><a href="Popper's_experiment" title="Popper's experiment">Popper</a></li></ul>
</div>
<ul><li><a href="Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a>
<ul><li><a href="Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul>
<div class="hlist">
<ul><li><a href="Schr%C3%B6dinger's_cat" title="Schrödinger's cat">Schrödinger's cat</a></li>
<li><a href="Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li>
<li><a href="Wheeler's_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Formulations</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">
<ul><li><a href="Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul>
<div class="hlist">
<ul><li><a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li>
<li><a href="Interaction_picture" title="Interaction picture">Interaction</a></li>
<li><a href="Matrix_mechanics" title="Matrix mechanics">Matrix</a></li>
<li><a href="Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li>
<li><a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li>
<li><a href="Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Equations</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">
<ul><li><a href="Dirac_equation" title="Dirac equation">Dirac</a></li>
<li><a href="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li>
<li><a href="Pauli_equation" title="Pauli equation">Pauli</a></li>
<li><a href="Rydberg_formula" title="Rydberg formula">Rydberg</a></li>
<li><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">
<ul><li><a href="Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li>
<li><a href="Consciousness_causes_collapse" title="Consciousness causes collapse">Consciousness causes collapse</a></li>
<li><a href="Consistent_histories" title="Consistent histories">Consistent histories</a></li>
<li><a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li>
<li><a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li>
<li><a href="Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li>
<li><a href="Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a></li>
<li><a href="Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li>
<li><a href="Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li>
<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>
<li><a href="Superdeterminism" title="Superdeterminism">Superdeterminism</a></li>
<li><a href="Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li>
<li><a href="Transactional_interpretation" title="Transactional interpretation">Transactional</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Advanced topics</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">
<ul><li><a href="Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li>
<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li>
<li><a href="Quantum_information_science" title="Quantum information science">Quantum information science</a></li>
<li><a href="Quantum_computing" title="Quantum computing">Quantum computing</a></li>
<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>
<li><a href="Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li>
<li><a href="Density_matrix" title="Density matrix">Density matrix</a></li>
<li><a href="Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li>
<li><a href="Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li>
<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">
<ul><li><a href="Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li>
<li><a href="John_Stewart_Bell" title="John Stewart Bell">Bell</a></li>
<li><a href="Hans_Bethe" title="Hans Bethe">Bethe</a></li>
<li><a href="Patrick_Blackett" title="Patrick Blackett">Blackett</a></li>
<li><a href="Felix_Bloch" title="Felix Bloch">Bloch</a></li>
<li><a href="David_Bohm" title="David Bohm">Bohm</a></li>
<li><a href="Niels_Bohr" title="Niels Bohr">Bohr</a></li>
<li><a href="Max_Born" title="Max Born">Born</a></li>
<li><a href="Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li>
<li><a href="Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li>
<li><a href="Arthur_Compton" title="Arthur Compton">Compton</a></li>
<li><a href="Paul_Dirac" title="Paul Dirac">Dirac</a></li>
<li><a href="Clinton_Davisson" title="Clinton Davisson">Davisson</a></li>
<li><a href="Peter_Debye" title="Peter Debye">Debye</a></li>
<li><a href="Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li>
<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>
<li><a href="Hugh_Everett_III" title="Hugh Everett III">Everett</a></li>
<li><a href="Vladimir_Fock" title="Vladimir Fock">Fock</a></li>
<li><a href="Enrico_Fermi" title="Enrico Fermi">Fermi</a></li>
<li><a href="Richard_Feynman" title="Richard Feynman">Feynman</a></li>
<li><a href="Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li>
<li><a href="Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li>
<li><a href="Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li>
<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>
<li><a href="Pascual_Jordan" title="Pascual Jordan">Jordan</a></li>
<li><a href="Hans_Kramers" title="Hans Kramers">Kramers</a></li>
<li><a href="Willis_Lamb" title="Willis Lamb">Lamb</a></li>
<li><a href="Lev_Landau" title="Lev Landau">Landau</a></li>
<li><a href="Max_von_Laue" title="Max von Laue">Laue</a></li>
<li><a href="Henry_Moseley" title="Henry Moseley">Moseley</a></li>
<li><a href="Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li>
<li><a href="Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li>
<li><a href="Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li>
<li><a href="Max_Planck" title="Max Planck">Planck</a></li>
<li><a href="Isidor_Isaac_Rabi" class="mw-redirect" title="Isidor Isaac Rabi">Rabi</a></li>
<li><a href="C._V._Raman" title="C. V. Raman">Raman</a></li>
<li><a href="Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li>
<li><a href="Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li>
<li><a href="Michelle_Simmons" title="Michelle Simmons">Simmons</a></li>
<li><a href="Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li>
<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li>
<li><a href="Hermann_Weyl" title="Hermann Weyl">Weyl</a></li>
<li><a href="Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li>
<li><a href="Eugene_Wigner" title="Eugene Wigner">Wigner</a></li>
<li><a href="Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li>
<li><a href="Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul>
</div></div></div></td>
