Schrödinger–Newton equation

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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Nonlinear modification of the Schrödinger equation</div> <p>The <b>Schrödinger–Newton equation</b>, sometimes referred to as the <b>Newton–Schrödinger</b> or <b>Schrödinger–Poisson equation</b>, is a nonlinear modification of the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> with a <a href="/wiki/Newtonian_gravity" class="mw-redirect" title="Newtonian gravity">Newtonian</a> gravitational potential, where the gravitational potential emerges from the treatment of the <a href="/wiki/Wave_function" title="Wave function">wave function</a> as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics.<sup id="cite_ref-vanMeter2011_1-0" class="reference"><a href="#cite_note-vanMeter2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson equation</a>. In the latter case it is also referred to in the plural form. </p><p>The Schrödinger–Newton equation was first considered by Ruffini and Bonazzola<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> in connection with self-gravitating <a href="/wiki/Boson_star" class="mw-redirect" title="Boson star">boson stars</a>. In this context of classical <a href="/wiki/General_relativity" title="General relativity">general relativity</a> it appears as the non-relativistic limit of either the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a> or the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> in a curved space-time together with the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</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> The equation also describes <a href="/wiki/Fuzzy_cold_dark_matter" title="Fuzzy cold dark matter">fuzzy dark matter</a> and approximates classical <a href="/wiki/Cold_dark_matter" title="Cold dark matter">cold dark matter</a> described by the <a href="/wiki/Vlasov_equation#The_Vlasov–Poisson_equation" title="Vlasov equation">Vlasov–Poisson equation</a> in the limit that the particle mass is large.<sup id="cite_ref-MoczLancaster2018_4-0" class="reference"><a href="#cite_note-MoczLancaster2018-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Later on it was proposed as a model to explain the <a href="/wiki/Wave_function_collapse" title="Wave function collapse">quantum wave function collapse</a> by <a href="/w/index.php?title=Lajos_Di%C3%B3si&action=edit&redlink=1" class="new" title="Lajos Diósi (page does not exist)">Lajos Diósi</a><sup id="cite_ref-Diosi1984_5-0" class="reference"><a href="#cite_note-Diosi1984-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a>,<sup id="cite_ref-Penrose1996_6-0" class="reference"><a href="#cite_note-Penrose1996-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Penrose1998_7-0" class="reference"><a href="#cite_note-Penrose1998-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Penrose2014_8-0" class="reference"><a href="#cite_note-Penrose2014-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> from whom the name "Schrödinger–Newton equation" originates. In this context, matter has quantum properties, while gravity remains classical even at the fundamental level. The Schrödinger–Newton equation was therefore also suggested as a way to test the necessity of <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a>.<sup id="cite_ref-Carlip2008_9-0" class="reference"><a href="#cite_note-Carlip2008-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>In a third context, the Schrödinger–Newton equation appears as a <a href="/w/index.php?title=Hartree_approximation&action=edit&redlink=1" class="new" title="Hartree approximation (page does not exist)">Hartree approximation</a> for the mutual gravitational interaction in a system of a large number of particles. In this context, a corresponding equation for the electromagnetic <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb</a> interaction was suggested by Philippe Choquard at the 1976 Symposium on Coulomb Systems in Lausanne to describe one-component plasmas. <a href="/wiki/Elliott_H._Lieb" title="Elliott H. Lieb">Elliott H. Lieb</a> provided the proof for the existence and uniqueness of a stationary ground state and referred to the equation as the <b>Choquard equation</b>.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a coupled system, the Schrödinger–Newton equations are the usual Schrödinger equation with a self-interaction <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a> <span class="mwe-math-element"><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 i\hbar \ {\frac {\partial \Psi }{\ \partial t\ }}=-{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\Psi \;+\;V\ \Psi \;+\;m\ \Phi \ \Psi \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mtext> </mtext> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Ψ</mi> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> <mi>V</mi> <mtext> </mtext> <mi mathvariant="normal">Ψ</mi> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> <mi>m</mi> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> <mtext> </mtext> <mi mathvariant="normal">Ψ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar \ {\frac {\partial \Psi }{\ \partial t\ }}=-{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\Psi \;+\;V\ \Psi \;+\;m\ \Phi \ \Psi \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52387db82e9a54475df32287b5c7919db4dfc665" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.264ex; height:5.843ex;" alt="{\displaystyle i\hbar \ {\frac {\partial \Psi }{\ \partial t\ }}=-{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\Psi \;+\;V\ \Psi \;+\;m\ \Phi \ \Psi \ ,}"></span> where <span class="texhtml mvar" style="font-style:italic;"> V </span> is an ordinary potential, and the gravitational potential <span class="mwe-math-element"><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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Phi \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a26804aef302555ab796e77dade502f3796a6d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.486ex; height:2.509ex;" alt="{\displaystyle \ \Phi \ ,}"></span> representing the interaction of the particle with its own gravitational field, satisfies the Poisson equation <span class="mwe-math-element"><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 \ \nabla ^{2}\Phi =4\pi \ G\ m\ |\Psi |^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \nabla ^{2}\Phi =4\pi \ G\ m\ |\Psi |^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3229aeac17ba5f8fff5171a8532922c5d6c9507" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.834ex; height:3.343ex;" alt="{\displaystyle \ \nabla ^{2}\Phi =4\pi \ G\ m\ |\Psi |^{2}~.