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<title>First quantization</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">First quantization</span></span>

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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>

<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><li><a href="Wave_function_collapse" title="Wave function collapse">Collapse</a></li></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>

<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>

<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="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="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="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="Lev_Landau" title="Lev Landau">Landau</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="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="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>

</tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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<p><b>First quantization</b> is a procedure for converting equations of classical particle equations into quantum wave equations. The companion concept of <a href="Second_quantization" title="Second quantization">second quantization</a> converts classical field equations in to quantum field equations.<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>

</p><p>However, this need not be the case. In particular, a fully quantum version of the theory can be created by interpreting the interacting fields and their associated potentials as operators of multiplication, provided the potential is written in the <a href="Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a> that are compatible with the <a href="Euclidean_space" title="Euclidean space">Euclidean</a> coordinates of standard <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> First quantization is appropriate for studying a single quantum-mechanical system (not to be confused with a single particle system, since a single quantum wave function describes the state of a single quantum system, which may have arbitrarily many complicated constituent parts, and whose evolution is given by just one uncoupled <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>) being controlled by <a href="Laboratory" title="Laboratory">laboratory</a> apparatuses that are governed by <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, for example an old fashioned voltmeter (one devoid of modern semiconductor devices, which rely on quantum theory—however though this is sufficient, it is not necessary), a simple thermometer, a magnetic field generator, and so on.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>Published in 1901, <a href="Max_Planck" title="Max Planck">Max Planck</a> deduced the existence and value of the constant now bearing his name from considering only <a href="Wien's_displacement_law" title="Wien's displacement law">Wien's displacement law</a>, <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, and <a href="Electromagnetic_theory" class="mw-redirect" title="Electromagnetic theory">electromagnetic theory</a>.<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> Four years later in 1905, <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a> went further to elucidate this constant and its deep connection to the stopping potential of electrons emitted in the <a href="Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The energy in the photoelectric effect depended not only on the number of incident photons (the intensity of light) but also the frequency of light, a novel phenomenon at the time. (This work would earn Einstein the 1921 <a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>) It can then be concluded that this was a key onset of quantization, that is the discretization of matter into fundamental constituents.

</p><p>About eight years later <a href="Niels_Bohr" title="Niels Bohr">Niels Bohr</a> in 1913, published his famous three part series where, essentially by fiat, he posits the quantization of the angular momentum in hydrogen and hydrogen like metals.<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><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><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Where in effect, the orbital angular 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 L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> of the (valence) electron, takes the form <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 L=l\hbar }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=l\hbar }</annotation>

</semantics>

</math></span><img src="./1b33fb2a9d803a939f102a5abb29055342693897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.681ex; height:2.176ex;" alt="{\displaystyle L=l\hbar }" loading="lazy"></span>, where <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 l}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

</semantics>

</math></span><img src="./829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span>, referred to as a quantum number, is presumed a whole number <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 0,\,1,\,2,\,3,\,\ldots \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0,\,1,\,2,\,3,\,\ldots \,}</annotation>

</semantics>

</math></span><img src="./075fe6d42907d1cc60cf9f16bed3abad474eb530.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.444ex; height:2.509ex;" alt="{\displaystyle 0,\,1,\,2,\,3,\,\ldots \,}" loading="lazy"></span>. In the original presentation, the orbital angular momentum of the electron was named <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 M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>, the <a href="Planck_constant" title="Planck constant">Planck constant</a> divided by two pi as <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 M_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{0}}</annotation>

</semantics>

</math></span><img src="./909a62f3ff41169372143733d3767afe0ad3b14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{0}}" loading="lazy"></span>, and the quantum number or "counting of number of passes between stationary points", as stated by Bohr originally as, <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 \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math></span><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span>. See references above for more detail.

