GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Variationsmethode (Quantenmechanik)</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Variationsmethode (Quantenmechanik)</span></h1>

</header>

</div>

<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>Die <b>Variationsmethode</b> ist in der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> ein <a href="N%C3%A4herungsverfahren" class="mw-redirect" title="Näherungsverfahren">Näherungsverfahren</a>, um eine obere Schranke für <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> einer quantenmechanischen <a href="Observable" title="Observable">Observablen</a> mit diskretem <a href="Spektrum_(Operatortheorie)" title="Spektrum (Operatortheorie)">Spektrum</a> zu finden.<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> Eine Verallgemeinerung der Methode führt auf das Min-Max-Prinzip.

</p><p>Eine verwandte Weiterentwicklung und Anwendung der klassischen Methode sind <i>variierte <a href="Quantenalgorithmus" title="Quantenalgorithmus">Quantenalgorithmen</a></i> (VAQ), um parametrisierte <a href="Quantenschaltung" title="Quantenschaltung">Quantenschaltkreise</a> zu trainieren. Der Ansatz hat das Potential, verschiedene Einschränkungen von Quantencomputern, z. B. <a href="Qubit" title="Qubit">Qubits</a> oder <a href="Rauschen_(Physik)" title="Rauschen (Physik)">Rauschen</a>, zu verbessern.<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><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>

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Grundzustand">Grundzustand</h3></div>

<p>Das Verfahren basiert darauf, dass der Eigenwert des Grundzustands eine untere Schranke für den <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> der <a href="Messung" title="Messung">Messung</a> der Observablen ist: Ist <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 g_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ce36142a0a1c6660e82bdf3ef3f1551317efe0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle g_{i}}" loading="lazy"></span> die Entartung eines Eigenwertes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, so lässt sich ein beliebiger Zustand als

</p>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\sum _{i}\sum _{j=1}^{g_{i}}c_{i,j}|\psi _{i,j}\rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/684dd79954e8b6f22d10d2e0db43f9a111877d88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.588ex; height:7.509ex;" alt="{\displaystyle |\psi \rangle =\sum _{i}\sum _{j=1}^{g_{i}}c_{i,j}|\psi _{i,j}\rangle }" loading="lazy"></span></dd></dl>

<p>schreiben, wobei die <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 _{i,j}\rangle }">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\psi _{i,j}\rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25097e7bf8c9b5d63c0621067ac606b3b0fd7bfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.999ex; height:3.009ex;" alt="{\displaystyle |\psi _{i,j}\rangle }" loading="lazy"></span> ein <a href="Vollst%C3%A4ndiges_Orthonormalsystem" class="mw-redirect" title="Vollständiges Orthonormalsystem">vollständiges Orthonormalsystem</a> bilden. Für den Erwartungswert des Zustands bei Messung einer 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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> mit Eigenwerten <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}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" loading="lazy"></span> gilt dann

</p>

</mrow>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \psi |H|\psi \rangle =\sum _{i}\sum _{j=1}^{g_{i}}E_{i}|c_{i,j}|^{2}\geq E_{0}\sum _{i}\sum _{j=1}^{g_{i}}|c_{i,j}|^{2}=E_{0}\langle \psi |\psi \rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e43d1f66f44422ab5f8035b60c7887f19f8c30e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.959ex; height:7.509ex;" alt="{\displaystyle \langle \psi |H|\psi \rangle =\sum _{i}\sum _{j=1}^{g_{i}}E_{i}|c_{i,j}|^{2}\geq E_{0}\sum _{i}\sum _{j=1}^{g_{i}}|c_{i,j}|^{2}=E_{0}\langle \psi |\psi \rangle }" loading="lazy"></span>.</dd></dl>

<p>Es lässt sich demnach eine obere Schranke für <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_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/411d268de7b1cf300d7481e3fe59f3b20887e0d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}" loading="lazy"></span> finden, wenn man für eine Schar von Zuständen <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 _{\alpha }\rangle }">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83a41aab295bb080a85fb405cc41f41ba481c886.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.349ex; height:2.843ex;" alt="{\displaystyle |\psi _{\alpha }\rangle }" loading="lazy"></span> den Erwartungswert berechnet und das Infimum sucht:

