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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Jordan-Wigner-Transformation</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>Mithilfe der <b>Jordan-Wigner-Transformation</b> können verschiedene eindimensionale <a href="Quantenmechanik" title="Quantenmechanik">quantenmechanische</a> Systeme aufeinander abgebildet werden. Genauer gesagt ist es möglich mit der Transformation eindimensionale <a href="Spin" title="Spin">Spin</a>-1/2-Ketten auf <a href="Fermion" title="Fermion">Fermionen</a> auf einer Kette abzubilden.
</p><p>Die Jordan-Wigner-Transformation bildet die <a href="Drehimpulsoperator#Spinoperator" class="mw-redirect" title="Drehimpulsoperator">Spin-1/2-Operatoren</a> und ihre Algebra (Algebra der <a href="Pauli-Matrix" class="mw-redirect" title="Pauli-Matrix">Pauli-Matrizen</a>) auf <a href="Erzeugungs-_und_Vernichtungsoperator" title="Erzeugungs- und Vernichtungsoperator">Erzeugungs- und Vernichtungsoperatoren</a> für Fermionen und deren Algebra ab. Mithilfe der Transformation kann die Äquivalenz zwischen dem <a href="Heisenbergmodell#1D-Heisenbergmodell" class="mw-redirect" title="Heisenbergmodell">eindimensionalen Heisenbergmodell</a> und Fermionen auf einem eindimensionalen Gitter mit nächster Nachbarwechselwirkung gezeigt werden.
</p><p>Die Transformation wurde 1928 von <a href="Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a> und <a href="Eugene_Wigner" class="mw-redirect" title="Eugene Wigner">Eugene Wigner</a> in der <i>Zeitschrift für Physik</i> veröffentlicht<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>. 1961 benutzten <a href="Elliott_Lieb" title="Elliott Lieb">Elliott Lieb</a>, T. Schultz, D. Mattis die Transformation bei der Einführung ihres exakt lösbaren eindimensionalen Spin-1/2-xy-Modells.<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>Die Jordan-Wigner-Transformation wurde auch auf zweidimensionale Spin-Systeme angewandt<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> und auf dreidimensionale Systeme. Die Anwendung auf zweidimensionale Systeme wurde als einer der Ersten von <a href="Eduardo_Fradkin" title="Eduardo Fradkin">Eduardo Fradkin</a> 1989 diskutiert.
</p><p><a href="Elliott_Lieb" title="Elliott Lieb">Elliott Lieb</a>, T. Schultz, <a href="Daniel_Mattis" title="Daniel Mattis">Daniel Mattis</a> wandten die Transformation 1964 auf die Transfermatrix im zweidimensionalen <a href="Isingmodell" class="mw-redirect" title="Isingmodell">Isingmodell</a> an und leiteten damit die zuvor von <a href="Lars_Onsager" title="Lars Onsager">Lars Onsager</a> gefundene exakte Lösung ab.<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Grundlegende_Idee">Grundlegende Idee</h2></div>
<p>Betrachtet man <a href="Drehimpulsoperator#Spinoperator" class="mw-redirect" title="Drehimpulsoperator">Spin-1/2-Operatoren</a> am Platz <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 j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>, so findet man, dass diese den grundlegenden <a href="Fockraum" title="Fockraum">kanonischen (Anti-)Vertauschungsrelationen</a> (Anti-Kommutatorrelationen) für Fermionen gehorchen:
</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 \{S_{j}^{+},S_{j}^{-}\}=1,\qquad \{S_{j}^{+},S_{j}^{+}\}=0=\{S_{j}^{-},S_{j}^{-}\},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>j</mi>
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<mo>=</mo>
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<mi>j</mi>
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<mo>−<!-- − --></mo>
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<mo>,</mo>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
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</msubsup>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{S_{j}^{+},S_{j}^{-}\}=1,\qquad \{S_{j}^{+},S_{j}^{+}\}=0=\{S_{j}^{-},S_{j}^{-}\},}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad1318915c74af5be71013ef439ee05c8ad5b06e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:46.217ex; height:3.509ex;" alt="{\displaystyle \{S_{j}^{+},S_{j}^{-}\}=1,\qquad \{S_{j}^{+},S_{j}^{+}\}=0=\{S_{j}^{-},S_{j}^{-}\},}" 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 \{A,B\}=AB+BA}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mi>A</mi>
<mi>B</mi>
<mo>+</mo>
