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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Observable</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Eine <b>Observable</b> (<span style="font-style:normal;font-weight:normal"><a href="Latein" title="Latein">lateinisch</a></span> <span lang="la-Latn" style="font-style:italic"><i>observabilis</i></span> ‚beobachtbar‘) ist in der <a href="Physik" title="Physik">Physik</a>, insbesondere der <a href="Quantenphysik" title="Quantenphysik">Quantenphysik</a>, der formale Name für eine <a href="Messgr%C3%B6%C3%9Fe" title="Messgröße">Messgröße</a> und den ihr zugeordneten <a href="Operator_(Mathematik)" title="Operator (Mathematik)">Operator</a> (siehe auch <a href="Hermitescher_Operator" title="Hermitescher Operator">hermitescher Operator</a>), die im <a href="Zustand_(Quantenmechanik)" title="Zustand (Quantenmechanik)">Zustandsraum</a>, einem <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a>, wirken.<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> Beispiele sind die <a href="Energie" title="Energie">Energie</a>, die Ortskoordinaten, die Koordinaten des <a href="Impuls_(Physik)" class="mw-redirect" title="Impuls (Physik)">Impulses</a> und die Komponenten des <a href="Spin" title="Spin">Spins</a> eines <a href="Teilchen" title="Teilchen">Teilchens</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Von-Neumannsche_Theorie">Von-Neumannsche Theorie</h2></div>
<p>Im traditionellen <a href="John_von_Neumann" title="John von Neumann">von-Neumannschen</a> mathematischen Formalismus der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> werden Observable durch <a href="Selbstadjungiert" class="mw-redirect" title="Selbstadjungiert">selbstadjungierte</a>, <a href="Dicht_definiert" class="mw-redirect" title="Dicht definiert">dicht definierte</a> <a href="Linearer_Operator" title="Linearer Operator">lineare Operatoren</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> auf einem <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"></span> dargestellt. Diese Theorie verallgemeinert die <a href="Born%E2%80%99sche_Wahrscheinlichkeitsinterpretation" class="mw-redirect" title="Born’sche Wahrscheinlichkeitsinterpretation">Bornsche Wahrscheinlichkeitsinterpretation</a>.
</p><p>Das Ergebnis einer <a href="Quantenmechanische_Messung" title="Quantenmechanische Messung">Messung</a> der Observablen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> eines quantenmechanischen Systems, dessen <a href="Zustand_(Quantenmechanik)" title="Zustand (Quantenmechanik)">Zustand</a> durch einen <a href="Normierter_Vektor" class="mw-redirect" title="Normierter Vektor">normierten Vektor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle \in {\mathcal {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle \in {\mathcal {H}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63efbb6c9f367b348c17dd8489d0bbe3e1f8e26f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.164ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle \in {\mathcal {H}}}" loading="lazy"></span> beschrieben wird (<a href="Wellenfunktion" title="Wellenfunktion">Wellenfunktion</a> in <a href="Bra-Ket-Notation" class="mw-redirect" title="Bra-Ket-Notation">Bra-Ket-Notation</a>), ist <a href="Zufall" title="Zufall">zufällig</a>. Die Wahrscheinlichkeit, mit der ein bestimmter Messwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> auftreten kann, ist gegeben durch die <a href="Wahrscheinlichkeitsverteilung" class="mw-redirect" title="Wahrscheinlichkeitsverteilung">Wahrscheinlichkeitsverteilung</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P[a]=\langle \Psi |\lambda _{A}(a)|\Psi \rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P[a]=\langle \Psi |\lambda _{A}(a)|\Psi \rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1f1a470d1013021bc0b75c29dbe2e2365957fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.946ex; height:2.843ex;" alt="{\displaystyle P[a]=\langle \Psi |\lambda _{A}(a)|\Psi \rangle }" loading="lazy"></span></dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{A}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{A}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a3bcf5b35ccd34136b0529d005ffa35e142f2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.509ex;" alt="{\displaystyle \lambda _{A}}" loading="lazy"></span> das <a href="Spektralma%C3%9F" title="Spektralmaß">Spektralmaß</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> nach dem <a href="Spektralsatz" title="Spektralsatz">Spektralsatz</a> bezeichnet.
</p><p>Wird der quantenmechanische Zustand des Systems allgemeiner durch einen <a href="Dichteoperator" title="Dichteoperator">Dichteoperator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> beschrieben, so ist die Wahrscheinlichkeitsverteilung des Messergebnisses gegeben durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P[a]=\operatorname {Spur} (\lambda _{A}(a)\,\rho )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>Spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P[a]=\operatorname {Spur} (\lambda _{A}(a)\,\rho )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a9ac3f513b88accd063f643754a2d39ff1bf09d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.414ex; height:2.843ex;" alt="{\displaystyle P[a]=\operatorname {Spur} (\lambda _{A}(a)\,\rho )}" loading="lazy"></span></dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spur} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Spur</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spur} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c4a170f5357bcbe7d169370100feaba1fd1319d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.789ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spur} }" loading="lazy"></span> die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spurabbildung</a> bezeichnet.