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<p>In various <a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">interpretations</a> of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <b>wave function collapse</b>, also called <b>reduction of the state vector</b>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> occurs when a <a href="Wave_function" title="Wave function">wave function</a>—initially in a <a href="Quantum_superposition" title="Quantum superposition">superposition</a> of several <a href="Eigenstates" class="mw-redirect" title="Eigenstates">eigenstates</a>—reduces to a single eigenstate due to <a href="Fundamental_interaction" title="Fundamental interaction">interaction</a> with the external world. This interaction is called an <a href="Observation_(physics)" class="mw-redirect" title="Observation (physics)"><i>observation</i></a> and is the essence of a <a href="Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurement in quantum mechanics</a>, which connects the wave function with classical <a href="Observable" title="Observable">observables</a> such as <a href="Position_(vector)" class="mw-redirect" title="Position (vector)">position</a> and <a href="Momentum" title="Momentum">momentum</a>. Collapse is one of the two processes by which <a href="Quantum_system" class="mw-redirect" title="Quantum system">quantum systems</a> evolve in time; the other is the continuous evolution governed by the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>.<sup id="cite_ref-Grundlagen_2-0" class="reference"><a href="#cite_note-Grundlagen-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>In the <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>, wave function collapse connects quantum to classical models, with a special <a href="Copenhagen_interpretation#Role_of_the_observer" title="Copenhagen interpretation">role for the observer</a>. By contrast, <a href="Objective-collapse_theory" title="Objective-collapse theory">objective-collapse</a> proposes an origin in physical processes. In the <a href="Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds interpretation</a>, collapse does not exist; all wave function outcomes occur while <a href="Quantum_decoherence" title="Quantum decoherence">quantum decoherence</a> accounts for the appearance of collapse.
</p><p>Historically, <a href="Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> was the first to use the idea of wave function reduction to explain quantum measurement.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-C._Kiefer-2002_4-0" class="reference"><a href="#cite_note-C._Kiefer-2002-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Mathematical_description">Mathematical description</h2></div>
<div role="note" class="hatnote navigation-not-searchable">For an explanation of the notation used, see <a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a>. For details on this formalism, see <a href="Quantum_state" title="Quantum state">Quantum state</a>.</div>
<p>In quantum mechanics each measurable physical quantity of a quantum system is called an <a href="Observable" title="Observable">observable</a> which, for example, could be the position <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">
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</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> and the momentum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
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<mi>p</mi>
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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> but also energy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
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<mi>E</mi>
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</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
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<mi>z</mi>
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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
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</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> components of spin (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{z}}">
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<annotation encoding="application/x-tex">{\displaystyle s_{z}}</annotation>
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</math></span><img src="./9fe17b6a1d1ce8d16e8753353f2b8b575ae4381d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.092ex; height:2.009ex;" alt="{\displaystyle s_{z}}" loading="lazy"></span>), and so on. The observable acts as a <a href="Linear_mapping" class="mw-redirect" title="Linear mapping">linear function</a> on the states of the system; its eigenvectors correspond to the quantum state (i.e. <a href="Quantum_state#Pure_states_of_wave_functions" title="Quantum state">eigenstate</a>) and the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> to the possible values of the observable. The collection of eigenstates/eigenvalue pairs represent all possible values of the observable. Writing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{i}}">
<semantics>
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<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle \phi _{i}}</annotation>
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</math></span><img src="./0182dbf29b54844c92fd9b0311778a02a38398ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.185ex; height:2.509ex;" alt="{\displaystyle \phi _{i}}" loading="lazy"></span> for an eigenstate and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>c</mi>
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<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation>
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</math></span><img src="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy"></span> for the corresponding observed value, any arbitrary state of the quantum system can be expressed as a vector using <a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>ψ<!-- ψ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .}</annotation>
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</math></span></span>
The kets <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|\phi _{i}\rangle \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \{|\phi _{i}\rangle \}}</annotation>
</semantics>
</math></span><img src="./c1b096484c065039acf661b662d7fe01f3a475e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.062ex; height:2.843ex;" alt="{\displaystyle \{|\phi _{i}\rangle \}}" loading="lazy"></span> specify the different available quantum "alternatives", i.e., particular quantum states.
</p><p>The <a href="Wave_function" title="Wave function">wave function</a> is a specific representation of a quantum state. Wave functions can therefore always be expressed as eigenstates of an observable though the converse is not necessarily true.
