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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">Der Impulsoperator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bd4c026f1b3413adc58b9b65e89e62bce92c85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}" loading="lazy"> ist in der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> der <a href="Operator_(Mathematik)" title="Operator (Mathematik)">Operator</a> zur <a href="Impuls" title="Impuls">Impuls</a>messung von <a href="Teilchen" title="Teilchen">Teilchen</a>. In der <a href="Ortsdarstellung" class="mw-redirect" title="Ortsdarstellung">Ortsdarstellung</a> ist der Impulsoperator in einer <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> gegeben durch:

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<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}_{x}=-\mathrm {i} \hbar {\frac {\partial }{\partial x}}={\frac {\mathrm {\hbar } }{i}}{\frac {\partial }{\partial x}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c854ccf5c92db0db0cc46d1d30be7bb51c99fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:21.69ex; height:5.509ex;" alt="{\displaystyle {\hat {p}}_{x}=-\mathrm {i} \hbar {\frac {\partial }{\partial x}}={\frac {\mathrm {\hbar } }{i}}{\frac {\partial }{\partial x}}}" loading="lazy"></dd></dl>

Dabei bezeichnet

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18f0f09f6fc40e634d34aed6e205ac0f7a40e062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }" loading="lazy"> die <a href="Imagin%C3%A4re_Einheit" class="mw-redirect" title="Imaginäre Einheit">Imaginäre Einheit</a></li>

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

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</math><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"> die <a href="Planck-Konstante#Reduziertes_Plancksches_Wirkungsquantum" title="Planck-Konstante">reduzierte Planck-Konstante</a> und</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48d5ed7cbe913e8e308fbe55483bd0549928db5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.484ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial x}}}" loading="lazy"> die <a href="Partielle_Ableitung" title="Partielle Ableitung">partielle Ableitung</a> in Richtung der Ortskoordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><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">.</li></ul>

Mit dem <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"> erhält man in drei Dimensionen den <a href="Vektor" title="Vektor">Vektor</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {p} }}=-\mathrm {i} \hbar \nabla }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eea20f77369d9ded9b8e3d488f01192f4e355c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.281ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {p} }}=-\mathrm {i} \hbar \nabla }" loading="lazy"></dd></dl>

Der physikalische <a href="Quantenmechanischer_Zustand" class="mw-redirect" title="Quantenmechanischer Zustand">Zustand</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi \,}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bb35f42ebab064f83f8a9dda692b3172f2ef437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.195ex; height:2.176ex;" alt="{\displaystyle \Psi \,}" loading="lazy"> eines Teilchens ist in der Quantenmechanik mathematisch durch einen zugehörigen Vektor eines <a href="Hilbertraum" title="Hilbertraum">Hilbertraumes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

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

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</math><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"> gegeben. Dieser Zustand wird folglich in der <a href="Bra-Ket" class="mw-redirect" title="Bra-Ket">Bra-Ket-Notation</a> durch den Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e77f6b1e903837c5765c9683da41dd93199621c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }" loading="lazy"> beschrieben. Die <a href="Observable" title="Observable">Observablen</a> werden durch <a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">selbstadjungierte Operatoren</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

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

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</math><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"> dargestellt. Speziell ist der Impuls-Operator die Zusammenfassung der drei Observablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {p} }}=({\hat {p}}_{1},{\hat {p}}_{2},{\hat {p}}_{3})}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {p} }}=({\hat {p}}_{1},{\hat {p}}_{2},{\hat {p}}_{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dea49175749c4322b1a8efbe7c0307774c9df8b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.702ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {p} }}=({\hat {p}}_{1},{\hat {p}}_{2},{\hat {p}}_{3})}" loading="lazy">, so dass

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<annotation encoding="application/x-tex">{\displaystyle E({\hat {p}}_{j})=\langle \Psi |{\hat {p}}_{j}\,|\Psi \rangle \,\quad j=1,2,3}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04dba6daa151d9e1e78620271c269361dc03a821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.649ex; height:3.176ex;" alt="{\displaystyle E({\hat {p}}_{j})=\langle \Psi |{\hat {p}}_{j}\,|\Psi \rangle \,\quad j=1,2,3}" loading="lazy"></dd></dl>

der Mittelwert (<a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a>) der Messergebnisse der j-ten Komponente des Impulses des Teilchens im Zustand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> ist.