<mi>B</mi>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{A,B\}=AB+BA}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8abd671fa759b555a01dd3e44d4ff372dee37c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.819ex; height:2.843ex;" alt="{\displaystyle \{A,B\}=AB+BA}" loading="lazy"></span>. Die Idee ist daher, die Spin-1/2-Operatoren als fermionische Operatoren zu betrachten. Allerdings erfüllen die Spin-1/2-Operatoren keine Anti-Kommutatorrelationen, sondern Kommutatorrelationen auf verschiedenen Gitterplätzen <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 j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> und <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 k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>k</mi>
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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>:
</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 [S_{j}^{+},S_{k}^{-}]=0=[S_{j}^{+},S_{k}^{+}]=[S_{j}^{-},S_{k}^{-}],}">
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<mo>−<!-- − --></mo>
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<mi>j</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
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<mo>,</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>+</mo>
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</msubsup>
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<mo>=</mo>
<mo stretchy="false">[</mo>
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<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
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<mo>,</mo>
<msubsup>
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<mo stretchy="false">]</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [S_{j}^{+},S_{k}^{-}]=0=[S_{j}^{+},S_{k}^{+}]=[S_{j}^{-},S_{k}^{-}],}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/626e2e508ae4f956531a0ae9a7dece1e684ab0f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:36.282ex; height:3.509ex;" alt="{\displaystyle [S_{j}^{+},S_{k}^{-}]=0=[S_{j}^{+},S_{k}^{+}]=[S_{j}^{-},S_{k}^{-}],}" 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 [A,B]=AB-BA}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>A</mi>
<mi>B</mi>
<mo>−<!-- − --></mo>
<mi>B</mi>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [A,B]=AB-BA}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a3b93b316dd0b6b0ab2c71e486c901ddfe6e79a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.788ex; height:2.843ex;" alt="{\displaystyle [A,B]=AB-BA}" loading="lazy"></span>.
</p><p>Jordan und Wigner haben erkannt, dass dies jedoch mit der Einführung eines Phasenoperators vor den Spin-1/2-Operatoren behoben werden kann. Es wird eine Wegorientierung definiert mit einem Phasenfaktor, der abhängig von der Anzahl der Up-Spins vor dem betrachteten Spin ist.
</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 c_{j}=e^{i\phi _{j}}S_{j}^{-}\qquad {\text{mit}}\quad \phi _{j}=\pi \sum _{k<j}S_{k}^{+}S_{k}^{-}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
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<msup>
<mi>e</mi>
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<mi>ϕ<!-- ϕ --></mi>
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<mi>j</mi>
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<msubsup>
<mi>S</mi>
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<mi>j</mi>
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<mspace width="2em"></mspace>
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<mtext>mit</mtext>
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<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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<munder>
<mo>∑<!-- ∑ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle c_{j}=e^{i\phi _{j}}S_{j}^{-}\qquad {\text{mit}}\quad \phi _{j}=\pi \sum _{k<j}S_{k}^{+}S_{k}^{-}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83977b786c0513db538509b623deb00f49225c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.999ex; height:5.843ex;" alt="{\displaystyle c_{j}=e^{i\phi _{j}}S_{j}^{-}\qquad {\text{mit}}\quad \phi _{j}=\pi \sum _{k<j}S_{k}^{+}S_{k}^{-}}" loading="lazy"></span></dd></dl>
<p>Ist an der Stelle <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 j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> ein Up-Spin, wird ein Phasenfaktor (−1) „aufgepickt“, bei einem Down-Spin passiert nichts (Phasenfaktor 1):
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\pi S_{j}^{+}S_{j}^{-}}=e^{i\pi n_{j}}=1-2n_{j}\qquad {\text{mit}}\quad n_{j}=S_{j}^{+}S_{j}^{-}}">
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<mrow class="MJX-TeXAtom-ORD">
