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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" 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">Approximation or recovery of classical mechanics in certain theories</div> <p>The <b>classical limit</b> or <b>correspondence limit</b> is the ability of a <a href="/wiki/Theoretical_physics" title="Theoretical physics">physical theory</a> to approximate or "recover" <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> when considered over special values of its parameters.<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> The classical limit is used with physical theories that predict non-classical behavior. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Quantum_theory">Quantum theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_limit&action=edit&section=1" title="Edit section: Quantum theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> postulate called the <a href="/wiki/Correspondence_principle" title="Correspondence principle">correspondence principle</a> was introduced to <a href="/wiki/Bohr_model" title="Bohr model">quantum theory</a> by <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a> normalized by the action of these systems becomes very small. Often, this is approached through "quasi-classical" techniques (cf. <a href="/wiki/WKB_approximation" title="WKB approximation">WKB approximation</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> </p><p>More rigorously,<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 mathematical operation involved in classical limits is a <a href="/wiki/Group_contraction" title="Group contraction">group contraction</a>, approximating physical systems where the relevant action is much larger than the reduced Planck constant <span class="texhtml mvar" style="font-style:italic;">ħ</span>, so the "deformation parameter" <span class="texhtml mvar" style="font-style:italic;">ħ</span>/<span class="texhtml mvar" style="font-style:italic;">S</span> can be effectively taken to be zero (cf. <a href="/wiki/Weyl_quantization" class="mw-redirect" title="Weyl quantization">Weyl quantization</a>.) Thus typically, quantum commutators (equivalently, <a href="/wiki/Moyal_bracket" title="Moyal bracket">Moyal brackets</a>) reduce to <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson brackets</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> in a <a href="/wiki/Group_contraction" title="Group contraction">group contraction</a>. </p><p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, due to <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg's</a> <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>, an <a href="/wiki/Electron" title="Electron">electron</a> can never be at rest; it must always have a non-zero <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>, a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot really have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can effectively appear to be at rest, and hence it appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics. The typical <a href="/wiki/Occupation_number" class="mw-redirect" title="Occupation number">occupation numbers</a> involved are huge: a macroscopic harmonic oscillator with <span class="texhtml mvar" style="font-style:italic;">ω</span> = 2 Hz, <span class="texhtml mvar" style="font-style:italic;">m</span> = 10 g, and maximum <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> <span class="texhtml mvar" style="font-style:italic;">x</span><sub>0</sub> = 10 cm, has <span class="texhtml"><i>S</i> ≈ <i>E</i>/<i>ω</i> ≈ <i>mωx</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">0</sub></span></span>/2 ≈ 10<sup>−4</sup> kg·m<sup>2</sup>/s</span> = <span class="texhtml mvar" style="font-style:italic;">ħn</span>, so that <span class="texhtml mvar" style="font-style:italic;">n</span> ≃ 10<sup>30</sup>. Further see <a href="/wiki/Coherent_states#The_wavefunction_of_a_coherent_state" class="mw-redirect" title="Coherent states">coherent states</a>. It is less clear, however, how the classical limit applies to chaotic systems, a field known as <a href="/wiki/Quantum_chaos" title="Quantum chaos">quantum chaos</a>. </p><p>Quantum mechanics and classical mechanics are usually treated with entirely different formalisms: quantum theory using <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, and classical mechanics using a representation in <a href="/wiki/Phase_space" title="Phase space">phase space</a>. One can bring the two into a common mathematical framework in various ways. In the <a href="/wiki/Phase_space_formulation" class="mw-redirect" title="Phase space formulation">phase space formulation</a> of quantum mechanics, which is statistical in nature, logical connections between quantum mechanics and classical statistical mechanics are made, enabling natural comparisons between them, including the violations of <a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem (Hamiltonian)</a> upon quantization.<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><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> </p><p>In a crucial paper (1933), <a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a><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> explained how classical mechanics is an <a href="/wiki/Emergence#Non-living,_physical_systems" title="Emergence">emergent phenomenon</a> of quantum mechanics: <a href="/wiki/Destructive_interference#Quantum_interference" class="mw-redirect" title="Destructive interference">destructive interference</a> among paths with non-<a href="/wiki/Extremal" class="mw-redirect" title="Extremal">extremal</a> macroscopic actions <span class="texhtml mvar" style="font-style:italic;">S</span> » <span class="texhtml mvar" style="font-style:italic;">ħ</span> obliterate amplitude contributions in the <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral</a> he introduced, leaving the extremal action <span class="texhtml mvar" style="font-style:italic;">S</span><sub>class</sub>, thus the classical action path as the dominant contribution, an observation further elaborated by <a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a> in his 1942 PhD dissertation.