<div class="mw-heading mw-heading2"><h2 id="Definition_und_Eigenschaften">Definition und Eigenschaften</h2></div>

<ul><li>Bei der <a href="Quantisierung_(Physik)#Quantenmechanik_(ab_1925)" title="Quantisierung (Physik)">kanonischen Quantisierung</a> deutet man die <a href="Phasenraum" title="Phasenraum">Phasenraum</a>koordinaten, also den Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><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"> und den Impuls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> des klassischen Systems, als selbstadjungierte Operatoren eines Hilbertraums und fordert für diese <a href="Ortsoperator" title="Ortsoperator">Orts-</a> und Impulsoperatoren die <a href="Kanonische_Vertauschungsrelation" title="Kanonische Vertauschungsrelation">kanonischen Vertauschungsrelationen</a>:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle [{\hat {x}}_{i},{\hat {p}}_{j}]=\mathrm {i} \,\hbar \,\delta _{ij}\,\quad [{\hat {x}}_{i},{\hat {x}}_{j}]=0=[{\hat {p}}_{i},{\hat {p}}_{j}]\ ,\quad i,j\in \{1,2,3\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1a04353a3455e6257b7d50cb3a22e87e5a4ddf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:56.035ex; height:3.176ex;" alt="{\displaystyle [{\hat {x}}_{i},{\hat {p}}_{j}]=\mathrm {i} \,\hbar \,\delta _{ij}\,\quad [{\hat {x}}_{i},{\hat {x}}_{j}]=0=[{\hat {p}}_{i},{\hat {p}}_{j}]\ ,\quad i,j\in \{1,2,3\}}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>in Analogie zu den <a href="Poisson-Klammer" title="Poisson-Klammer">Poisson-Klammern</a> der <a href="Hamiltonsche_Mechanik" title="Hamiltonsche Mechanik">Hamiltonschen Formulierung</a></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \{x_{i},p_{j}\}=\delta _{ij}\,\quad \{x_{i},x_{j}\}=0=\{p_{i},p_{j}\}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d002cfc2407a4e6d0540fb97e8bc813a8853469c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.413ex; height:3.009ex;" alt="{\displaystyle \{x_{i},p_{j}\}=\delta _{ij}\,\quad \{x_{i},x_{j}\}=0=\{p_{i},p_{j}\}\,.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Der Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

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</math><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"> ist aus <a href="Dimensionsanalyse" title="Dimensionsanalyse">Dimensionsgründen</a> erforderlich, denn Ort mal Impuls hat die Dimension eines <a href="Drehimpuls" title="Drehimpuls">Drehimpulses</a> oder einer <a href="Wirkung_(Physik)" title="Wirkung (Physik)">Wirkung</a>. Die imaginäre Einheit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {i}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdfe15f9e7151dab6f1af398f23ee2e840bddf35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle {\rm {i}}}" loading="lazy"> muss auftreten, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}_{i}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d530f9a088a3c0ee1a3c2dba9eb6e2cba97c790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.509ex;" alt="{\displaystyle {\hat {x}}_{i}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}_{j}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cf6fda6079ff5a778426c7d59058db9855aef78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:2.359ex; height:3.009ex;" alt="{\displaystyle {\hat {p}}_{j}}" loading="lazy"> <a href="Selbstadjungiert" class="mw-redirect" title="Selbstadjungiert">selbstadjungiert</a> sind und ihr <a href="Kommutator_(Mathematik)" title="Kommutator (Mathematik)">Kommutator</a> daher bei <a href="Adjunktion_(Algebra)" title="Adjunktion (Algebra)">Adjunktion</a> sein Vorzeichen wechselt.</dd></dl>

<ul><li>Aus den kanonischen Vertauschungsrelationen folgt, dass die drei Komponenten des Impulses gemeinsam messbar sind und dass ihr <a href="Spektrum_(Operatortheorie)" title="Spektrum (Operatortheorie)">Spektrum</a> (Bereich der möglichen Messwerte) aus dem gesamten Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> besteht. Die möglichen Impulse sind also nicht quantisiert, sondern <a href="Kontinuum_(Physik)" title="Kontinuum (Physik)">kontinuierlich</a>.</li>