</p><p>While it would be later shown that this assumption is not entirely correct, it in fact ends up being rather close to the correct expression for the orbital angular momentum operator's (eigenvalue) quantum number for large values of the quantum number <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 l}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

</semantics>

</math></span><img src="./829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span>, and indeed this was part of Bohr's own assumption. Regard the consequence of Bohr's assumption <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 L^{2}=l^{2}\hbar ^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}=l^{2}\hbar ^{2}}</annotation>

</semantics>

</math></span><img src="./e54cc4d6def03b2963389b8d39bbf3a269fa41ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.859ex; height:2.676ex;" alt="{\displaystyle L^{2}=l^{2}\hbar ^{2}}" loading="lazy"></span>, and compare it with the correct version known today as <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 L^{2}=l(l+1)\hbar ^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}=l(l+1)\hbar ^{2}}</annotation>

</semantics>

</math></span><img src="./1b02d03b0e612672eaee910358adf0dc6a9b32b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.31ex; height:3.176ex;" alt="{\displaystyle L^{2}=l(l+1)\hbar ^{2}}" loading="lazy"></span>. Clearly for large <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 l}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

</semantics>

</math></span><img src="./829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span>, there is little difference, just as well as for <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 l=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l=0}</annotation>

</semantics>

</math></span><img src="./66485a3e3da13d226eb36a131bf1fc7e16403a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=0}" loading="lazy"></span>, the equivalence is exact. Without going into further historical detail, it suffices to stop here and regard this era of the history of quantization to be the "<a href="Old_quantum_theory" title="Old quantum theory">old quantum theory</a>", meaning a period in the history of physics where the corpuscular nature of subatomic particles began to play an increasingly important role in understanding the results of physical experiments, whose mandatory conclusion was the discretization of key physical observable quantities. However, unlike the era below described as the era of <b>first quantization</b>, this era was based solely on purely classical arguments such as <a href="Wien's_displacement_law" title="Wien's displacement law">Wien's displacement law</a>, <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a>, <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, and the <a href="Electromagnetic_theory" class="mw-redirect" title="Electromagnetic theory">electromagnetic theory</a>. In fact, the observation of the <a href="Balmer_series" title="Balmer series">Balmer series</a> of hydrogen in the history of spectroscopy dates as far back as 1885.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p><p>Nonetheless, the watershed events that would come to denote the era of <b>first quantization</b> took place in the vital years spanning 1925–1928. Simultaneously the authors <a href="Max_Born" title="Max Born">Max Born</a> and <a href="Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a> in December 1925,<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> together with <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a> also in December 1925,<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> then <a href="Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> in January 1926,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> following that, <a href="Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> together with Born and Jordan in August 1926,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and finally Dirac in 1928.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The results of these publications were three theoretical formalisms, two of which proved to be equivalent; that of Born, Heisenberg and Jordan was equivalent to that of Schrödinger, while Dirac's 1928 theory came to be regarded as the relativistic version of the prior two. Lastly, it is worth mentioning the publication of Heisenberg and Pauli in 1929,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> which can be regarded as the first attempt at "<a href="Second_quantization" title="Second quantization">second quantization</a>", a term used verbatim by Pauli in a 1943 publication of the <a href="American_Physical_Society" title="American Physical Society">American Physical Society</a>.<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>

</p><p>For purposes of clarification and understanding of the terminology as it evolved over history, it suffices to end with the major publication that helped recognize the equivalence of the <a href="Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> of Born, Heisenberg, and Jordan 1925–1926 with the wave equation of Schrödinger in 1926. The collected and expanded works of <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> showed that the two theories were mathematically equivalent,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and it is this realization that is today understood as <b>first quantization</b>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Qualitative_mathematical_preliminaries">Qualitative mathematical preliminaries</h2></div>

<p>To understand the term <b>first quantization</b> one must first understand what it means for something to be quantum in the first place. The classical theory of Newton is a second order <a href="Nonlinear" class="mw-redirect" title="Nonlinear">nonlinear</a> <a href="Differential_equation" title="Differential equation">differential equation</a> that gives the deterministic trajectory of a system of <a href="Mass" title="Mass">mass</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 m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span>. The <a href="Acceleration" title="Acceleration">acceleration</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 a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>, in <a href="Newton's_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> of motion, <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 F=ma}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=ma}</annotation>

</semantics>

</math></span><img src="./1ca4e42b7d6d66f52294364928cb5f7c590f514c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.109ex; height:2.176ex;" alt="{\displaystyle F=ma}" loading="lazy"></span>, is the second derivative of the system's position as a function of time. Therefore, it is natural to seek solutions of the Newton equation that are at least second order <a href="Differentiable" class="mw-redirect" title="Differentiable">differentiable</a>.