</p>
<div class="mw-heading mw-heading3"><h3 id="Collapse">Collapse</h3></div>
<p>To account for the experimental result that repeated measurements of a quantum system give the same results, the theory postulates a "collapse" or "reduction of the state vector" upon observation,<sup id="cite_ref-GriffithsSchroeter3rd_5-0" class="reference"><a href="#cite_note-GriffithsSchroeter3rd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">: <span title="Page / location: 566
Quotation: "to account for the fact that an immediately repeated measurement yields the same result, we are forced to assume that the act of measurement collapses the wave function,"" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">566</span> </sup> abruptly converting an arbitrary state into a single component eigenstate of the observable:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle \mapsto |\psi '\rangle =|\phi _{i}\rangle .}">
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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle \mapsto |\psi '\rangle =|\phi _{i}\rangle .}</annotation>
</semantics>
</math></span><img src="./d8af82f845ca9eb3163a8f42034bd98246631ab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.294ex; height:5.509ex;" alt="{\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle \mapsto |\psi '\rangle =|\phi _{i}\rangle .}" loading="lazy"></span></dd></dl>
<p>where the arrow represents a measurement of the observable corresponding to the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">
<semantics>
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<mi>ϕ<!-- ϕ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
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</math></span><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> basis.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
For any single event, only one eigenvalue is measured, chosen randomly from among the possible values.
</p>
<div class="mw-heading mw-heading3"><h3 id="Meaning_of_the_expansion_coefficients">Meaning of the expansion coefficients</h3></div>
<p>The <a href="Complex_number" title="Complex number">complex</a> coefficients <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{c_{i}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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</msub>
<mo fence="false" stretchy="false">}</mo>
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<annotation encoding="application/x-tex">{\displaystyle \{c_{i}\}}</annotation>
</semantics>
</math></span><img src="./9c181688b279d6a1e343259c49156d91e8345fb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.131ex; height:2.843ex;" alt="{\displaystyle \{c_{i}\}}" loading="lazy"></span> in the expansion of a quantum state in terms of eigenstates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|\phi _{i}\rangle \}}">
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<annotation encoding="application/x-tex">{\displaystyle \{|\phi _{i}\rangle \}}</annotation>
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</math></span><img src="./c1b096484c065039acf661b662d7fe01f3a475e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.062ex; height:2.843ex;" alt="{\displaystyle \{|\phi _{i}\rangle \}}" loading="lazy"></span>,
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .}">
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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .}</annotation>
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can be written as an (complex) overlap of the corresponding eigenstate and the quantum state:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}=\langle \phi _{i}|\psi \rangle .}">
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<mi>c</mi>
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<annotation encoding="application/x-tex">{\displaystyle c_{i}=\langle \phi _{i}|\psi \rangle .}</annotation>
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They are called the <a href="Probability_amplitude" title="Probability amplitude">probability amplitudes</a>. The <a href="Absolute_value#Complex_numbers" title="Absolute value">square modulus</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |c_{i}|^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>i</mi>
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<msup>
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<annotation encoding="application/x-tex">{\displaystyle |c_{i}|^{2}}</annotation>
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</math></span><img src="./7616cf4bbd92dbb939ef4eccd47e711faa4086f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.154ex; height:3.343ex;" alt="{\displaystyle |c_{i}|^{2}}" loading="lazy"></span> is the probability that a measurement of the observable yields the eigenstate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi _{i}\rangle }">
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<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle |\phi _{i}\rangle }</annotation>
</semantics>
</math></span><img src="./afe78aae7ac03e9cfc121eb4917e6a1d15a441d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.737ex; height:2.843ex;" alt="{\displaystyle |\phi _{i}\rangle }" loading="lazy"></span>. The sum of the probability over all possible outcomes must be one:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi |\psi \rangle =\sum _{i}|c_{i}|^{2}=1.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
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<mi>c</mi>
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<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle \langle \psi |\psi \rangle =\sum _{i}|c_{i}|^{2}=1.}</annotation>
</semantics>
</math></span><img src="./74e9b1662bf601c77e66d03559ae974bff52c5c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.385ex; height:5.509ex;" alt="{\displaystyle \langle \psi |\psi \rangle =\sum _{i}|c_{i}|^{2}=1.}" loading="lazy"></span></dd></dl>
<p>As examples, individual counts in a <a href="Double_slit_experiment" class="mw-redirect" title="Double slit experiment">double slit experiment</a> with electrons appear at random locations on the detector; after many counts are summed the distribution shows a wave interference pattern.<sup id="cite_ref-Bach_Pope_Liou_Batelaan_2013_p=033018_8-0" class="reference"><a href="#cite_note-Bach_Pope_Liou_Batelaan_2013_p=033018-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In a <a href="Stern-Gerlach_experiment" class="mw-redirect" title="Stern-Gerlach experiment">Stern-Gerlach experiment</a> with silver atoms, each particle appears in one of two areas unpredictably, but the final conclusion has equal numbers of events in each area.