}"></span> Because of the back coupling of the wave-function into the potential, it is a <a href="/wiki/Nonlinear_system" title="Nonlinear system">nonlinear system</a>. </p><p>Replacing <span class="mwe-math-element"><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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Phi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cceb6296aeb0b11e9529b1efa8aab6e651147a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.839ex; height:2.176ex;" alt="{\displaystyle \ \Phi \ }"></span> with the solution to the Poisson equation produces the integro-differential form of the Schrödinger–Newton equation: <span class="mwe-math-element"><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 i\hbar \ {\frac {\ \partial \Psi \ }{\partial t}}=\left[\ -{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\;+\;V\;-\;G\ m^{2}\int {\frac {\ |\Psi (t,\mathbf {y} )|^{2}}{\ |\mathbf {x} -\mathbf {y} |\ }}\;\mathrm {d} ^{3}\mathbf {y} \ \right]\Psi ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ψ</mi> <mtext> </mtext> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> <mi>V</mi> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> <mi>G</mi> <mtext> </mtext> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar \ {\frac {\ \partial \Psi \ }{\partial t}}=\left[\ -{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\;+\;V\;-\;G\ m^{2}\int {\frac {\ |\Psi (t,\mathbf {y} )|^{2}}{\ |\mathbf {x} -\mathbf {y} |\ }}\;\mathrm {d} ^{3}\mathbf {y} \ \right]\Psi ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd66d8b8dcefaa4421747280ee434a9f91329645" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.986ex; height:7.509ex;" alt="{\displaystyle i\hbar \ {\frac {\ \partial \Psi \ }{\partial t}}=\left[\ -{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\;+\;V\;-\;G\ m^{2}\int {\frac {\ |\Psi (t,\mathbf {y} )|^{2}}{\ |\mathbf {x} -\mathbf {y} |\ }}\;\mathrm {d} ^{3}\mathbf {y} \ \right]\Psi ~.}"></span> It is obtained from the above system of equations by integration of the Poisson equation under the assumption that the potential must vanish at infinity. </p><p>Mathematically, the Schrödinger–Newton equation is a special case of the <a href="/wiki/Hartree_equation" title="Hartree equation">Hartree equation</a> for <span class="texhtml"><i>n</i> = 2</span> . The equation retains most of the properties of the linear Schrödinger equation. In particular, it is invariant under constant phase shifts, leading to conservation of probability and exhibits full <a href="/wiki/Galilei_invariance" class="mw-redirect" title="Galilei invariance">Galilei invariance</a>. In addition to these symmetries, a simultaneous transformation <span class="mwe-math-element"><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 m\to \mu \ m\ ,\qquad t\to \mu ^{-5}t\ ,\qquad \mathbf {x} \to \mu ^{-3}\mathbf {x} \ ,\qquad \psi (t,\mathbf {x} )\to \mu ^{9/2}\psi (\mu ^{5}t,\mu ^{3}\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">→</mo> <mi>μ</mi> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mi>t</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">→</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>t</mi> <mo>,</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\to \mu \ m\ ,\qquad t\to \mu ^{-5}t\ ,\qquad \mathbf {x} \to \mu ^{-3}\mathbf {x} \ ,\qquad \psi (t,\mathbf {x} )\to \mu ^{9/2}\psi (\mu ^{5}t,\mu ^{3}\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65f1954104e43921fd3bcb3f3eb7214e24ee75e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.495ex; height:3.343ex;" alt="{\displaystyle m\to \mu \ m\ ,\qquad t\to \mu ^{-5}t\ ,\qquad \mathbf {x} \to \mu ^{-3}\mathbf {x} \ ,\qquad \psi (t,\mathbf {x} )\to \mu ^{9/2}\psi (\mu ^{5}t,\mu ^{3}\mathbf {x} )}"></span> maps solutions of the Schrödinger–Newton equation to solutions.<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 id="cite_ref-Giulini2011_12-0" class="reference"><a href="#cite_note-Giulini2011-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The stationary equation, which can be obtained in the usual manner via a separation of variables, possesses an infinite family of normalisable solutions of which only the stationary ground state is stable.<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><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><sup id="cite_ref-Harrison2003_15-0" class="reference"><a href="#cite_note-Harrison2003-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_semi-classical_and_quantum_gravity">Relation to semi-classical and quantum gravity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=2" title="Edit section: Relation to semi-classical and quantum gravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Schrödinger–Newton equation can be derived under the assumption that gravity remains classical, even at the fundamental level, and that the right way to couple quantum matter to gravity is by means of the <a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">semiclassical Einstein equations</a>. In this case, a Newtonian gravitational potential term is added to the Schrödinger equation, where the source of this gravitational potential is the expectation value of the mass density operator or mass flux-current.