<li>Die Ortsdarstellung ist durch die Spektraldarstellung des Ortsoperators definiert. Der Hilbertraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}=L^{2}(\mathbb {R} ^{3};\mathbb {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=L^{2}(\mathbb {R} ^{3};\mathbb {C} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c2745e113c1d688892b2ba9d338f3d67ff3221c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.953ex; height:3.176ex;" alt="{\displaystyle {\mathcal {H}}=L^{2}(\mathbb {R} ^{3};\mathbb {C} )}" loading="lazy"> ist der Raum der <a href="Quadratintegrierbar" class="mw-redirect" title="Quadratintegrierbar">quadratintegrierbaren</a>, <a href="Komplexe_Funktion" class="mw-redirect" title="Komplexe Funktion">komplexen Funktionen</a> des <a href="Ortsraum" title="Ortsraum">Ortsraums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3};}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3};}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae613b45e932009c0ba932442199b5e65b9bffc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3};}" loading="lazy"> jeder Zustand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> ist durch eine Orts<a href="Wellenfunktion" title="Wellenfunktion">wellenfunktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {x} )}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dfd81f896f8c1b273912cb4bb83997908658269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.733ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {x} )}" loading="lazy"> gegeben. Die Ortsoperatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},{\hat {x}}_{3})}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},{\hat {x}}_{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fd8dd3495f52ad0fcb94de77a5c1d908ee4cf01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.538ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},{\hat {x}}_{3})}" loading="lazy"> sind die Multiplikationsoperatoren mit den Koordinatenfunktionen, d. h. der Ortsoperator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}_{i}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d530f9a088a3c0ee1a3c2dba9eb6e2cba97c790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.509ex;" alt="{\displaystyle {\hat {x}}_{i}}" loading="lazy"> wirkt auf Ortswellenfunktionen durch die Multiplikation der Wellenfunktion mit der Koordinatenfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle ({\hat {x}}_{i}\,\psi )(\mathbf {x} )=x_{i}\,\psi (\mathbf {x} )\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7c7fb1dc3337ea375eb17d5d1d34450e05525a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.441ex; height:2.843ex;" alt="{\displaystyle ({\hat {x}}_{i}\,\psi )(\mathbf {x} )=x_{i}\,\psi (\mathbf {x} )\,.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Der mathematische Satz von Stone und von Neumann<a href="#cite_note-1">[1]</a> besagt dann, dass bei geeigneter Wahl von Phasen der Impulsoperator, der in den kanonischen Vertauschungsrelationen auftritt, auf Ortswellenfunktionen als <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> wirkt:</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle ({\hat {p}}_{j}\psi )(\mathbf {x} )=-{\rm {i}}\,\hbar \,\left({\frac {\partial }{\partial x_{j}}}\psi \right)(\mathbf {x} )\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4695bc333243087e90a99e9b620f2d9ee14c1d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.802ex; height:6.343ex;" alt="{\displaystyle ({\hat {p}}_{j}\psi )(\mathbf {x} )=-{\rm {i}}\,\hbar \,\left({\frac {\partial }{\partial x_{j}}}\psi \right)(\mathbf {x} )\,.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Sein Erwartungswert ist:</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle E({\hat {p}}_{j})=\langle \Psi |{\hat {p}}_{j}\,|\Psi \rangle =\int {\overline {\psi (\mathbf {x} )}}\,\left(-\mathrm {i} \,\hbar {\frac {\partial }{\partial x_{j}}}\psi (\mathbf {x} )\right)\,\mathrm {d} ^{3}x\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd64fb04adab2bdd0894d0f5c8a829b14fd6a9cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.811ex; height:6.343ex;" alt="{\displaystyle E({\hat {p}}_{j})=\langle \Psi |{\hat {p}}_{j}\,|\Psi \rangle =\int {\overline {\psi (\mathbf {x} )}}\,\left(-\mathrm {i} \,\hbar {\frac {\partial }{\partial x_{j}}}\psi (\mathbf {x} )\right)\,\mathrm {d} ^{3}x\,.}" loading="lazy"></dd></dl></dd></dl>