<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> (Further see <a href="/wiki/Quantum_decoherence" title="Quantum decoherence">quantum decoherence</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="Time-evolution_of_expectation_values">Time-evolution of expectation values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_limit&action=edit&section=2" title="Edit section: Time-evolution of expectation values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ehrenfest_theorem" title="Ehrenfest theorem">Ehrenfest theorem</a></div> <p>One simple way to compare classical to quantum mechanics is to consider the time-evolution of the <i>expected</i> position and <i>expected</i> momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the <a href="/wiki/Ehrenfest_theorem" title="Ehrenfest theorem">Ehrenfest theorem</a>. For a one-dimensional quantum particle moving in a 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, the Ehrenfest theorem says<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> <dl><dd><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 m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>;</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mrow> <mo>⟨</mo> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d5e514f985a955617f92c1c99cc8e1493f51a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.903ex; height:5.509ex;" alt="{\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .}"></span></dd></dl> <p>Although the first of these equations is consistent with the classical mechanics, the second is not: If the pair <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 (\langle X\rangle ,\langle P\rangle )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>P</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\langle X\rangle ,\langle P\rangle )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a208b4fe89bcbccd675e91ada6ab2f734def8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.187ex; height:2.843ex;" alt="{\displaystyle (\langle X\rangle ,\langle P\rangle )}"></span> were to satisfy Newton's second law, the right-hand side of the second equation would have read </p> <dl><dd><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 {\frac {d}{dt}}\langle p\rangle =-V'\left(\left\langle X\right\rangle \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>V</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mo>⟨</mo> <mi>X</mi> <mo>⟩</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\langle p\rangle =-V'\left(\left\langle X\right\rangle \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111e4cac8a53a827ffc92edf505f01f1d22a789d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.364ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}\langle p\rangle =-V'\left(\left\langle X\right\rangle \right)}"></span>.</dd></dl> <p>But in most cases, </p> <dl><dd><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 \left\langle V'(X)\right\rangle \neq V'(\left\langle X\right\rangle )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>≠</mo> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow> <mo>⟨</mo> <mi>X</mi> <mo>⟩</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle V'(X)\right\rangle \neq V'(\left\langle X\right\rangle )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2188696ef5c10aa37b82e3c5492cb76562072c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.499ex; height:3.009ex;" alt="{\displaystyle \left\langle V'(X)\right\rangle \neq V'(\left\langle X\right\rangle )}"></span>.</dd></dl> <p>If for example, 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is cubic, then <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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff31fe992a31b4c954a933f1be91e2739a1c0ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.602ex; height:2.509ex;" alt="{\displaystyle V'}"></span> is quadratic, in which case, we are talking about the distinction between <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 \langle X^{2}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle X^{2}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c709b0821e6cdff54da65be02b03d0708bc185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.86ex; height:3.176ex;" alt="{\displaystyle \langle X^{2}\rangle }"></span> 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 \langle X\rangle ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>X</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle X\rangle ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11f0def5713baa82e390942e3dc4a78956dd6c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.844ex; height:3.176ex;" alt="{\displaystyle \langle X\rangle ^{2}}"></span>, which differ 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 (\Delta X)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Delta X)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d0ad84bcf442f30c44c35506a863c980f2291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.779ex; height:3.176ex;" alt="{\displaystyle (\Delta X)^{2}}"></span>. </p><p>An exception occurs in case when the classical equations of motion are linear, that is, when <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is quadratic 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 V'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff31fe992a31b4c954a933f1be91e2739a1c0ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.602ex; height:2.509ex;" alt="{\displaystyle V'}"></span> is linear. In that special case, <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'\left(\left\langle X\right\rangle \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mo>⟨</mo> <mi>X</mi> <mo>⟩</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V'\left(\left\langle X\right\rangle \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4334cb83aa21b56c0941d539772062e8c6cd79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.587ex; height:3.009ex;" alt="{\displaystyle V'\left(\left\langle X\right\rangle \right)}"></span> 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 \left\langle V'(X)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle V'(X)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db61e58cc63308d55d3cea8ab0ea1c35213082f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.2ex; height:3.009ex;" alt="{\displaystyle \left\langle V'(X)\right\rangle }"></span> do agree. In particular, for a free particle or a quantum harmonic oscillator, the expected position and expected momentum exactly follows solutions of Newton's equations. </p><p>For general systems, the best we can hope for is that the expected position and momentum will <i>approximately</i> follow the classical trajectories. If the wave function is highly concentrated around a point <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 x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, then <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'\left(\left\langle X\right\rangle \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <mo>⟨</mo> <mi>X</mi> <mo>⟩</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V'\left(\left\langle X\right\rangle \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4334cb83aa21b56c0941d539772062e8c6cd79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.587ex; height:3.009ex;" alt="{\displaystyle V'\left(\left\langle X\right\rangle \right)}"></span> 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 \left\langle V'(X)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle V'(X)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db61e58cc63308d55d3cea8ab0ea1c35213082f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.2ex; height:3.009ex;" alt="{\displaystyle \left\langle V'(X)\right\rangle }"></span> will be <i>almost</i> the same, since both will be approximately equal to <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'(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V'(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597d8921d945c06cd6f9c31c4547c1fbc824b9ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:3.009ex;" alt="{\displaystyle V'(x_{0})}"></span>. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least <i>for as long as</i> the wave function remains highly localized in position.<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><p>Now, if the initial state is very localized in position, it will be very spread out in momentum, and thus we expect that the wave function will rapidly spread out, and the connection with the classical trajectories will be lost. When the Planck constant is small, however, it is possible to have a state that is well localized in <i>both</i> position and momentum. The small uncertainty in momentum ensures that the particle <i>remains</i> well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories for a long time. </p> <div class="mw-heading mw-heading2"><h2 id="Relativity_and_other_deformations">Relativity and other deformations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_limit&action=edit&section=3" title="Edit section: Relativity and other deformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other familiar deformations in physics involve: </p> <ul><li>The deformation of classical Newtonian into relativistic mechanics (<a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>), with deformation parameter <span class="texhtml"><i>v</i>/<i>c</i></span>; the classical limit involves small speeds, so <span class="texhtml"><i>v</i>/<i>c</i> → 0</span>, and the systems appear to obey Newtonian mechanics.</li> <li>Similarly for the deformation of Newtonian gravity into <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, with deformation parameter Schwarzschild-radius/characteristic-dimension, we find that objects once again appear to obey classical mechanics (flat space), when the mass of an object times the square of the <a href="/wiki/Planck_length" class="mw-redirect" title="Planck length">Planck length</a> is much smaller than its size and the sizes of the problem addressed. See <a href="/wiki/Newtonian_limit" title="Newtonian limit">Newtonian limit</a>.</li> <li>Wave optics might also be regarded as a deformation of <a href="/wiki/Geometrical_optics" title="Geometrical optics">ray optics</a> for deformation parameter <span class="texhtml"><i>λ</i>/<i>a</i></span>.</li> <li>Likewise, <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> deforms to <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> with deformation parameter <span class="texhtml">1/<i>N</i></span>.