</p>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{0}\leq \inf _{\alpha }{\frac {\langle \psi _{\alpha }|H|\psi _{\alpha }\rangle }{\langle \psi _{\alpha }|\psi _{\alpha }\rangle }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e9b6b28502458d32d905941849f79d26eca06a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.504ex; height:6.509ex;" alt="{\displaystyle E_{0}\leq \inf _{\alpha }{\frac {\langle \psi _{\alpha }|H|\psi _{\alpha }\rangle }{\langle \psi _{\alpha }|\psi _{\alpha }\rangle }}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Angeregte_Zustände"><span id="Angeregte_Zust.C3.A4nde"></span>Angeregte Zustände</h3></div>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\psi _{0}\rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6c1c429af2a5f67ca03dabc80a872dfd9a3768b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.119ex; height:2.843ex;" alt="{\displaystyle |\psi _{0}\rangle }" loading="lazy"></span> die Eigenfunktion zu einem (nicht entarteten) Grundzustand mit 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 E_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/411d268de7b1cf300d7481e3fe59f3b20887e0d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}" loading="lazy"></span>, so lässt sich für einen beliebigen Zustand <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 }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc27f1893b769a08cd6b296e115a29e61cab675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }" loading="lazy"></span> schreiben

</p>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H|\psi \rangle =c_{0}E_{0}|\psi _{0}\rangle +\varepsilon |\varphi \rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0bdc94b12d0f3ad0abf77402e8691968ec56272.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.172ex; height:2.843ex;" alt="{\displaystyle H|\psi \rangle =c_{0}E_{0}|\psi _{0}\rangle +\varepsilon |\varphi \rangle }" loading="lazy"></span>,</dd></dl>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\varphi \rangle \perp |\psi _{0}\rangle }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5690845fcf5897b86872e0006b4730f0c7ae5551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.289ex; height:2.843ex;" alt="{\displaystyle |\varphi \rangle \perp |\psi _{0}\rangle }" loading="lazy"></span>. Zerlegt man <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 |\varphi \rangle }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66b05bc04322b5f34a11c4d6985ac7e350d6f75e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.072ex; height:2.843ex;" alt="{\displaystyle |\varphi \rangle }" loading="lazy"></span> wie oben in Eigenzustände, erhält man unter der Nebenbedingung <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 \varphi |\psi _{0}\rangle =0}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \varphi |\psi _{0}\rangle =0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07816d503a214969e63fea9df6f2c04d2b25f83d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle \langle \varphi |\psi _{0}\rangle =0}" loading="lazy"></span>

</p>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1}\leq \inf _{\alpha }{\frac {\langle \varphi _{\alpha }|H|\varphi _{\alpha }\rangle }{\langle \varphi _{\alpha }|\varphi _{\alpha }\rangle }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5758e96c07c1567afbf10e140a643f922e12ebd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.518ex; height:6.509ex;" alt="{\displaystyle E_{1}\leq \inf _{\alpha }{\frac {\langle \varphi _{\alpha }|H|\varphi _{\alpha }\rangle }{\langle \varphi _{\alpha }|\varphi _{\alpha }\rangle }}}" loading="lazy"></span>,</dd></dl>

<p>da in der Summe der Wert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31a682d568ee6a5fe51d76423186057f625ada5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=0}" loading="lazy"></span> fehlt.

</p><p>Die Suche nach weiteren Eigenzuständen erfolgt analog, wobei dann unter Orthogonalität zu mehreren Teilräumen, die die niedrigeren Eigenwerte aufspannen, zu minimieren ist.

</p>

<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="Variationsrechnung" title="Variationsrechnung">Variationsrechnung</a> und <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a></div>

<div class="mw-heading mw-heading3"><h3 id="Klassiker_oder_ältere_Werke"><span id="Klassiker_oder_.C3.A4ltere_Werke"></span>Klassiker oder ältere Werke</h3></div>

<ul><li><a href="P%C3%A1l_Gomb%C3%A1s" title="Pál Gombás">P. Gombás</a>: <cite style="font-style:italic">Theorie und Lösungsmethoden des Mehrteilchenproblems der Wellenmechanik</cite>. Birkhäuser Basel, Basel 1950, ISBN 978-3-0348-6957-7, <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/10.1007/978-3-0348-6956-0">10.1007/978-3-0348-6956-0</a></span>.<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:Variationsmethode+%28Quantenmechanik%29&rft.au=P.+Gomb%C3%A1s&rft.btitle=Theorie+und+L%C3%B6sungsmethoden+des+Mehrteilchenproblems+der+Wellenmechanik&rft.date=1950&rft.doi=10.1007%2F978-3-0348-6956-0&rft.genre=book&rft.isbn=9783034869577&rft.place=Basel&rft.pub=Birkh%C3%A4user+Basel" style="display:none"> </span></li>