</p><p><a href="Quantum_mechanics" title="Quantum mechanics">Quantum theory</a> differs dramatically in that it replaces physical observables such as the position of the system, the time at which that observation is made, the mass, and the velocity of the system at the instant of observation with the notion of operator observables. Operators as observables change the notion of what is measurable and brings to the table the unavoidable conclusion of the Max Born probability theory. In this framework of nondeterminism, the probability of finding the system in a particular observable state is given by a dynamic probability density that is defined as the <a href="Absolute_value" title="Absolute value">absolute value</a> squared of the solution to the <a href="Schrodinger_equation" class="mw-redirect" title="Schrodinger equation">Schrödinger equation</a>. The fact that probability densities are integrable and normalizable to unity imply that the solutions to the Schrödinger equation must be square integrable. The vector space of infinite sequences, whose square summed up is a convergent series, is known as <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 \ell ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math></span><img src="./91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy"></span> (pronounced "little ell two"). It is in one-to-one correspondence with the infinite dimensional vector space of square-integrable functions, <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 L^{2}(\mathbb {R} ^{d})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} ^{d})}</annotation>

</semantics>

</math></span><img src="./326ea18b5cd9f7d2bb32599cdc7319e3d0066490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.216ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{d})}" loading="lazy"></span>, from the <a href="Euclidean_space" title="Euclidean space">Euclidean space</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 \mathbb {R} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation>

</semantics>

</math></span><img src="./a713426956296f1668fce772df3c60b9dde8a685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}" loading="lazy"></span> to the <a href="Complex_plane" title="Complex plane">complex plane</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 \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span>. For this reason, <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 \ell ^{2}}">

<msup>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation>

</semantics>

</math></span><img src="./91f1f909abd70bd3d8fff0f7ae1ac23052387e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" loading="lazy"></span> 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 L^{2}(\mathbb {R} ^{d})}">

<msup>

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</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} ^{d})}</annotation>

</semantics>

</math></span><img src="./326ea18b5cd9f7d2bb32599cdc7319e3d0066490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.216ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{d})}" loading="lazy"></span> are often referred to indiscriminately as "the" Hilbert space. This is rather misleading because <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 \mathbb {R} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation>

</semantics>

</math></span><img src="./a713426956296f1668fce772df3c60b9dde8a685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}" loading="lazy"></span> is also a Hilbert space when equipped and <a href="Complete_metric_space" title="Complete metric space">completed</a> under the Euclidean <a href="Inner_product_space" title="Inner product space">inner product</a>, albeit a finite dimensional space.

</p>

<div class="mw-heading mw-heading2"><h2 id="Types_of_systems">Types of systems</h2></div>

<p>Both the Newton theory and the Schrödinger theory have a mass parameter in them and they can thus describe the evolution of a collection of masses or a single constituent system with a single total mass, as well as an idealized single particle with idealized single mass system. Below are examples of different types of systems.

</p>

<div class="mw-heading mw-heading3"><h3 id="One-particle_systems">One-particle systems</h3></div>

<p>In general, the one-particle state could be described by a complete set of quantum numbers denoted by <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 \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math></span><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"></span>. For example, the three <a href="Quantum_numbers" class="mw-redirect" title="Quantum numbers">quantum numbers</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 n,l,m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n,l,m}</annotation>

</semantics>

</math></span><img src="./19c0d3f19eb315d24ed7cf1cb29e46699d5f472e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.196ex; height:2.509ex;" alt="{\displaystyle n,l,m}" loading="lazy"></span> associated to an electron in a <a href="Coulomb's_law" title="Coulomb's law">coulomb potential</a>, like the <a href="Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>, form a complete set (ignoring spin). Hence, the state is called <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 |\nu \rangle }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\nu \rangle }</annotation>