</p><p>This statistical aspect of quantum measurements differs fundamentally from <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>. In quantum mechanics the only information we have about a system is its wave function and measurements of its wave function can only give statistical information.<sup id="cite_ref-GriffithsSchroeter3rd_5-1" class="reference"><a href="#cite_note-GriffithsSchroeter3rd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 17">: 17 </span></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div>
<p>The two terms "reduction of the state vector" (or "state reduction" for short) and "wave function collapse" are used to describe the same concept. A <a href="Quantum_state" title="Quantum state">quantum state</a> is a mathematical description of a quantum system; a <a href="Quantum_state_vector" class="mw-redirect" title="Quantum state vector">quantum state vector</a> uses Hilbert space vectors for the description.<sup id="cite_ref-messiah_9-0" class="reference"><a href="#cite_note-messiah-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 159">: 159 </span></sup> Reduction of the state vector replaces the full state vector with a single eigenstate of the observable.
</p><p>The term "wave function" is typically used for a different mathematical representation of the quantum state, one that uses spatial coordinates also called the "position representation".<sup id="cite_ref-messiah_9-1" class="reference"><a href="#cite_note-messiah-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 324">: 324 </span></sup> When the wave function representation is used, the "reduction" is called "wave function collapse".
</p>
<div class="mw-heading mw-heading2"><h2 id="The_measurement_problem">The measurement problem</h2></div>
<p>The Schrödinger equation describes quantum systems but does not describe their measurement. Solution to the equations include all possible observable values for measurements, but measurements only result in one definite outcome. This difference is called the <a href="Measurement_problem" title="Measurement problem">measurement problem</a> of quantum mechanics. To predict measurement outcomes from quantum solutions, the orthodox interpretation of quantum theory postulates wave function collapse and uses the <a href="Born_rule" title="Born rule">Born rule</a> to compute the probable outcomes.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Despite the widespread quantitative success of these postulates scientists remain dissatisfied and have sought more detailed physical models. Rather than suspending the Schrödinger equation during the process of measurement, the measurement apparatus should be included and governed by the laws of quantum mechanics.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 127">: 127 </span></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Physical_approaches_to_collapse">Physical approaches to collapse</h2></div>
<p>Quantum theory offers no dynamical description of the "collapse" of the wave function. Viewed as a statistical theory, no description is expected. As Fuchs and Peres put it, "collapse is something that happens in our description of the system, not to the system itself".<sup id="cite_ref-FuchsPeresNo_12-0" class="reference"><a href="#cite_note-FuchsPeresNo-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>
</p><p>Various <a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">interpretations of quantum mechanics</a> attempt to provide a physical model for collapse.<sup id="cite_ref-Stamatescu_13-0" class="reference"><a href="#cite_note-Stamatescu-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 816">: 816 </span></sup> Three treatments of collapse can be found among the common interpretations. The first group includes hidden-variable theories like <a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm theory</a>; here random outcomes only result from unknown values of hidden variables. Results from <a href="Bell_test" title="Bell test">tests</a> of <a href="Bell's_theorem" title="Bell's theorem">Bell's theorem</a> shows that these variables would need to be non-local. The second group models measurement as quantum entanglement between the quantum state and the measurement apparatus. This results in a simulation of classical statistics called quantum decoherence. This group includes the <a href="Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds interpretation</a> and <a href="Consistent_histories" title="Consistent histories">consistent histories</a> models. The third group postulates additional, but as yet undetected, physical basis for the randomness; this group includes for example the <a href="Objective-collapse_interpretation" class="mw-redirect" title="Objective-collapse interpretation">objective-collapse interpretations</a>. While models in all groups have contributed to better understanding of quantum theory, no alternative explanation for individual events has emerged as more useful than collapse followed by statistical prediction with the Born rule.<sup id="cite_ref-Stamatescu_13-1" class="reference"><a href="#cite_note-Stamatescu-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 819">: 819 </span></sup>
</p><p>The significance ascribed to the wave function varies from interpretation to interpretation and even within an interpretation (such as the <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>). If the wave function merely encodes an observer's knowledge of the universe, then the wave function collapse corresponds to the receipt of new information. This is somewhat analogous to the situation in classical physics, except that the classical "wave function" does not necessarily obey a wave equation. If the wave function is physically real, in some sense and to some extent, then the collapse of the wave function is also seen as a real process, to the same extent.