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</msubsup>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msubsup>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>π<!-- π --></mi>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mn>2</mn>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mspace width="2em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>mit</mtext>
</mrow>
<mspace width="1em"></mspace>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msubsup>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{i\pi S_{j}^{+}S_{j}^{-}}=e^{i\pi n_{j}}=1-2n_{j}\qquad {\text{mit}}\quad n_{j}=S_{j}^{+}S_{j}^{-}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d9271a37707e6e526990930e60c84012b780d40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:47.497ex; height:4.176ex;" alt="{\displaystyle e^{i\pi S_{j}^{+}S_{j}^{-}}=e^{i\pi n_{j}}=1-2n_{j}\qquad {\text{mit}}\quad n_{j}=S_{j}^{+}S_{j}^{-}}" loading="lazy"></span>
</p><p>Die so definierten fermionischen Operatoren erfüllen die Anti-Kommutatorrelationen auf verschiedenen Plätzen <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 j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> und <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 k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>:
</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 \{c_{j},c_{k}^{\dagger }\}=\delta _{jk},\qquad \{c_{j}^{\dagger },c_{k}^{\dagger }\}=0=\{c_{j},c_{k}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
</mrow>
</msubsup>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="2em"></mspace>
<mo fence="false" stretchy="false">{</mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
</mrow>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
</mrow>
</msubsup>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mn>0</mn>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{c_{j},c_{k}^{\dagger }\}=\delta _{jk},\qquad \{c_{j}^{\dagger },c_{k}^{\dagger }\}=0=\{c_{j},c_{k}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5abe37c2d2664371f100d875dc7e06beaecaedca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.101ex; height:3.843ex;" alt="{\displaystyle \{c_{j},c_{k}^{\dagger }\}=\delta _{jk},\qquad \{c_{j}^{\dagger },c_{k}^{\dagger }\}=0=\{c_{j},c_{k}\}}" loading="lazy"></span></dd></dl>
<p>Besonders hilfreich sind folgende Zusammenhänge für die Abbildung zwischen verschiedenen Modellen:
</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 S_{j}^{+}S_{j+1}^{-}=\pm c_{j}^{\dagger }c_{j+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msubsup>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msubsup>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
</mrow>
</msubsup>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{j}^{+}S_{j+1}^{-}=\pm c_{j}^{\dagger }c_{j+1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87010cf947afdccd325cdd8ea1ea8b07bce21489.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.36ex; height:3.843ex;" alt="{\displaystyle S_{j}^{+}S_{j+1}^{-}=\pm c_{j}^{\dagger }c_{j+1}}" loading="lazy"></span></dd>
<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 S_{z}=S_{j}^{+}S_{j}^{-}-{\frac {1}{2}}=c_{j}^{\dagger }c_{j}-{\frac {1}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo>=</mo>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msubsup>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
</mrow>
</msubsup>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{z}=S_{j}^{+}S_{j}^{-}-{\frac {1}{2}}=c_{j}^{\dagger }c_{j}-{\frac {1}{2}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00ff83c162fe834371a34ee40bfe45cda822179e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.252ex; height:5.176ex;" alt="{\displaystyle S_{z}=S_{j}^{+}S_{j}^{-}-{\frac {1}{2}}=c_{j}^{\dagger }c_{j}-{\frac {1}{2}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>
<div class="mw-heading mw-heading3"><h3 id="1D-Heisenberg-Modell">1D-Heisenberg-Modell</h3></div>
<p>Zur Veranschaulichung der Jordan-Wigner-Transformation wird sie auf das <a href="Heisenbergmodell#1D-Heisenbergmodell" class="mw-redirect" title="Heisenbergmodell">eindimensionale Heisenberg-Modell</a> angewandt. Die nötigen Produkte der verschiedenen Operatoren sind bereits im vorherigen Abschnitt aufgelistet. Der Hamiltonian <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 H_{\text{Heis}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Heis</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{\text{Heis}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/641c5b257d6c530ed66e1a2d247ffe9810aa3803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.231ex; height:2.509ex;" alt="{\displaystyle H_{\text{Heis}}}" loading="lazy"></span> des 1D-Heisenberg Modells kann demnach geschrieben werden als:
</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 {\begin{aligned}H_{\text{Heis}}&=-J\sum _{n=1}^{N}{\vec {S}}_{n}\cdot {\vec {S}}_{n+1}=-J\sum _{n=1}^{N}\left[{\frac {1}{2}}(S_{n}^{+}S_{n+1}^{-}+S_{n}^{-}S_{n+1}^{+})+S_{n}^{z}S_{n+1}^{z}\right]\\&=-J\sum _{i=1}^{N}\left[{\frac {1}{2}}\left(c_{i}^{\dagger }c_{i+1}+{\text{h.c}}\right)+\left((c_{i}^{\dagger }c_{i}-{\frac {1}{2}})(c_{i+1}^{\dagger }c_{i+1}-{\frac {1}{2}})\right)\right]\\&=H_{0}+H_{J}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Heis</mtext>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>J</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</munderover>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>S</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>S</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
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<mo>−<!-- − --></mo>
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<mrow class="MJX-TeXAtom-ORD">
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<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</msubsup>
<msubsup>
<mi>S</mi>
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<mi>n</mi>
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<mrow>
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<mrow class="MJX-TeXAtom-ORD">
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<msubsup>
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<mrow class="MJX-TeXAtom-ORD">
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<mtr>
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<mtd>
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<mrow class="MJX-TeXAtom-ORD">
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<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>J</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{\text{Heis}}&=-J\sum _{n=1}^{N}{\vec {S}}_{n}\cdot {\vec {S}}_{n+1}=-J\sum _{n=1}^{N}\left[{\frac {1}{2}}(S_{n}^{+}S_{n+1}^{-}+S_{n}^{-}S_{n+1}^{+})+S_{n}^{z}S_{n+1}^{z}\right]\\&=-J\sum _{i=1}^{N}\left[{\frac {1}{2}}\left(c_{i}^{\dagger }c_{i+1}+{\text{h.c}}\right)+\left((c_{i}^{\dagger }c_{i}-{\frac {1}{2}})(c_{i+1}^{\dagger }c_{i+1}-{\frac {1}{2}})\right)\right]\\&=H_{0}+H_{J}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8386bb726e804c82a42065a4e334f5395c411564.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:71.174ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}H_{\text{Heis}}&=-J\sum _{n=1}^{N}{\vec {S}}_{n}\cdot {\vec {S}}_{n+1}=-J\sum _{n=1}^{N}\left[{\frac {1}{2}}(S_{n}^{+}S_{n+1}^{-}+S_{n}^{-}S_{n+1}^{+})+S_{n}^{z}S_{n+1}^{z}\right]\\&=-J\sum _{i=1}^{N}\left[{\frac {1}{2}}\left(c_{i}^{\dagger }c_{i+1}+{\text{h.c}}\right)+\left((c_{i}^{\dagger }c_{i}-{\frac {1}{2}})(c_{i+1}^{\dagger }c_{i+1}-{\frac {1}{2}})\right)\right]\\&=H_{0}+H_{J}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Die Transformation zeigt also die Äquivalenz des 1D-Heisenberg Modells mit spinlosen Fermionen auf dem Gitter mit periodischen Randbedingungen und lediglich nächster Nachbarwechselwirkung. Der erste Term <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 H_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43910602a221b7a4c373791f94793e3008622070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}" loading="lazy"></span> beschreibt wechselwirkungsfreie Fermionen und der zweite Term <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 H_{J}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>J</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{J}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30fde27785df33dcc9bf641803afc26d15f8c575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.204ex; height:2.509ex;" alt="{\displaystyle H_{J}}" loading="lazy"></span> ist der Wechselwirkungsterm mit einer Wechselwirkung <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 U=-J}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U=-J}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b049c7e7f83ec6a5dcb42990137f8a5dd7fa4e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.161ex; height:2.343ex;" alt="{\displaystyle U=-J}" loading="lazy"></span> gegeben über die <a href="Kopplungskonstante" title="Kopplungskonstante">Kopplungskonstante</a> des Heisenbergmodells.