</p><p>Der <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>A</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle A\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/471a7c588a72faea10a4d02f6134e670467962a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.552ex; height:2.843ex;" alt="{\displaystyle \langle A\rangle }" loading="lazy"></span> des Messergebnisses, also der Erwartungswert der Wahrscheinlichkeitsverteilung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>, ist gegeben durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Psi |A|\Psi \rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle \Psi |A|\Psi \rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8ba02d71ed0803b7f908621fb89bd72c8fd88a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.462ex; height:2.843ex;" alt="{\displaystyle \langle \Psi |A|\Psi \rangle }" loading="lazy"></span> bzw. durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spur} (A\,\rho )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Spur</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mspace width="thinmathspace"></mspace>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spur} (A\,\rho )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0509c6b2a8fc19f7ece6b8ba5fbe570ae785d8ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.931ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spur} (A\,\rho )}" loading="lazy"></span>.
</p><p>Die <a href="Standardabweichung_(Wahrscheinlichkeitstheorie)" class="mw-redirect" title="Standardabweichung (Wahrscheinlichkeitstheorie)">Standardabweichung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{A}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{A}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/399e9d09fb7fc7fc1f286c97d3bde5ef6a814fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.792ex; height:2.009ex;" alt="{\displaystyle \sigma _{A}}" loading="lazy"></span>, auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebe8d9735d44ca6e2d6c3871c549f9e423231b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.679ex; height:2.176ex;" alt="{\displaystyle \Delta A}" loading="lazy"></span>, der Observablen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> ist die Wurzel aus der mittleren quadratischen Abweichung der Einzelwerte <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> vom Erwartungswert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>A</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle A\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/471a7c588a72faea10a4d02f6134e670467962a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.552ex; height:2.843ex;" alt="{\displaystyle \langle A\rangle }" loading="lazy"></span>, also
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{A}={\sqrt {\langle \Psi |(A-\langle A\rangle )^{2}|\Psi \rangle }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>A</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{A}={\sqrt {\langle \Psi |(A-\langle A\rangle )^{2}|\Psi \rangle }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aab19dd751cd086fbc6177882e0ff1efdce37106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.933ex; height:4.843ex;" alt="{\displaystyle \sigma _{A}={\sqrt {\langle \Psi |(A-\langle A\rangle )^{2}|\Psi \rangle }}}" loading="lazy"></span> bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {{\text{Spur}}((A-\langle A\rangle )^{2}\rho )}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Spur</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>A</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {{\text{Spur}}((A-\langle A\rangle )^{2}\rho )}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcec16a0053f113d3890058484268d34fad0abe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.124ex; height:4.843ex;" alt="{\displaystyle {\sqrt {{\text{Spur}}((A-\langle A\rangle )^{2}\rho )}}}" loading="lazy"></span></dd></dl>
<p>Im Spezialfall, dass das Spektrum von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> diskret und einfach ist, sind die möglichen Messergebnisse die <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>. Die Wahrscheinlichkeit, den Eigenwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> als Messergebnis zu finden, lautet dann <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle \phi _{a}|\Psi \rangle |^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<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 |\langle \phi _{a}|\Psi \rangle |^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57ceabd68df4c178640f5a9ccf6c2ee86cce3913.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.099ex; height:3.343ex;" alt="{\displaystyle |\langle \phi _{a}|\Psi \rangle |^{2}}" loading="lazy"></span> bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi _{a}|\rho \phi _{a}\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>ρ<!-- ρ --></mi>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle \phi _{a}|\rho \phi _{a}\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8996b9ca1b8b8a13499bfe3a1bc9689a10f57b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.633ex; height:2.843ex;" alt="{\displaystyle \langle \phi _{a}|\rho \phi _{a}\rangle }" loading="lazy"></span>, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{a}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi _{a}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb6c41e8e95dca929d19d147485f691f9764f0bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \phi _{a}}" loading="lazy"></span> einen normierten <a href="Eigenvektor" class="mw-redirect" title="Eigenvektor">Eigenvektor</a> zum Eigenwert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> bezeichnet.