<ul><li>In der Impulsdarstellung wirkt der Impulsoperator multiplikativ auf quadratintegrierbare Impulswellenfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\psi }}(\mathbf {p} )}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7c69b0620d1956b8db4ef99fab5833715043596.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.89ex; height:3.176ex;" alt="{\displaystyle {\tilde {\psi }}(\mathbf {p} )}" loading="lazy">:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle ({\hat {p}}_{j}\,{\tilde {\psi }})(\mathbf {p} )=p_{j}\,{\tilde {\psi }}(\mathbf {p} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52a06d8394e1f6fc52309249f7a821ff0f14459f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.811ex; height:3.509ex;" alt="{\displaystyle ({\hat {p}}_{j}\,{\tilde {\psi }})(\mathbf {p} )=p_{j}\,{\tilde {\psi }}(\mathbf {p} )}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>und der Ortsoperator wirkt als Differentialoperator:</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle ({\hat {x}}_{i}\,{\tilde {\psi }})(\mathbf {p} )=\mathrm {i} \,\hbar \,\left({\frac {\partial }{\partial p_{i}}}{\tilde {\psi }}\right)(\mathbf {p} )\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/757ffb6c376f2b8e4878bdb11df8c35aec4ca5f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.285ex; height:6.176ex;" alt="{\displaystyle ({\hat {x}}_{i}\,{\tilde {\psi }})(\mathbf {p} )=\mathrm {i} \,\hbar \,\left({\frac {\partial }{\partial p_{i}}}{\tilde {\psi }}\right)(\mathbf {p} )\,.}" loading="lazy"></dd></dl></dd></dl>

<ul><li>Die Orts- und Impulsoperatoren sind <a href="Linearkombination" title="Linearkombination">Linearkombinationen</a> von <a href="Erzeugungs-_und_Vernichtungsoperator" title="Erzeugungs- und Vernichtungsoperator">Erzeugungs- und Vernichtungsoperatoren</a>:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {x}}_{i}=l_{i}{\frac {a_{i}+a_{i}^{\dagger }}{\sqrt {2}}}\,\quad {\hat {p}}_{j}={\frac {\hbar }{l_{j}}}{\frac {a_{j}-a_{j}^{\dagger }}{{\sqrt {2}}\,\mathrm {i} }}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db129ac523673b7c3f866fd528d46756184984f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.177ex; height:7.676ex;" alt="{\displaystyle {\hat {x}}_{i}=l_{i}{\frac {a_{i}+a_{i}^{\dagger }}{\sqrt {2}}}\,\quad {\hat {p}}_{j}={\frac {\hbar }{l_{j}}}{\frac {a_{j}-a_{j}^{\dagger }}{{\sqrt {2}}\,\mathrm {i} }}\,.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Dabei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{1},l_{2},l_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle l_{1},l_{2},l_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85523c9d374cfb6a83d646a0682b6046ebaffd70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.31ex; height:2.509ex;" alt="{\displaystyle l_{1},l_{2},l_{3}}" loading="lazy"> frei wählbare Längen (größer Null) und die Erzeugungs- und Vernichtungsoperatoren genügen den kanonischen Vertauschungsrelationen:</dd></dl>

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<div class="mw-heading mw-heading2"><h2 id="Warum_ist_der_Impulsoperator_in_Ortsdarstellung_ein_Differentialoperator?">Warum ist der Impulsoperator in Ortsdarstellung ein Differentialoperator?</h2></div>

Nach dem <a href="Noether-Theorem" title="Noether-Theorem">Noether-Theorem</a> gehört zu jeder kontinuierlichen <a href="Symmetrie_(Physik)" title="Symmetrie (Physik)">Symmetrie</a> der <a href="Wirkung_(Physik)" title="Wirkung (Physik)">Wirkung</a> eine <a href="Erhaltungsgr%C3%B6%C3%9Fe" class="mw-redirect" title="Erhaltungsgröße">Erhaltungsgröße</a>. Umgekehrt impliziert jede Erhaltungsgröße die Existenz einer (mindestens infinitesimalen) Symmetrie der Wirkung. Beispielsweise ist der Impuls genau dann erhalten, wenn die Wirkung translationsinvariant ist. In der Hamiltonschen Formulierung erzeugt die Erhaltungsgröße die Symmetrietransformation im Phasenraum durch ihre Poisson-Klammer, der Impuls erzeugt Verschiebungen.

Auf eine Wellenfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> angewendet, ergibt jede Verschiebung um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><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"> die verschobene Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T_{a}\,\psi )}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c986b3f6f14ec27a5d0b4911de1e1d9a341f6bb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.169ex; height:2.843ex;" alt="{\displaystyle (T_{a}\,\psi )}" loading="lazy">, die an jeder Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><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"> den Wert hat, den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> am Urbild <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-a}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e32f42f9f52615f0186e3c8f2c25d7cd6f7bd6aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.4ex; height:2.176ex;" alt="{\displaystyle x-a}" loading="lazy"> hatte,