</p>
<div class="mw-heading mw-heading3"><h3 id="1D-XY-Modell">1D-XY-Modell</h3></div>
<p>Ein weiteres Beispiel ist das eindimensionale <a href="XY-Modell" title="XY-Modell">XY-Modell</a> als Spezialfall des 1D-Heisenberg-Modells. Der Hamiltonian <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 H_{\text{Heis}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Heis</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{\text{Heis}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/641c5b257d6c530ed66e1a2d247ffe9810aa3803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.231ex; height:2.509ex;" alt="{\displaystyle H_{\text{Heis}}}" loading="lazy"></span> des XY-Modells kann geschrieben werden als:
</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 {\begin{aligned}H_{\text{XY}}&=-J\sum _{n=1}^{N}\left[{\frac {1}{2}}(S_{n}^{+}S_{n+1}^{-}+S_{n}^{-}S_{n+1}^{+})\right]\\&=-J\sum _{i=1}^{N}{\frac {1}{2}}\left(c_{i}^{\dagger }c_{i+1}+c_{i+1}^{\dagger }c_{i}\right)=H_{0}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
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<msub>
<mi>H</mi>
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<mtext>XY</mtext>
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</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>J</mi>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
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<mo>[</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
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</msubsup>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mo>−<!-- − --></mo>
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</msubsup>
<msubsup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
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<mo>]</mo>
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<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
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<munderover>
<mo>∑<!-- ∑ --></mo>
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<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
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</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
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<mrow>
<mo>(</mo>
<mrow>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
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</msubsup>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>+</mo>
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<mo>+</mo>
<msubsup>
<mi>c</mi>
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<mi>i</mi>
<mo>+</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
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<msub>
<mi>c</mi>
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<mi>i</mi>
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<mo>)</mo>
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<mo>=</mo>
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<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{\text{XY}}&=-J\sum _{n=1}^{N}\left[{\frac {1}{2}}(S_{n}^{+}S_{n+1}^{-}+S_{n}^{-}S_{n+1}^{+})\right]\\&=-J\sum _{i=1}^{N}{\frac {1}{2}}\left(c_{i}^{\dagger }c_{i+1}+c_{i+1}^{\dagger }c_{i}\right)=H_{0}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3def6e1c871dc7cab6c36e0d4ada8018da045205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:41.562ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}H_{\text{XY}}&=-J\sum _{n=1}^{N}\left[{\frac {1}{2}}(S_{n}^{+}S_{n+1}^{-}+S_{n}^{-}S_{n+1}^{+})\right]\\&=-J\sum _{i=1}^{N}{\frac {1}{2}}\left(c_{i}^{\dagger }c_{i+1}+c_{i+1}^{\dagger }c_{i}\right)=H_{0}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Die Jordan-Wigner Transformation bildet das Spin-System also auf wechselwirkungsfreie spinlose Fermionen ab. Für dieses System kann man die <a href="Zustandssumme" title="Zustandssumme">Zustandssumme</a> exakt angeben.
</p>
<div class="mw-heading mw-heading3"><h3 id="Quanteninformationstheorie">Quanteninformationstheorie</h3></div>
<p>Die Transformation wurde in der <a href="Quanteninformatik" title="Quanteninformatik">Quanteninformationstheorie</a> benutzt, um ein System wechselwirkender <a href="Qubit" title="Qubit">Qubits</a> auf ein äquivalentes System wechselwirkender Fermionen abzubilden und umgekehrt.<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> Außerdem konnte damit durch <a href="Raymond_Laflamme" title="Raymond Laflamme">Raymond Laflamme</a> und Kollegen<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> das Problem der Simulation fermionischer quantenmechanischer Systeme in Quantencomputern gelöst werden, ein Problem das in der Pionierarbeit von <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a> von 1982<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> noch offen war.
</p>
<div class="mw-heading mw-heading2"><h2 id="Quellen">Quellen</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">P. Jordan and E. Wigner, <i>Über das Paulische Äquivalenzverbot</i>, Zeitschrift für Physik 47, No. 9. (1928), pp. 631–651, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01331938">10.1007/BF01331938</a></span>.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Lieb, Schultz, Mattis, Annals of Physics, Band 16, 1961, S. 407</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Oleg Derzho, Jordan-Wigner fermionization for spin-1/2 systems in two dimensions: A brief review, Journal of Physical Studies, Band 5, 2001, S. 49–64, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0101188">Arxiv</a></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Lieb, Schultz, Mattis, Review of Modern Physics, Band 36, 1964, S. 856</span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><a href="Michael_Nielsen" title="Michael Nielsen">Michael Nielsen</a>, The fermionic canonical commutation relations and the Jordan-Wigner transform, 2005 <a rel="nofollow" class="external text" href="http://michaelnielsen.org/blog/complete-notes-on-fermions-and-the-jordan-wigner-transform/">Online</a> als <i>Complete notes on fermions and the Jordan-Wigner transform.</i> </span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">R. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, R. Laflamme, <i>Simulating physical phenomena by quantum networks</i>, Physical Review A, Band 65, 2002, S. 042323, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0108146">Arxiv</a></span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Richard Feynman, Simulating physics with computers, Int. J. Theor. Phys., Band 21, 1982, S. 467–488</span>
</li>
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