</li></ul> <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=Classical_limit&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Classical_probability_density" title="Classical probability density">Classical probability density</a></li> <li><a href="/wiki/Ehrenfest_theorem" title="Ehrenfest theorem">Ehrenfest theorem</a></li> <li><a href="/wiki/Madelung_equations" title="Madelung equations">Madelung equations</a></li> <li><a href="/wiki/Fresnel_integral" title="Fresnel integral">Fresnel integral</a></li> <li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Quantum decoherence</a></li> <li><a href="/wiki/Quantum_limit" title="Quantum limit">Quantum limit</a></li> <li><a href="/wiki/Semiclassical_physics" title="Semiclassical physics">Semiclassical physics</a></li> <li><a href="/wiki/Wigner%E2%80%93Weyl_transform" title="Wigner–Weyl transform">Wigner–Weyl transform</a></li> <li><a href="/wiki/WKB_approximation" title="WKB approximation">WKB approximation</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=Classical_limit&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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="CITEREFBohm1989" class="citation book cs1"><a href="/wiki/David_Bohm" title="David Bohm">Bohm, D.</a> (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hEHCAgAAQBAJ"><i>Quantum Theory</i></a>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486659695" title="Special:BookSources/9780486659695"><bdi>9780486659695</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=9780486659695&rft.aulast=Bohm&rft.aufirst=D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhEHCAgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1977" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, L. D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E. M.</a> (1977). <i>Quantum Mechanics: Non-Relativistic Theory</i>. 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L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". <i><a href="/w/index.php?title=Asia_Pacific_Physics_Newsletter&action=edit&redlink=1" class="new" title="Asia Pacific Physics Newsletter (page does not exist)">Asia Pacific Physics Newsletter</a></i>. <b>1</b>: 37–46. <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/1104.5269">1104.5269</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS2251158X12000069">10.1142/S2251158X12000069</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:119230734">119230734</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Asia+Pacific+Physics+Newsletter&rft.atitle=Quantum+Mechanics+in+Phase+Space&rft.volume=1&rft.pages=37-46&rft.date=2012&rft_id=info%3Aarxiv%2F1104.5269&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119230734%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS2251158X12000069&rft.aulast=Curtright&rft.aufirst=T.+L.&rft.au=Zachos%2C+C.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrackenWood2006" class="citation journal cs1">Bracken, A.; Wood, J. (2006). "Semiquantum versus semiclassical mechanics for simple nonlinear systems". <i><a href="/wiki/Physical_Review_A" title="Physical Review A">Physical Review A</a></i>. <b>73</b> (1): 012104. <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/quant-ph/0511227">quant-ph/0511227</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/2006PhRvA..73a2104B">2006PhRvA..73a2104B</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%2FPhysRevA.73.012104">10.1103/PhysRevA.73.012104</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:14444752">14444752</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Semiquantum+versus+semiclassical+mechanics+for+simple+nonlinear+systems&rft.volume=73&rft.issue=1&rft.pages=012104&rft.date=2006&rft_id=info%3Aarxiv%2Fquant-ph%2F0511227&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14444752%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.73.012104&rft_id=info%3Abibcode%2F2006PhRvA..73a2104B&rft.aulast=Bracken&rft.aufirst=A.&rft.au=Wood%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Conversely, in the lesser-known <a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">approach presented in 1932 by Koopman and von Neumann</a>, the dynamics of classical mechanics have been formulated in terms of an <a href="/wiki/Operator_(physics)" title="Operator (physics)">operational</a> formalism in <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, a formalism used conventionally for quantum mechanics. <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoopmanvon_Neumann1932" class="citation journal cs1"><a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Koopman, B. O.</a>; <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, J.</a> (1932). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076203">"Dynamical Systems of Continuous Spectra"</a>. <i><a href="/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America" title="Proceedings of the National Academy of Sciences of the United States of America">Proceedings of the National Academy of Sciences of the United States of America</a></i>. <b>18</b> (3): 255–263. <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/1932PNAS...18..255K">1932PNAS...18..255K</a>. <a href="/wiki/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/10.1073%2Fpnas.18.3.255">10.1073/pnas.18.3.255</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076203">1076203</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16587673">16587673</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=Dynamical+Systems+of+Continuous+Spectra&rft.volume=18&rft.issue=3&rft.pages=255-263&rft.date=1932&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1076203%23id-name%3DPMC&rft_id=info%3Apmid%2F16587673&rft_id=info%3Adoi%2F10.1073%2Fpnas.18.3.255&rft_id=info%3Abibcode%2F1932PNAS...18..255K&rft.aulast=Koopman&rft.aufirst=B.