</p>
<div class="mw-heading mw-heading3"><h3 id="Quantum_decoherence">Quantum decoherence</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Quantum_decoherence" title="Quantum decoherence">Quantum decoherence</a></div>
<p>Quantum decoherence explains why a system interacting with an environment transitions from being a <a href="Quantum_state#Pure_states_as_rays_in_a_complex_Hilbert_space" title="Quantum state">pure state</a>, exhibiting superpositions, to a <a href="Quantum_state#Mixed_states" title="Quantum state">mixed state</a>, an incoherent combination of classical alternatives.<sup id="cite_ref-Stanford1_14-0" class="reference"><a href="#cite_note-Stanford1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible in the same sense as in the <a href="Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>: the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the <a href="Classical_limit" title="Classical limit">classical limit</a> of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.<sup id="cite_ref-Schlosshauer_15-0" class="reference"><a href="#cite_note-Schlosshauer-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stanford1_14-1" class="reference"><a href="#cite_note-Stanford1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>
</p><p>The form of decoherence known as <a href="Einselection" title="Einselection">environment-induced superselection</a> proposes that when a quantum system interacts with the environment, the superpositions <i>apparently</i> reduce to mixtures of classical alternatives. The combined wave function of the system and environment continue to obey the Schrödinger equation throughout this <i>apparent</i> collapse.<sup id="cite_ref-Zurek_17-0" class="reference"><a href="#cite_note-Zurek-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> More importantly, this is not enough to explain <i>actual</i> wave function collapse, as decoherence does not reduce it to a single eigenstate.<sup id="cite_ref-Schlosshauer_15-1" class="reference"><a href="#cite_note-Schlosshauer-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stanford1_14-2" class="reference"><a href="#cite_note-Stanford1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>
<p>The concept of wavefunction collapse was introduced by <a href="Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> in his 1927 paper on the <a href="Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", and incorporated into the <a href="Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">mathematical formulation of quantum mechanics</a> by <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a>, in his 1932 treatise <i>Mathematische Grundlagen der Quantenmechanik</i>.<sup id="cite_ref-C._Kiefer-2002_4-1" class="reference"><a href="#cite_note-C._Kiefer-2002-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Heisenberg did not try to specify exactly what the collapse of the wavefunction meant. However, he emphasized that it should not be understood as a physical process.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Niels Bohr never mentions wave function collapse in his published work, but he repeatedly cautioned that we must give up a "pictorial representation". Despite the differences between Bohr and Heisenberg, their views are often grouped together as the "Copenhagen interpretation", of which wave function collapse is regarded as a key feature.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>
</p><p><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a>'s influential 1932 work <i><a href="Mathematical_Foundations_of_Quantum_Mechanics" title="Mathematical Foundations of Quantum Mechanics">Mathematical Foundations of Quantum Mechanics</a></i> took a more formal approach, developing an "ideal" measurement scheme<sup id="cite_ref-HartleQMCosmology_20-0" class="reference"><a href="#cite_note-HartleQMCosmology-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SchlosshauerReview_21-0" class="reference"><a href="#cite_note-SchlosshauerReview-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">: <span title="Page / location: 1270
Quotation: "Note that von Neumann’s scheme is in sharp contrast to the Copenhagen interpretation, where measurement is not treated as a system-apparatus interaction described by the usual quantum-mechanical formalism, but instead as an independent component of the theory, to be represented entirely in fundamentally classical terms."" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">1270</span> </sup> that postulated that there were two processes of wave function change:
</p>
<ol><li>The <a href="Probability" title="Probability">probabilistic</a>, non-<a href="Unitary_transformation" title="Unitary transformation">unitary</a>, <a href="Local_realism" class="mw-redirect" title="Local realism">non-local</a>, discontinuous change brought about by observation and <a href="Quantum_measurement" class="mw-redirect" title="Quantum measurement">measurement</a> (state reduction or collapse).</li>
<li>The <a href="Deterministic" class="mw-redirect" title="Deterministic">deterministic</a>, unitary, continuous <a href="Time_evolution" title="Time evolution">time evolution</a> of an isolated system that obeys the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>.</li></ol>
<p>In 1957 <a href="Hugh_Everett_III" title="Hugh Everett III">Hugh Everett III</a> proposed a model of quantum mechanics that dropped von Neumann's first postulate. Everett observed that the measurement apparatus was also a quantum system and its quantum interaction with the system under observation should determine the results. He proposed that the discontinuous change is instead a splitting of a wave function representing the universe.<sup id="cite_ref-SchlosshauerReview_21-1" class="reference"><a href="#cite_note-SchlosshauerReview-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1288">: 1288 </span></sup> While Everett's approach rekindled interest in foundational quantum mechanics, it left core issues unresolved. Two key issues relate to origin of the observed classical results: what causes quantum systems to appear classical and to resolve with the observed probabilities of the <a href="Born_rule" title="Born rule">Born rule</a>.<sup id="cite_ref-SchlosshauerReview_21-2" class="reference"><a href="#cite_note-SchlosshauerReview-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1290">: 1290 </span></sup><sup id="cite_ref-HartleQMCosmology_20-1" class="reference"><a href="#cite_note-HartleQMCosmology-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 5">: 5 </span></sup>