<li><a href="Wolfgang_Yourgrau" title="Wolfgang Yourgrau">Wolfgang Yourgrau</a>, <a href="Stanley_Mandelstam" title="Stanley Mandelstam">Stanley Mandelstam</a>: <cite style="font-style:italic">Variational Principles in Dynamics and Quantum Theory</cite>. Dover Publications, New York 1979, ISBN 978-0-486-63773-0.<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:Variationsmethode+%28Quantenmechanik%29&rft.au=Wolfgang+Yourgrau%2C+Stanley+Mandelstam&rft.btitle=Variational+Principles+in+Dynamics+and+Quantum+Theory&rft.date=1979&rft.genre=book&rft.isbn=9780486637730&rft.place=New+York&rft.pub=Dover+Publications" style="display:none"> </span></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">P. Gombás: <cite style="font-style:italic">Theorie und Lösungsmethoden des Mehrteilchenproblems der Wellenmechanik</cite>. Birkhäuser Basel, Basel 1950, ISBN 978-3-0348-6957-7, <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/10.1007/978-3-0348-6956-0">10.1007/978-3-0348-6956-0</a></span> (<a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007%2F978-3-0348-6956-0">springer.com</a> [abgerufen am 24. Januar 2023]).<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:Variationsmethode+%28Quantenmechanik%29&rft.au=P.+Gomb%C3%A1s&rft.btitle=Theorie+und+L%C3%B6sungsmethoden+des+Mehrteilchenproblems+der+Wellenmechanik&rft.date=1950&rft.doi=10.1007%2F978-3-0348-6956-0&rft.genre=book&rft.isbn=9783034869577&rft.place=Basel&rft.pub=Birkh%C3%A4user+Basel" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, Patrick J. Coles: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Variational quantum algorithms</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Nature Reviews Physics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>3</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>9</span>, 12. August 2021, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%222522-5820%22&key=cql">2522-5820</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>625–644</span>, <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/10.1038/s42254-021-00348-9">10.1038/s42254-021-00348-9</a></span> (englisch, <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s42254-021-00348-9">nature.com</a> [abgerufen am 30. Januar 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsmethode+%28Quantenmechanik%29&rft.atitle=Variational+quantum+algorithms&rft.au=M.+Cerezo%2C+Andrew+Arrasmith%2C+Ryan+Babbush%2C+...&rft.date=2021-08-12&rft.doi=10.1038%2Fs42254-021-00348-9&rft.genre=journal&rft.issn=2522-5820&rft.issue=9&rft.jtitle=Nature+Reviews+Physics&rft.pages=625-644&rft.volume=3" style="display:none"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, Jeremy L. O’Brien: <cite class="lang" lang="en" dir="auto" style="font-style:italic">A variational eigenvalue solver on a photonic quantum processor</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Nature Communications</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>5</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, 23. Juli 2014, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%222041-1723%22&key=cql">2041-1723</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>4213</span>, <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/10.1038/ncomms5213">10.1038/ncomms5213</a></span> (englisch, <a rel="nofollow" class="external text" href="https://www.nature.com/articles/ncomms5213">nature.com</a> [abgerufen am 30. Januar 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsmethode+%28Quantenmechanik%29&rft.atitle=A+variational+eigenvalue+solver+on+a+photonic+quantum+processor&rft.au=Alberto+Peruzzo%2C+Jarrod+McClean%2C+Peter+Shadbolt%2C+...&rft.date=2014-07-23&rft.doi=10.1038%2Fncomms5213&rft.genre=journal&rft.issn=2041-1723&rft.issue=1&rft.jtitle=Nature+Communications&rft.pages=4213&rft.volume=5" style="display:none"> </span></span>

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2023-02-02" href="https://de.wikipedia.org/wiki/?title=Variationsmethode_(Quantenmechanik)&oldid=230463292">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>