</semantics>

</math></span><img src="./2b3c62380dab0fda4781459fed96895501fc5f57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.784ex; height:2.843ex;" alt="{\displaystyle |\nu \rangle }" loading="lazy"></span> and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using <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 _{\nu }(\mathbf {r} )=\langle \mathbf {r} |\nu \rangle }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }(\mathbf {r} )=\langle \mathbf {r} |\nu \rangle }</annotation>

</semantics>

</math></span><img src="./b0917a7d81179b56faaae9e488d09b8887cfd09e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.417ex; height:2.843ex;" alt="{\displaystyle \psi _{\nu }(\mathbf {r} )=\langle \mathbf {r} |\nu \rangle }" loading="lazy"></span>. All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state <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="{\textstyle |\psi \rangle =\sum _{\nu }|\nu \rangle \langle \nu |\psi \rangle }">

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle |\psi \rangle =\sum _{\nu }|\nu \rangle \langle \nu |\psi \rangle }</annotation>

</semantics>

</math></span><img src="./6e001859594e28172cdd1cc366b437976f4692fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.093ex; height:3.009ex;" alt="{\textstyle |\psi \rangle =\sum _{\nu }|\nu \rangle \langle \nu |\psi \rangle }" loading="lazy"></span> obtaining the completeness relation:

</p>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu }|\nu \rangle \langle \nu |=\mathbf {I} }</annotation>

</semantics>

</math></span><img src="./10ed91e9d3b1f39b8f37736354d1619d24b4d0ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.422ex; height:5.509ex;" alt="{\displaystyle \sum _{\nu }|\nu \rangle \langle \nu |=\mathbf {I} }" loading="lazy"></span></dd></dl>

<p>Many have felt that all the properties of the particle could be known using this vector basis, which is expressed here using the Dirac <a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a>. However this need not be true.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Many-particle_systems">Many-particle systems</h3></div>

<p>When turning to <i>N</i>-particle systems, i.e., systems containing <i>N</i> <a href="Identical_particles" class="mw-redirect" title="Identical particles">identical particles</a> i.e. particles characterized by the same physical parameters such as <a href="Mass" title="Mass">mass</a>, <a href="Electric_charge" title="Electric charge">charge</a> and <a href="Spin_(physics)" title="Spin (physics)">spin</a>, an extension of the single-particle state function <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 (\mathbf {r} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} )}</annotation>

</semantics>

</math></span><img src="./a5942742e26cc0a33549c3cb11c79e14f76fcfa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.425ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {r} )}" loading="lazy"></span> to the <i>N</i>-particle state function <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 (\mathbf {r} _{1},\mathbf {r} _{2},...,\mathbf {r} _{N})}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} _{1},\mathbf {r} _{2},...,\mathbf {r} _{N})}</annotation>

</semantics>

</math></span><img src="./fb93ecdd3d616493a3b4a943cde2771c1addea8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.632ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {r} _{1},\mathbf {r} _{2},...,\mathbf {r} _{N})}" loading="lazy"></span> is necessary.<sup id="cite_ref-Merzbacher_22-0" class="reference"><a href="#cite_note-Merzbacher-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> A fundamental difference between classical and quantum mechanics concerns the concept of <a href="Identical_particles" class="mw-redirect" title="Identical particles">indistinguishability</a> of identical particles. Only two species of particles are thus possible in quantum physics, the so-called <a href="Bosons" class="mw-redirect" title="Bosons">bosons</a> and <a href="Fermions" class="mw-redirect" title="Fermions">fermions</a> which obey the rules:

</p>

<msub>

</mrow>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} _{1},...,\mathbf {r} _{j},...,\mathbf {r} _{k},...,\mathbf {r_{N}} )=+\psi (\mathbf {r} _{1},...,\mathbf {r} _{k},...,\mathbf {r} _{j},...,\mathbf {r} _{N})}</annotation>