</p><p>Beginning in 1970 <a href="H._Dieter_Zeh" title="H. Dieter Zeh">H. Dieter Zeh</a> sought a detailed quantum decoherence model for the discontinuous change without postulating collapse. Further work by <a href="Wojciech_H._Zurek" title="Wojciech H. Zurek">Wojciech H. Zurek</a> in 1980 lead eventually to a large number of papers on many aspects of the concept.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Decoherence assumes that every quantum system interacts quantum mechanically with its environment and such interaction is not separable from the system, a concept called an "open system".<sup id="cite_ref-SchlosshauerReview_21-3" class="reference"><a href="#cite_note-SchlosshauerReview-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1273">: 1273 </span></sup> Decoherence has been shown to work very quickly and within a minimal environment, but as yet it has not succeeded in a providing a detailed model replacing the collapse postulate of orthodox quantum mechanics.<sup id="cite_ref-SchlosshauerReview_21-4" class="reference"><a href="#cite_note-SchlosshauerReview-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1302">: 1302 </span></sup>
</p><p>By explicitly dealing with the interaction of object and measuring instrument, von Neumann<sup id="cite_ref-Grundlagen_2-1" class="reference"><a href="#cite_note-Grundlagen-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> described a quantum mechanical measurement scheme consistent with wave function collapse. However, he did not prove the <i>necessity</i> of such a collapse. Von Neumann's projection postulate was conceived based on experimental evidence available during the 1930s, in particular <a href="Compton_scattering" title="Compton scattering">Compton scattering</a>. Later work refined the notion of measurements into the more easily discussed <i>first kind</i>, that will give the same value when immediately repeated, and the <i>second kind</i> that give different values when repeated.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
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<ul><li><a href="Arrow_of_time" title="Arrow of time">Arrow of time</a></li>
<li><a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations of quantum mechanics</a></li>
<li><a href="Quantum_decoherence" title="Quantum decoherence">Quantum decoherence</a></li>
<li><a href="Quantum_interference" class="mw-redirect" title="Quantum interference">Quantum interference</a></li>
<li><a href="Quantum_Zeno_effect" title="Quantum Zeno effect">Quantum Zeno effect</a></li>
<li><a href="Schr%C3%B6dinger's_cat" title="Schrödinger's cat">Schrödinger's cat</a></li>
<li><a href="Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a></li>
<li><a href="Wave_function_collapse_(algorithm)" class="mw-redirect" title="Wave function collapse (algorithm)">Wave function collapse (algorithm)</a></li></ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-Stanford1-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stanford1_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stanford1_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Stanford1_14-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFFine2020" class="citation encyclopaedia cs1">Fine, Arthur (2020). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/qm-decoherence/">"The Role of Decoherence in Quantum Mechanics"</a>. <i>Stanford Encyclopedia of Philosophy</i>. Center for the Study of Language and Information, Stanford University website<span class="reference-accessdate">. Retrieved <span class="nowrap">11 April</span> 2021</span>.</cite></span>
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<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFWojciech_H._Zurek2003" class="citation journal cs1">Wojciech H. Zurek (2003). "Decoherence, einselection, and the quantum origins of the classical". <i>Reviews of Modern Physics</i>. <b>75</b> (3): 715. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0105127">quant-ph/0105127</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003RvMP...75..715Z">2003RvMP...75..715Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.75.715">10.1103/RevModPhys.75.715</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14759237">14759237</a>.</cite></span>
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<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFG._Jaeger2017" class="citation journal cs1">G. Jaeger (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe19100513">""Wave-Packet Reduction" and the Quantum Character of the Actualization of Potentia"</a>. <i>Entropy</i>. <b>19</b> (10): 13. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017Entrp..19..513J">2017Entrp..19..513J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe19100513">10.3390/e19100513</a></span>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2144%2F41814">2144/41814</a></span>.</cite></span>
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<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFHenrik_Zinkernagel2016" class="citation journal cs1">Henrik Zinkernagel (2016). "Niels Bohr on the wave function and the classical/quantum divide". <i>Studies in History and Philosophy of Modern Physics</i>. <b>53</b>: <span class="nowrap">9–</span>19. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1603.00353">1603.00353</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016SHPMP..53....9Z">2016SHPMP..53....9Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.shpsb.2015.11.001">10.1016/j.shpsb.2015.11.001</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18890207">18890207</a>. <q>Among Bohr scholars it is common to assert that Bohr never mentions the wave function collapse (see e.g. Howard, 2004 and Faye, 2008). It is true that in Bohr's published writings, he does not discuss the status or existence of this standard component in the popular image of the Copenhagen interpretation.</q></cite></span>
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<li id="cite_note-HartleQMCosmology-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-HartleQMCosmology_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HartleQMCosmology_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Hartle, James B. <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/1805.12246.pdf">"The quantum mechanics of cosmology."</a> Notes from the lectures by the author at the 7th Jerusalem Winter School 1990 on Quantum Cosmology and Baby Universes. arXiv:1805.12246 (2018).</span>