</p><p><b>Beispiele:</b>
</p>
<ul><li>Der Observablen „Ort“ eines Teilchens in einer Dimension entspricht (in Ortsdarstellung) der Multiplikationsoperator mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> über dem <a href="Lp-Raum" title="Lp-Raum">Lebesgue-Raum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8722fb232f689925a4baa0e4ba478e43ee346672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}" loading="lazy"></span>, der <a href="Ortsoperator" title="Ortsoperator">Ortsoperator</a>.</li>
<li>Der Observablen „Impuls“ eines Teilchens in einer Dimension entspricht (in Ortsdarstellung) der <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathrm {i} \hbar {\tfrac {\mathrm {d} }{\mathrm {d} x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -\mathrm {i} \hbar {\tfrac {\mathrm {d} }{\mathrm {d} x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa7237248b75b0872534f2be956d23b2a7229dfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.452ex; height:3.843ex;" alt="{\displaystyle -\mathrm {i} \hbar {\tfrac {\mathrm {d} }{\mathrm {d} x}}}" loading="lazy"></span> über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8722fb232f689925a4baa0e4ba478e43ee346672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}" loading="lazy"></span>; genauer gesagt dessen selbstadjungierte <a href="Fortsetzung_(Mathematik)" title="Fortsetzung (Mathematik)">Fortsetzung</a>, der <a href="Impulsoperator" title="Impulsoperator">Impulsoperator</a>. Hierbei bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"></span> die <a href="Reduzierte_Planck-Konstante" class="mw-redirect" title="Reduzierte Planck-Konstante">reduzierte Planck-Konstante</a>.</li>
<li>Der Observablen „Energie“ entspricht der <a href="Hamiltonoperator" title="Hamiltonoperator">Hamiltonoperator</a>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Beschreibung_durch_POVM">Beschreibung durch POVM</h2></div>
<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Positive_Operator_Valued_Probability_Measure" title="Positive Operator Valued Probability Measure">Positive Operator Valued Probability Measure (POVM)</a></i></div>
<p>Die Beschreibung von Zeitmessungen passt nicht in den traditionellen von-Neumann’schen Formalismus, z. B. der Ankunftszeit eines Teilchens in einem <a href="Teilchendetektor" title="Teilchendetektor">Detektor</a>. Eine genauere realistische formale Modellierung realer Experimente zeigt, dass auch die meisten realen Messungen an Quantensystemen nicht genau durch von-Neumann’sche Observable beschrieben werden. Diese Defekte behebt die allgemeinere Beschreibung quantenmechanischer Observablen durch POVM.
</p>
<div class="mw-heading mw-heading2"><h2 id="Zusammenhang_mit_dem_Kommutator">Zusammenhang mit dem Kommutator</h2></div>
<p>Abhängig vom Wert ihres <a href="Kommutator_(Mathematik)" title="Kommutator (Mathematik)">Kommutators</a> (genauer: vom Wert des Kommutators ihrer Operatoren) bezeichnet man zwei Observable als:
</p>
<ul><li>kommutierende bzw. vertauschende Observable, wenn ihr Kommutator den Wert 0 hat. Sie sind <a href="Kommensurabilit%C3%A4t_(Quantenmechanik)" title="Kommensurabilität (Quantenmechanik)">kommensurabel</a>, d. h. sie können gleichzeitig beliebig genau gemessen werden. Siehe auch <a href="Vollst%C3%A4ndiger_Satz_kommutierender_Observablen" title="Vollständiger Satz kommutierender Observablen">Vollständiger Satz kommutierender Observablen</a>.</li>
<li>inkommensurable bzw. nicht vertauschende Observable, wenn ihr Kommutator einen Wert ungleich 0 hat; sie können <i>nicht</i> gleichzeitig beliebig genau gemessen werden.
<ul><li><a href="Komplement%C3%A4re_Observablen" title="Komplementäre Observablen">komplementäre Observablen</a>, wenn ihr Kommutator einen Wert von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} \hbar }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
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<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \hbar }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61935b6a2126471fe9bb7f1ab6033a73d3734def.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.953ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} \hbar }" loading="lazy"></span> aufweist. Ein Beispiel hierfür sind Ort und Impuls.</li></ul></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<div class="sieheauch" role="navigation" style="font-style:italic;"><span class="sieheauch-text">Siehe auch</span>: <a href="Mathematische_Formulierung_der_Quantenmechanik" title="Mathematische Formulierung der Quantenmechanik">Mathematische Formulierung der Quantenmechanik</a></div>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Jürgen Audretsch: <cite style="font-style:italic">Verschränkte Systeme</cite>. Wiley-VCH, Weinheim 2005, ISBN 3-527-40452-X.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Observable&rft.au=J%C3%BCrgen+Audretsch&rft.btitle=Verschr%C3%A4nkte+Systeme&rft.date=2005&rft.genre=book&rft.isbn=352740452X&rft.place=Weinheim&rft.pub=Wiley-VCH" style="display:none"> </span></span>
</li>
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