<sup id="cite_ref-jones1995_16-0" class="reference"><a href="#cite_note-jones1995-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In this regard, <i>if</i> gravity is fundamentally classical, the Schrödinger–Newton equation is a fundamental one-particle equation, which can be generalised to the case of many particles (see below). </p><p>If, on the other hand, the gravitational field is quantised, the fundamental Schrödinger equation remains linear. The Schrödinger–Newton equation is then only valid as an approximation for the gravitational interaction in systems of a large number of particles and has no effect on the centre of mass.<sup id="cite_ref-Bahrami2014_17-0" class="reference"><a href="#cite_note-Bahrami2014-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Many-body_equation_and_centre-of-mass_motion">Many-body equation and centre-of-mass motion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=3" title="Edit section: Many-body equation and centre-of-mass motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the Schrödinger–Newton equation is considered as a fundamental equation, there is a corresponding <i>N</i>-body equation that was already given by Diósi<sup id="cite_ref-Diosi1984_5-1" class="reference"><a href="#cite_note-Diosi1984-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and can be derived from semiclassical gravity in the same way as the one-particle equation: <span class="mwe-math-element"><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 {\begin{aligned}i\hbar {\frac {\partial }{\partial t}}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N})={\bigg (}&-\sum _{j=1}^{N}{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+\sum _{j\neq k}V_{jk}{\big (}|\mathbf {x} _{j}-\mathbf {x} _{k}|{\big )}\\&-G\sum _{j,k=1}^{N}m_{j}m_{k}\int \mathrm {d} ^{3}\mathbf {y} _{1}\cdots \mathrm {d} ^{3}\mathbf {y} _{N}\,{\frac {|\Psi (t,\mathbf {y} _{1},\dots ,\mathbf {y} _{N})|^{2}}{|\mathbf {x} _{j}-\mathbf {y} _{k}|}}{\bigg )}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≠</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>G</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∫</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}i\hbar {\frac {\partial }{\partial t}}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N})={\bigg (}&-\sum _{j=1}^{N}{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+\sum _{j\neq k}V_{jk}{\big (}|\mathbf {x} _{j}-\mathbf {x} _{k}|{\big )}\\&-G\sum _{j,k=1}^{N}m_{j}m_{k}\int \mathrm {d} ^{3}\mathbf {y} _{1}\cdots \mathrm {d} ^{3}\mathbf {y} _{N}\,{\frac {|\Psi (t,\mathbf {y} _{1},\dots ,\mathbf {y} _{N})|^{2}}{|\mathbf {x} _{j}-\mathbf {y} _{k}|}}{\bigg )}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bf2fbdd41bb4ad5eb53ef6c549fbec3b9da48f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:97.641ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}i\hbar {\frac {\partial }{\partial t}}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N})={\bigg (}&-\sum _{j=1}^{N}{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+\sum _{j\neq k}V_{jk}{\big (}|\mathbf {x} _{j}-\mathbf {x} _{k}|{\big )}\\&-G\sum _{j,k=1}^{N}m_{j}m_{k}\int \mathrm {d} ^{3}\mathbf {y} _{1}\cdots \mathrm {d} ^{3}\mathbf {y} _{N}\,{\frac {|\Psi (t,\mathbf {y} _{1},\dots ,\mathbf {y} _{N})|^{2}}{|\mathbf {x} _{j}-\mathbf {y} _{k}|}}{\bigg )}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N}).\end{aligned}}}"></span> The potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{jk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{jk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282ea1ccacbd67b453b5193572b1949dc87e3a10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.121ex; height:2.843ex;" alt="{\displaystyle V_{jk}}"></span> contains all the mutual linear interactions, e.g. electrodynamical Coulomb interactions, while the gravitational-potential term is based on the assumption that all particles perceive the same gravitational potential generated by all the <a href="/wiki/Marginal_distribution" title="Marginal distribution">marginal distributions</a> for all the particles together. </p><p>In a <a href="/wiki/Born%E2%80%93Oppenheimer_approximation" title="Born–Oppenheimer approximation">Born–Oppenheimer</a>-like approximation, this <i>N</i>-particle equation can be separated into two equations, one describing the relative motion, the other providing the dynamics of the centre-of-mass wave-function. For the relative motion, the gravitational interaction does not play a role, since it is usually weak compared to the other interactions represented by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{jk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{jk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282ea1ccacbd67b453b5193572b1949dc87e3a10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.121ex; height:2.843ex;" alt="{\displaystyle V_{jk}}"></span>. But it has a significant influence on the centre-of-mass motion. While <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{jk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{jk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282ea1ccacbd67b453b5193572b1949dc87e3a10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.121ex; height:2.843ex;" alt="{\displaystyle V_{jk}}"></span> only depends on relative coordinates and therefore does not contribute to the centre-of-mass dynamics at all, the nonlinear Schrödinger–Newton interaction does contribute. In the aforementioned approximation, the centre-of-mass wave-function satisfies the following