</li>
<li id="cite_note-SchlosshauerReview-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-SchlosshauerReview_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SchlosshauerReview_21-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-SchlosshauerReview_21-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-SchlosshauerReview_21-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-SchlosshauerReview_21-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFSchlosshauer2005" class="citation journal cs1">Schlosshauer, Maximilian (2005-02-23). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.76.1267">"Decoherence, the measurement problem, and interpretations of quantum mechanics"</a>. <i>Reviews of Modern Physics</i>. <b>76</b> (4): <span class="nowrap">1267–</span>1305. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0312059">quant-ph/0312059</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004RvMP...76.1267S">2004RvMP...76.1267S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.76.1267">10.1103/RevModPhys.76.1267</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</a>.</cite></span>
</li>
<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFCamilleri2009" class="citation journal cs1">Camilleri, Kristian (2009-12-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S1355219809000562">"A history of entanglement: Decoherence and the interpretation problem"</a></span>. <i>Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics</i>. On The History Of The Quantum. <b>40</b> (4): <span class="nowrap">290–</span>302. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SHPMP..40..290C">2009SHPMP..40..290C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.shpsb.2009.09.003">10.1016/j.shpsb.2009.09.003</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1355-2198">1355-2198</a>.</cite></span>
</li>
<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">
<cite id="CITEREFW._Pauli1958" class="citation book cs1 cs1-prop-foreign-lang-source">W. Pauli (1958). "Die allgemeinen Prinzipien der Wellenmechanik". In S. Flügge (ed.). <i>Handbuch der Physik</i> (in German). Vol. V. Berlin: Springer-Verlag. p. 73.</cite></span>
</li>
<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">
<cite id="CITEREFL._LandauR._Peierls1931" class="citation journal cs1 cs1-prop-foreign-lang-source">L. Landau & R. Peierls (1931). "Erweiterung des Unbestimmtheitsprinzips für die relativistische Quantentheorie". <i><a href="Zeitschrift_f%C3%BCr_Physik" title="Zeitschrift für Physik">Zeitschrift für Physik</a></i> (in German). <b>69</b> (<span class="nowrap">1–</span>2): <span class="nowrap">56–</span>69. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1931ZPhy...69...56L">1931ZPhy...69...56L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01391513">10.1007/BF01391513</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123160388">123160388</a>.</cite>)</span>
</li>
<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Discussions of measurements of the second kind can be found in most treatments on the foundations of quantum mechanics, for instance, <cite id="CITEREFJ._M._Jauch1968" class="citation book cs1">J. M. Jauch (1968). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/foundationsofqua0000jauc"><i>Foundations of Quantum Mechanics</i></a></span>. Addison-Wesley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/foundationsofqua0000jauc/page/165">165</a>.</cite>; <cite id="CITEREFB._d'Espagnat1976" class="citation book cs1">B. d'Espagnat (1976). <i>Conceptual Foundations of Quantum Mechanics</i>. W. A. Benjamin. pp. 18, 159.</cite>; and <cite id="CITEREFW._M._de_Muynck2002" class="citation book cs1">W. M. de Muynck (2002). <i>Foundations of Quantum Mechanics: An Empiricist Approach</i>. Kluwer Academic Publishers. section 3.2.4.</cite></span>
</li>
</ol></div>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><span class="noviewer" typeof="mw:File"></span> Quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/Wave_function_collapse" class="extiw external" title="wikiquote:Special:Search/Wave function collapse">Wave function collapse</a> at Wikiquote</li></ul>
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</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics328" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Quantum_mechanics328" style="font-size:114%;margin:0 4em"><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li>
<li><a href="History_of_quantum_mechanics" title="History of quantum mechanics">History</a>
<ul><li><a href="Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li>
<li><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li>
<li><a href="Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li>
<li><a href="Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Born_rule" title="Born rule">Born rule</a></li>
<li><a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li>
<li><a href="Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li>
<li><a href="Density_matrix" title="Density matrix">Density matrix</a></li>
<li><a href="Energy_level" title="Energy level">Energy level</a>
<ul><li><a href="Ground_state" title="Ground state">Ground state</a></li>
<li><a href="Excited_state" title="Excited state">Excited state</a></li>
<li><a href="Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li>
<li><a href="Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li>
<li><a href="Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li>
<li><a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li>
<li><a href="Wave_interference" title="Wave interference">Interference</a></li>
<li><a href="Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li>
<li><a href="Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li>
<li><a href="Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li>
<li><a href="Quantum_state" title="Quantum state">Quantum state</a></li>
<li><a href="Quantum_superposition" title="Quantum superposition">Superposition</a></li>
<li><a href="Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li>
<li><a href="Scattering#Theory" title="Scattering">Scattering theory</a></li>
<li><a href="Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li>
<li><a href="Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li>
<li><a href="Wave_function" title="Wave function">Wave function</a>
<ul>
<li><a href="Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li>
<li><a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li>