</semantics>

</math></span><img src="./155e84c73b394782a20eb072d0c62b7e3700d8f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:60.894ex; height:3.009ex;" alt="{\displaystyle \psi (\mathbf {r} _{1},...,\mathbf {r} _{j},...,\mathbf {r} _{k},...,\mathbf {r_{N}} )=+\psi (\mathbf {r} _{1},...,\mathbf {r} _{k},...,\mathbf {r} _{j},...,\mathbf {r} _{N})}" loading="lazy"></span> (bosons),</dd></dl>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} _{1},...,\mathbf {r} _{j},...,\mathbf {r} _{k},...,\mathbf {r_{N}} )=-\psi (\mathbf {r} _{1},...,\mathbf {r} _{k},...,\mathbf {r} _{j},...,\mathbf {r} _{N})}</annotation>

</semantics>

</math></span><img src="./64e6f7fda332c7850d584a78c6c4ec362b5b8151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:60.894ex; height:3.009ex;" alt="{\displaystyle \psi (\mathbf {r} _{1},...,\mathbf {r} _{j},...,\mathbf {r} _{k},...,\mathbf {r_{N}} )=-\psi (\mathbf {r} _{1},...,\mathbf {r} _{k},...,\mathbf {r} _{j},...,\mathbf {r} _{N})}" loading="lazy"></span> (fermions).</dd></dl>

<p>Where we have interchanged two coordinates <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 (\mathbf {r} _{j},\mathbf {r} _{k})}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {r} _{j},\mathbf {r} _{k})}</annotation>

</semantics>

</math></span><img src="./3955adb82b04e0479ac372060293a75a04cbba08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.046ex; height:3.009ex;" alt="{\displaystyle (\mathbf {r} _{j},\mathbf {r} _{k})}" loading="lazy"></span> of the state function. The usual wave function is obtained using the <a href="Slater_determinant" title="Slater determinant">Slater determinant</a> and the identical particles theory. Using this basis, it is possible to solve any many-particle problem that can be clearly and accurately described by a single wave function single system-wide diagonalizable state. From this perspective, first quantization is not a truly multi-particle theory but the notion of "system" need not consist of a single particle either.

</p>

<ul><li><a href="Canonical_quantization" title="Canonical quantization">Canonical quantization</a></li>

<li><a href="Geometric_quantization" title="Geometric quantization">Geometric quantization</a></li>

<li><a href="Quantization_(physics)" title="Quantization (physics)">Quantization</a></li>

<li><a href="Second_quantization" title="Second quantization">Second quantization</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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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">This statement is not unique since it can be argued that the mathematically imprecise notation of Dirac, even still today, can elucidate the equivalence.</span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Just as well, the "testing ground" of hydrogen can also be seen as strong evidence for a conclusion of equivalence.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFDuckSudarshan1998" class="citation book cs1">Duck, Ian; Sudarshan, E C G (1998). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/worldscibooks/10.1142/3457"><i>Pauli and the Spin-Statistics Theorem</i></a>. WORLD SCIENTIFIC. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F3457">10.1142/3457</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-02-3114-9</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFDirac1950" class="citation journal cs1">Dirac, P. A. M. (1950). <a rel="nofollow" class="external text" href="https://doi.org/10.4153%2Fcjm-1950-012-1">"Generalized Hamiltonian Dynamics"</a>. <i>Canadian Journal of Mathematics</i>. <b>2</b>: <span class="nowrap">129–</span>148. <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/10.4153%2Fcjm-1950-012-1">10.4153/cjm-1950-012-1</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0008-414X">0008-414X</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:119748805">119748805</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFPlanck1901" class="citation journal cs1">Planck, Max (1901). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19013090310">"Ueber das Gesetz der Energieverteilung im Normalspectrum"</a>. <i>Annalen der Physik</i>. <b>309</b> (3): <span class="nowrap">553–</span>563. <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/1901AnP...309..553P">1901AnP...309..553P</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/10.1002%2Fandp.19013090310">10.1002/andp.19013090310</a></span>.</cite></span>

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