nonlinear Schrödinger equation: <span class="mwe-math-element"><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 i\hbar {\frac {\partial \psi _{c}(t,\mathbf {R} )}{\partial t}}=\left({\frac {\hbar ^{2}}{2M}}\nabla ^{2}-G\int \mathrm {d} ^{3}\mathbf {R'} \,\int \mathrm {d} ^{3}\mathbf {y} \,\int \mathrm {d} ^{3}\mathbf {z} \,{\frac {|\psi _{c}(t,\mathbf {R'} )|^{2}\,\rho _{c}(\mathbf {y} )\rho _{c}(\mathbf {z} )}{\left|\mathbf {R} -\mathbf {R'} -\mathbf {y} +\mathbf {z} \right|}}\right)\psi _{c}(t,\mathbf {R} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>G</mi> <mo>∫</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">R</mi> <mo>′</mo> </msup> </mrow> <mspace width="thinmathspace" /> <mo>∫</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mspace width="thinmathspace" /> <mo>∫</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">R</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">R</mi> <mo>′</mo> </msup> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial \psi _{c}(t,\mathbf {R} )}{\partial t}}=\left({\frac {\hbar ^{2}}{2M}}\nabla ^{2}-G\int \mathrm {d} ^{3}\mathbf {R'} \,\int \mathrm {d} ^{3}\mathbf {y} \,\int \mathrm {d} ^{3}\mathbf {z} \,{\frac {|\psi _{c}(t,\mathbf {R'} )|^{2}\,\rho _{c}(\mathbf {y} )\rho _{c}(\mathbf {z} )}{\left|\mathbf {R} -\mathbf {R'} -\mathbf {y} +\mathbf {z} \right|}}\right)\psi _{c}(t,\mathbf {R} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0263a0c87d83ae96ad25edf8c21da9d6c080337" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:85.763ex; height:7.509ex;" alt="{\displaystyle i\hbar {\frac {\partial \psi _{c}(t,\mathbf {R} )}{\partial t}}=\left({\frac {\hbar ^{2}}{2M}}\nabla ^{2}-G\int \mathrm {d} ^{3}\mathbf {R'} \,\int \mathrm {d} ^{3}\mathbf {y} \,\int \mathrm {d} ^{3}\mathbf {z} \,{\frac {|\psi _{c}(t,\mathbf {R'} )|^{2}\,\rho _{c}(\mathbf {y} )\rho _{c}(\mathbf {z} )}{\left|\mathbf {R} -\mathbf {R'} -\mathbf {y} +\mathbf {z} \right|}}\right)\psi _{c}(t,\mathbf {R} ),}"></span> where <span class="texhtml mvar" style="font-style:italic;">M</span> is the total mass, <span class="texhtml"><b>R</b></span> is the relative coordinate, <span class="mwe-math-element"><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 _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999407536cc7a4b1cf5e29bf39ad6b702e8f7426" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.457ex; height:2.509ex;" alt="{\displaystyle \psi _{c}}"></span> the centre-of-mass wave-function, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/523e8c709cfeaef7ed497ce5be80144fdf2e9b70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle \rho _{c}}"></span> is the mass density of the many-body system (e.g. a molecule or a rock) relative to its centre of mass.<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> </p><p>In the limiting case of a wide wave-function, i.e. where the width of the centre-of-mass distribution is large compared to the size of the considered object, the centre-of-mass motion is approximated well by the Schrödinger–Newton equation for a single particle. The opposite case of a narrow wave-function can be approximated by a harmonic-oscillator potential, where the Schrödinger–Newton dynamics leads to a rotation in phase space.<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>In the context where the Schrödinger–Newton equation appears as a Hartree approximation, the situation is different. In this case the full <i>N</i>-particle wave-function is considered a product state of <i>N</i> single-particle wave-functions, where each of those factors obeys the Schrödinger–Newton equation. The dynamics of the centre-of-mass, however, remain strictly linear in this picture. This is true in general: nonlinear Hartree equations never have an influence on the centre of mass. </p> <div class="mw-heading mw-heading2"><h2 id="Significance_of_effects">Significance of effects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=4" title="Edit section: Significance of effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A rough order-of-magnitude estimate of the regime where effects of the Schrödinger–Newton equation become relevant can be obtained by a rather simple reasoning.<sup id="cite_ref-Carlip2008_9-1" class="reference"><a href="#cite_note-Carlip2008-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For a spherically symmetric <a href="/wiki/Normal_distribution" title="Normal distribution">Gaussian</a>, <span class="mwe-math-element"><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 (t=0,r)=(\pi \sigma ^{2})^{-3/4}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>π</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (t=0,r)=(\pi \sigma ^{2})^{-3/4}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3205526bed36c3a802c1f8715cffdfd5c1b7dc39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.601ex; height:6.343ex;" alt="{\displaystyle \Psi (t=0,r)=(\pi \sigma ^{2})^{-3/4}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}}}\right),}"></span> the free linear Schrödinger equation has the solution <span class="mwe-math-element"><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 (t,r)=(\pi \sigma ^{2})^{-3/4}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)^{-3/2}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>π</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>t</mi> </mrow> <mrow> <mi>m</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>t</mi> </mrow> <mrow> <mi>m</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (t,r)=(\pi \sigma ^{2})^{-3/4}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)^{-3/2}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c924f8748507686b2295b3c3c99b5203368d188" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:61.864ex; height:10.176ex;" alt="{\displaystyle \Psi (t,r)=(\pi \sigma ^{2})^{-3/4}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)^{-3/2}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)}}\right).