+O.&rft.au=von+Neumann%2C+J.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1076203&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMauro2003" class="citation arxiv cs1">Mauro, D. (2003). "Topics in Koopman-von Neumann Theory". <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/quant-ph/0301172">quant-ph/0301172</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Topics+in+Koopman-von+Neumann+Theory&rft.date=2003&rft_id=info%3Aarxiv%2Fquant-ph%2F0301172&rft.aulast=Mauro&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBracken2003" class="citation journal cs1">Bracken, A. J. (2003). "Quantum mechanics as an approximation to classical mechanics in Hilbert space". <i><a href="/wiki/Journal_of_Physics_A" title="Journal of Physics A">Journal of Physics A</a></i>. <b>36</b> (23): L329–L335. <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/quant-ph/0210164">quant-ph/0210164</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F36%2F23%2F101">10.1088/0305-4470/36/23/101</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:15505801">15505801</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+A&rft.atitle=Quantum+mechanics+as+an+approximation+to+classical+mechanics+in+Hilbert+space&rft.volume=36&rft.issue=23&rft.pages=L329-L335&rft.date=2003&rft_id=info%3Aarxiv%2Fquant-ph%2F0210164&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15505801%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F36%2F23%2F101&rft.aulast=Bracken&rft.aufirst=A.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac1933" class="citation journal cs1"><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac, P.A.M.</a> (1933). <a rel="nofollow" class="external text" href="http://www.ifi.unicamp.br/~cabrera/teaching/aula%2015%202010s1.pdf">"The Lagrangian in quantum mechanics"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/w/index.php?title=Physikalische_Zeitschrift_der_Sowjetunion&action=edit&redlink=1" class="new" title="Physikalische Zeitschrift der Sowjetunion (page does not exist)">Physikalische Zeitschrift der Sowjetunion</a></i>. <b>3</b>: 64–72.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift+der+Sowjetunion&rft.atitle=The+Lagrangian+in+quantum+mechanics&rft.volume=3&rft.pages=64-72&rft.date=1933&rft.aulast=Dirac&rft.aufirst=P.A.M.&rft_id=http%3A%2F%2Fwww.ifi.unicamp.br%2F~cabrera%2Fteaching%2Faula%252015%25202010s1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman1942" class="citation thesis cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, R. P.</a> (1942). <i>The Principle of Least Action in Quantum Mechanics</i> (Ph.D. Dissertation). <a href="/wiki/Princeton_University" title="Princeton University">Princeton University</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=The+Principle+of+Least+Action+in+Quantum+Mechanics&rft.inst=Princeton+University&rft.date=1942&rft.aulast=Feynman&rft.aufirst=R.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span> <dl><dd>Reproduced in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman2005" class="citation book cs1">Feynman, R. P. (2005). Brown, L. M. (ed.). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/feynmansthesisne00feyn_0"><i>Feynman's Thesis: a New Approach to Quantum Theory</i></a></span>. <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-380-4" title="Special:BookSources/978-981-256-380-4"><bdi>978-981-256-380-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feynman%27s+Thesis%3A+a+New+Approach+to+Quantum+Theory&rft.pub=World+Scientific&rft.date=2005&rft.isbn=978-981-256-380-4&rft.aulast=Feynman&rft.aufirst=R.+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffeynmansthesisne00feyn_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></dd></dl> </span></li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2013">Hall 2013</a> Section 3.7.5</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2013">Hall 2013</a> p. 78</span> </li> </ol></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2013" class="citation cs2">Hall, Brian C. (2013), <i>Quantum Theory for Mathematicians</i>, Graduate Texts in Mathematics, vol. 267, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1461471158" title="Special:BookSources/978-1461471158"><bdi>978-1461471158</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory+for+Mathematicians&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2013&rft.isbn=978-1461471158&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+limit" class="Z3988"></span></li></ul>    </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=Classical_limit&oldid=1225362730">https://en.wikipedia.org/w/index.php?title=Classical_limit&oldid=1225362730</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 href="/wiki/Category:Concepts_in_physics" title="Category:Concepts in physics">Concepts in physics</a></li><li><a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Quantum mechanics</a></li><li><a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Theory of relativity</a></li><li><a href="/wiki/Category:Philosophy_of_science" title="Category:Philosophy of science">Philosophy of science</a></li><li><a href="/wiki/Category:Emergence" title="Category:Emergence">Emergence</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_American_English_from_January_2019" title="Category:Use American English from January 2019">Use American English from January 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 23 May 2024, at 23:47<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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