<li><a href="Interaction_picture" title="Interaction picture">Interaction</a></li>
<li><a href="Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li>
<li><a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li>
<li><a href="Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li>
<li><a href="Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li>
<li><a href="Dirac_equation" title="Dirac equation">Dirac</a></li>
<li><a href="Weyl_equation" title="Weyl equation">Weyl</a></li>
<li><a href="Majorana_equation" title="Majorana equation">Majorana</a></li>
<li><a href="Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li>
<li><a href="Pauli_equation" title="Pauli equation">Pauli</a></li>
<li><a href="Rydberg_formula" title="Rydberg formula">Rydberg</a></li>
<li><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li>
<li><a href="Consciousness_causes_collapse" title="Consciousness causes collapse">Consciousness causes collapse</a></li>
<li><a href="Consistent_histories" title="Consistent histories">Consistent histories</a></li>
<li><a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li>
<li><a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li>
<li><a href="Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li>
<li><a href="Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a>
<ul><li><a href="Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a>
<ul><li><a href="Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li>
<li><a href="Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li>
<li><a href="Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li>
<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>
<li><a href="Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li>
<li><a href="Transactional_interpretation" title="Transactional interpretation">Transactional</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bell_test" title="Bell test">Bell test</a></li>
<li><a href="Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li>
<li><a href="Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li>
<li><a href="Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li>
<li><a href="Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li>
<li><a href="Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li>
<li><a href="Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li>
<li><a href="Popper's_experiment" title="Popper's experiment">Popper</a></li>
<li><a href="Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li>
<li><a href="Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li>
<li><a href="Wheeler's_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Nanotechnology" title="Nanotechnology">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Quantum_biology" title="Quantum biology">Quantum biology</a></li>
<li><a href="Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li>
<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>
<li><a href="Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li>
<li><a href="Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li>
<li><a href="Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li>
<li><a href="Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li>
<li><a href="Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li>
<li><a href="Quantum_mind" title="Quantum mind">Quantum mind</a></li>
<li><a href="Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li>
<li><a href="Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_engineering" title="Quantum engineering">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li>
<li><a href="Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li>
<li><a href="Quantum_bus" title="Quantum bus">Quantum bus</a></li>
<li><a href="Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a>
<ul><li><a href="Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li>
<li><a href="Quantum_channel" title="Quantum channel">Quantum channel</a></li>
<li><a href="Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li>
<li><a href="Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li>
<li><a href="Quantum_computing" title="Quantum computing">Quantum computing</a>
<ul><li><a href="Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li>
<li><a href="Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li>
<li><a href="Quantum_optics#Quantum_electronics" title="Quantum optics">Quantum electronics</a></li>
<li><a href="Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li>
<li><a href="Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li>
<li><a href="Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li>
<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>
<li><a href="Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li>
<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>
<li><a href="Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li>
<li><a href="Quantum_machine" title="Quantum machine">Quantum machine</a></li>
<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li>
<li><a href="Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li>
<li><a href="Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li>
<li><a href="Quantum_network" title="Quantum network">Quantum network</a></li>
<li><a href="Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li>
<li><a href="Quantum_optics" title="Quantum optics">Quantum optics</a></li>
<li><a href="Quantum_programming" title="Quantum programming">Quantum programming</a></li>
<li><a href="Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li>
<li><a href="Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li>
<li><a href="Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li>
<li><a href="Casimir_effect" title="Casimir effect">Casimir effect</a></li>
<li><a href="Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li>
<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a>
<ul><li><a href="History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li>
<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li>
<li><a href="Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Schr%C3%B6dinger's_cat" title="Schrödinger's cat">Schrödinger's cat</a>
<ul><li><a href="Schr%C3%B6dinger's_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li>
<li><a href="Wigner's_friend" title="Wigner's friend">Wigner's friend</a></li>
<li><a href="Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li>
<li><a href="Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul>
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