}"></span> The peak of the radial probability density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi r^{2}|\Psi |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}|\Psi |^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54b3251ace39efc73d4bd1044cb6bafb68e185d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.753ex; height:3.343ex;" alt="{\displaystyle 4\pi r^{2}|\Psi |^{2}}"></span> can be found at <span class="mwe-math-element"><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 r_{p}=\sigma {\sqrt {1+{\frac {\hbar ^{2}t^{2}}{m^{2}\sigma ^{4}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{p}=\sigma {\sqrt {1+{\frac {\hbar ^{2}t^{2}}{m^{2}\sigma ^{4}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80d8c4e4668db24f5f87941727e8cdaf463a062" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.825ex; height:7.509ex;" alt="{\displaystyle r_{p}=\sigma {\sqrt {1+{\frac {\hbar ^{2}t^{2}}{m^{2}\sigma ^{4}}}}}.}"></span> Now we set the acceleration <span class="mwe-math-element"><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 {\ddot {r}}_{p}={\frac {\hbar ^{2}}{m^{2}r_{p}^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {r}}_{p}={\frac {\hbar ^{2}}{m^{2}r_{p}^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63cc35f44948c3bf74cdb92050053db23716b3d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.488ex; height:6.676ex;" alt="{\displaystyle {\ddot {r}}_{p}={\frac {\hbar ^{2}}{m^{2}r_{p}^{3}}}}"></span> of this peak probability equal to the acceleration due to Newtonian gravity: <span class="mwe-math-element"><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 {\ddot {r}}=-{\frac {Gm}{r^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>m</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {r}}=-{\frac {Gm}{r^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fdb604f138313305826c9c8a7304ab28ca6ac4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.548ex; height:5.676ex;" alt="{\displaystyle {\ddot {r}}=-{\frac {Gm}{r^{2}}},}"></span> using that <span class="mwe-math-element"><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_{p}=\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{p}=\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cdcc84ee0f9320e2e6b3d7525b2bb9fe9c995e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.536ex; height:2.343ex;" alt="{\displaystyle r_{p}=\sigma }"></span> at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"></span>. This yields the relation <span class="mwe-math-element"><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 m^{3}\sigma ={\frac {\hbar ^{2}}{G}}\approx 1.7\times 10^{-58}~{\text{m}}\,{\text{kg}}^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>G</mi> </mfrac> </mrow> <mo>≈</mo> <mn>1.7</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>58</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>kg</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{3}\sigma ={\frac {\hbar ^{2}}{G}}\approx 1.7\times 10^{-58}~{\text{m}}\,{\text{kg}}^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2054857f5f7ba3d153e58e0fb93b6d1db84cef2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.12ex; height:5.843ex;" alt="{\displaystyle m^{3}\sigma ={\frac {\hbar ^{2}}{G}}\approx 1.7\times 10^{-58}~{\text{m}}\,{\text{kg}}^{3},}"></span> which allows us to determine a critical width for a given mass value and conversely. We also recognise the scaling law mentioned above. Numerical simulations<sup id="cite_ref-Giulini2011_12-1" class="reference"><a href="#cite_note-Giulini2011-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-vanMeter2011_1-1" class="reference"><a href="#cite_note-vanMeter2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> show that this equation gives a rather good estimate of the mass regime above which effects of the Schrödinger–Newton equation become significant. </p><p>For an atom the critical width is around 10<sup>22</sup> metres, while it is already down to 10<sup>−31</sup> metres for a mass of one microgram. The regime where the mass is around 10<sup>10</sup> <a href="/wiki/Atomic_mass_unit" class="mw-redirect" title="Atomic mass unit">atomic mass units</a> while the width is of the order of micrometers is expected to allow an experimental test of the Schrödinger–Newton equation in the future. A possible candidate are <a href="/wiki/Interferometry" title="Interferometry">interferometry</a> experiments with heavy molecules, which currently reach masses up to <span class="nowrap"><span data-sort-value="7004100000000000000♠"></span>10<span style="margin-left:.25em;">000</span></span> atomic mass units. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_wave_function_collapse">Quantum wave function collapse</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=5" title="Edit section: Quantum wave function collapse"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The idea that gravity causes (or somehow influences) the <a href="/wiki/Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">wavefunction collapse</a> dates back to the 1960s and was originally proposed by <a href="/wiki/Frigyes_K%C3%A1rolyh%C3%A1zy" title="Frigyes Károlyházy">Károlyházy</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The Schrödinger–Newton equation was proposed in this context by Diósi.<sup id="cite_ref-Diosi1984_5-2" class="reference"><a href="#cite_note-Diosi1984-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> There the equation provides an estimation for the "line of demarcation" between microscopic (quantum) and macroscopic (classical) objects. The stationary ground state has a width of <span class="mwe-math-element"><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 a_{0}\approx {\frac {\hbar ^{2}}{Gm^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>G</mi> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}\approx {\frac {\hbar ^{2}}{Gm^{3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cfe067f397c0366610156836a1430df898f740" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.787ex; height:6.009ex;" alt="{\displaystyle a_{0}\approx {\frac {\hbar ^{2}}{Gm^{3}}}.}"></span> For a well-localised homogeneous sphere, i.e. a sphere with a centre-of-mass wave-function that is narrow compared to the radius of the sphere, Diósi finds as an estimate for the width of the ground-state centre-of-mass wave-function <span class="mwe-math-element"><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 a_{0}^{(R)}\approx a_{0}^{1/4}R^{3/4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>≈</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msubsup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}^{(R)}\approx a_{0}^{1/4}R^{3/4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5f708225a77227f3502be692b7a5dd8cb8ffd5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.124ex; height:3.676ex;" alt="{\displaystyle a_{0}^{(R)}\approx a_{0}^{1/4}R^{3/4}.}"></span> Assuming a usual density around 1000 kg/m<sup>3</sup>, a critical radius can be calculated for which <span class="mwe-math-element"><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_{0}^{(R)}\approx R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>≈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}^{(R)}\approx R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e05e89b6861fe98e6acb1f95f8c72c3f27728f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.851ex; height:3.676ex;" alt="{\displaystyle a_{0}^{(R)}\approx R}"></span>. This critical radius is around a tenth of a micrometer. </p><p><a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a> proposed that the Schrödinger–Newton equation mathematically describes the basis states involved in a gravitationally induced <a href="/wiki/Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">wavefunction collapse</a> scheme.<sup id="cite_ref-Penrose1996_6-1" class="reference"><a href="#cite_note-Penrose1996-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Penrose1998_7-1" class="reference"><a href="#cite_note-Penrose1998-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Penrose2014_8-1" class="reference"><a href="#cite_note-Penrose2014-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Penrose suggests that a superposition of two or more quantum states having a significant amount of mass displacement ought to be unstable and reduce to one of the states within a finite time. He hypothesises that there exists a "preferred" set of states that could collapse no further, specifically, the stationary states of the Schrödinger–Newton equation. A macroscopic system can therefore never be in a spatial superposition, since the nonlinear gravitational self-interaction immediately leads to a collapse to a stationary state of the Schrödinger–Newton equation. According to Penrose's idea, when a quantum particle is measured, there is an interplay of this nonlinear collapse and environmental <a href="/wiki/Quantum_decoherence" title="Quantum decoherence">decoherence</a>. The gravitational interaction leads to the reduction of the environment to one distinct state, and decoherence leads to the localisation of the particle, e.g. as a dot on a screen. </p> <div class="mw-heading mw-heading3"><h3 id="Problems_and_open_matters">Problems and open matters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=6" title="Edit section: Problems and open matters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Three major problems occur with this interpretation of the Schrödinger–Newton equation as the cause of the wave-function collapse: </p> <ol><li>Excessive residual probability far from the collapse point</li> <li>Lack of any apparent reason for the <a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li>Promotion of the previously strictly hypothetical <a href="/wiki/Wave_function" title="Wave function">wave function</a> to an observable (real) quantity.</li></ol> <p>First, numerical studies<sup id="cite_ref-Giulini2011_12-2" class="reference"><a href="#cite_note-Giulini2011-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Harrison2003_15-1" class="reference"><a href="#cite_note-Harrison2003-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-vanMeter2011_1-2" class="reference"><a href="#cite_note-vanMeter2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> agreeingly find that when a wave packet "collapses" to a stationary solution, a small portion of it seems to run away to infinity. This would mean that even a completely collapsed quantum system still can be found at a distant location. Since the solutions of the linear Schrödinger equation tend towards infinity even faster, this only indicates that the Schrödinger–Newton equation alone is not sufficient to explain the wave-function collapse. If the environment is taken into account, this effect might disappear and therefore not be present in the scenario described by Penrose. </p><p>A second problem, also arising in Penrose's proposal, is the origin of the <a href="/wiki/Born_rule" title="Born rule">Born rule</a>: To solve the <a href="/wiki/Quantum_measurement_problem" class="mw-redirect" title="Quantum measurement problem">measurement problem</a>, a mere explanation of why a wave-function collapses – e.g., to a dot on a screen – is not enough. A good model for the collapse process <i>also</i> has to explain why the dot appears on different positions of the screen, with probabilities that are determined by the squared absolute-value of the wave-function. It might be possible that a model based on Penrose's idea could provide such an explanation, but there is as yet no known reason that Born's rule would naturally arise from it. </p><p>Thirdly, since the gravitational potential is linked to the wave-function in the picture of the Schrödinger–Newton equation, the wave-function must be interpreted as a real object. Therefore, at least in principle, it becomes a measurable quantity. Making use of the nonlocal nature of entangled quantum systems, this could be used to send signals faster than light, which is generally thought to be in contradiction with causality. It is, however, not clear whether this problem can be resolved by applying the right collapse prescription, yet to be found, consistently to the full quantum system. Also, since gravity is such a weak interaction, it is not clear that such an experiment can be actually performed within the parameters given in our universe (see the referenced discussion<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> about a similar thought experiment proposed by Eppley & Hannah<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>). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">Nonlinear Schrödinger equation</a></li> <li><a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</a></li> <li><a href="/wiki/Penrose_interpretation" title="Penrose interpretation">Penrose interpretation</a></li> <li><a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dinger%E2%80%93Newton_equation&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-vanMeter2011-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-vanMeter2011_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-vanMeter2011_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-vanMeter2011_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFvan_Meter2011" class="citation cs2">van Meter, J. 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"Why Eppley and Hannah's thought experiment fails". <i>Physical Review D</i>. <b>73</b> (6): 064025. <a href="/wiki/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/gr-qc/0601127">gr-qc/0601127</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006PhRvD..73f4025M">2006PhRvD..73f4025M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevd.73.064025">10.1103/physrevd.73.064025</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" 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"The necessity of quantizing the gravitational field", <i>Foundations of Physics</i>, <b>7</b> (1–2): 51–68, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977FoPh....7...51E">1977FoPh....7...51E</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00715241">10.1007/BF00715241</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123251640">123251640</a></cite><span 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scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Publications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Fluxions</a></i> (1671)</li> <li><i><a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De Motu</a></i> (1684)</li> <li><i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> (1687)</li> <li><i><a href="/wiki/Opticks" title="Opticks">Opticks</a></i> (1704)</li> <li><i><a href="/wiki/The_Queries" class="mw-redirect" title="The Queries">Queries</a></i> (1704)</li> <li><i><a href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica</a></i> (1707)</li> <li><i><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De Analysi</a></i> (1711)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Other writings</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones</a></i> (1661–1665)</li> <li>"<a href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">standing on the shoulders of giants</a>" (1675)</li> <li><i><a href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</a></i> (c. 1680)</li> <li>"<a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a>" (1713; <i>"<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">hypotheses non fingo</a>"</i> )</li> <li><i><a href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">Ancient Kingdoms Amended</a></i> (1728)</li> <li><i><a href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">Corruptions of Scripture</a></i> (1754)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Contributions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a> <ul><li><a href="/wiki/Fluxion" title="Fluxion">fluxion</a></li></ul></li> <li><a href="/wiki/Impact_depth" title="Impact depth">Impact depth</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a></li> <li><a href="/wiki/Newton_disc" title="Newton disc">Newton disc</a></li> <li><a href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</a> <ul><li><a href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a></li></ul></li> <li><a href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</a></li> <li><a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</a></li> <li><a href="/wiki/Newton_scale" title="Newton scale">Newton scale</a></li> <li><a href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</a></li> <li><a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></li> <li><a href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</a></li> <li><a href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</a></li> <li><a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</a></li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> <ul><li><a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized</a></li> <li><a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li></ul></li> <li><a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a></li> <li><a class="mw-selflink selflink">Schrödinger–Newton equation</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> <ul><li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a></li></ul></li> <li><a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a></li> <li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</a> <ul><li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Apollonius's problem</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">truncated Newton method</a></li></ul></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</a></li> <li><a href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</a></li> <li><a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a></li> <li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</a></li> <li><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a></li> <li><a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a></li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></li> <li><a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> <ul><li><a href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">generalized Gauss–Newton method</a></li></ul></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a></li> <li><a href="/wiki/Power_number" title="Power number">Newton number</a> <ul><li><a href="/wiki/Kissing_number" title="Kissing number">kissing number problem</a></li></ul></li> <li><a href="/wiki/Difference_quotient" title="Difference quotient">Newton's quotient</a></li> <li><a href="/wiki/Parallelogram_of_force" title="Parallelogram of force">Parallelogram of force</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</a></li> <li><a href="/wiki/Absolute_space_and_time#Newton" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Luminiferous_aether" title="Luminiferous aether">Luminiferous aether</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Newtonian series</a> <ul><li><a href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">table</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Personal life</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</a> (birthplace)</li> <li><a href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</a> (home)</li> <li><a href="/wiki/Early_life_of_Isaac_Newton" title="Early life of Isaac Newton">Early life</a></li> <li><a href="/wiki/Later_life_of_Isaac_Newton" title="Later life of Isaac Newton">Later life</a></li> <li><a href="/wiki/Isaac_Newton%27s_apple_tree" title="Isaac Newton's apple tree">Apple tree</a></li> <li><a href="/wiki/Religious_views_of_Isaac_Newton" title="Religious views of Isaac Newton">Religious views</a></li> <li><a href="/wiki/Isaac_Newton%27s_occult_studies" title="Isaac Newton's occult studies">Occult studies</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Relations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Catherine_Barton" title="Catherine Barton">Catherine Barton</a> (niece)</li> <li><a href="/wiki/John_Conduitt" title="John Conduitt">John Conduitt</a> (nephew-in-law)</li> <li><a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> (professor)</li> <li><a href="/wiki/William_Clarke_(apothecary)" title="William Clarke (apothecary)">William Clarke</a> (mentor)</li> <li><a href="/wiki/Benjamin_Pulleyn" title="Benjamin Pulleyn">Benjamin Pulleyn</a> (tutor)</li> <li><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> (student)</li> <li><a href="/wiki/William_Whiston" title="William Whiston">William Whiston</a> (student)</li> <li><a href="/wiki/John_Keill" title="John Keill">John Keill</a> (disciple)</li> <li><a href="/wiki/William_Stukeley" title="William Stukeley">William Stukeley</a> (friend)</li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> (friend)</li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> (friend)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">Depictions</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(Blake)" title="Newton (Blake)"><i>Newton</i> by Blake</a> (monotype)</li> <li><a href="/wiki/Newton_(Paolozzi)" title="Newton (Paolozzi)"><i>Newton</i> by Paolozzi</a> (sculpture)</li> <li><i><a href="/wiki/Isaac_Newton_Gargoyle" title="Isaac Newton Gargoyle">Isaac Newton Gargoyle</a></i></li> <li><i><a href="/wiki/Astronomers_Monument" title="Astronomers Monument">Astronomers Monument</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">Namesake</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(unit)" title="Newton (unit)">Newton (unit)</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</a></li> <li><a href="/wiki/Institute_of_Physics_Isaac_Newton_Medal" class="mw-redirect" title="Institute of Physics Isaac Newton Medal">Isaac Newton Medal</a></li> <li><a href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</a></li> <li><a href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</a></li> <li><a href="/wiki/XMM-Newton" title="XMM-Newton">XMM-Newton</a></li> <li><a href="/wiki/Sir_Isaac_Newton_Sixth_Form" title="Sir Isaac Newton Sixth Form">Sir Isaac Newton Sixth Form</a></li> <li><a href="/wiki/Statal_Institute_of_Higher_Education_Isaac_Newton" title="Statal Institute of Higher Education Isaac Newton">Statal Institute of Higher Education Isaac Newton</a></li> <li><a href="/wiki/Newton_International_Fellowship" title="Newton International Fellowship">Newton International Fellowship</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><div class="div-col"> <div class="CategoryTreeTag" data-ct-options="{"mode":20,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeBullet"><a class="CategoryTreeToggle" data-ct-title="Isaac_Newton" aria-expanded="false"></a> </span> <bdi dir="ltr"><a href="/wiki/Category:Isaac_Newton" title="Category:Isaac Newton">Isaac Newton</a></bdi></div><div class="CategoryTreeChildren" style="display:none"></div></div></div> </div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Schrödinger–Newton_equation&oldid=1187087362">https://en.wikipedia.org/w/index.php?title=Schrödinger–Newton_equation&oldid=1187087362</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a 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Schrödinger–Newton equation - Wikipedia