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<annotation encoding="application/x-tex">{\displaystyle (T_{a}\,\psi )(x)=\psi (x-a)=\sum _{n=0}^{\infty }{{\frac {1}{n!}}\left(-a{\frac {\partial }{\partial x}}\right)^{n}}\psi =\exp \left(-a{\frac {\partial }{\partial x}}\right)\psi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2336dbc1bcc605cd819280c61cea446cdcba591d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.629ex; height:6.843ex;" alt="{\displaystyle (T_{a}\,\psi )(x)=\psi (x-a)=\sum _{n=0}^{\infty }{{\frac {1}{n!}}\left(-a{\frac {\partial }{\partial x}}\right)^{n}}\psi =\exp \left(-a{\frac {\partial }{\partial x}}\right)\psi }" loading="lazy"> (also: über <a href="Taylorreihe" title="Taylorreihe">Taylorreihe</a> zu einer formalen <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a>).</dd></dl>

Der infinitesimale <a href="Stark_stetige_Halbgruppe#Infinitesimaler_Erzeuger" title="Stark stetige Halbgruppe">Erzeuger</a> dieser einparametrigen Schar von Verschiebungen definiert also bis auf einen Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathrm {i} /\hbar }">

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<annotation encoding="application/x-tex">{\displaystyle T_{a}\,\psi =\exp \left(-a{\frac {\partial }{\partial x}}\right)\psi =\exp {\left(-{\rm {i}}\,a{\frac {{\hat {p}}_{x}}{\hbar }}\right)}\,\psi \,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8f51894b3af59da73a72230a6ab861c388f662d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.3ex; height:6.176ex;" alt="{\displaystyle T_{a}\,\psi =\exp \left(-a{\frac {\partial }{\partial x}}\right)\psi =\exp {\left(-{\rm {i}}\,a{\frac {{\hat {p}}_{x}}{\hbar }}\right)}\,\psi \,.}" loading="lazy"></dd></dl>

Dabei tritt der Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

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

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</math><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"> aus Dimensionsgründen auf, denn das Produkt von Impuls und Ort hat die Dimension eines Drehimpulses oder einer Wirkung. Die <a href="Imagin%C3%A4re_Einheit" class="mw-redirect" title="Imaginäre Einheit">imaginäre Einheit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18f0f09f6fc40e634d34aed6e205ac0f7a40e062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }" loading="lazy"> ist erforderlich, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{a}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54b73a473771885bb5d025ecd2fb10469ab859d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.459ex; height:2.509ex;" alt="{\displaystyle T_{a}}" loading="lazy"> ein <a href="Unit%C3%A4rer_Operator" title="Unitärer Operator">unitärer Operator</a> ist und der Impuls <a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">selbstadjungiert</a> sein soll. Leitet man die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle \left(\exp {\left(-{\rm {i}}\,{\frac {{\hat {p}}_{j}\,a^{j}}{\hbar }}\right)}\,\psi \right)(x)=\psi (x-a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bd191d1fc11de35b7085980f005881c29155cb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.023ex; height:7.509ex;" alt="{\displaystyle \left(\exp {\left(-{\rm {i}}\,{\frac {{\hat {p}}_{j}\,a^{j}}{\hbar }}\right)}\,\psi \right)(x)=\psi (x-a)}" loading="lazy"></dd></dl>

nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle a^{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f15cfdb5ef6fdf7bc318b0e40ffc3b0f917d9092.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.14ex; height:2.676ex;" alt="{\displaystyle a^{j}}" loading="lazy"> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90d476e5e765a5d77bbcff32e4584579207ec7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" loading="lazy"> ab, so ergibt sich der Impulsoperator als Ableitung nach dem Ort,

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<annotation encoding="application/x-tex">{\displaystyle ({\hat {p}}_{j}\,\psi )(x)=\left.\mathrm {i} \,\hbar \,{\frac {\partial }{\partial a^{j}}}\right|_{a=0}\psi (x-a)=-\mathrm {i} \,\hbar {\frac {\partial }{\partial x^{j}}}\psi (x)\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b973e8384776c8cddfdfeb017dcb1bdfd19bfdc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.136ex; height:6.009ex;" alt="{\displaystyle ({\hat {p}}_{j}\,\psi )(x)=\left.\mathrm {i} \,\hbar \,{\frac {\partial }{\partial a^{j}}}\right|_{a=0}\psi (x-a)=-\mathrm {i} \,\hbar {\frac {\partial }{\partial x^{j}}}\psi (x)\,.}" loading="lazy"></dd></dl>

Dass der Impulsoperator im Ortsraum diese Form annimmt, lässt sich auch ohne die Kenntnis des zugehörigen unitären Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{a}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54b73a473771885bb5d025ecd2fb10469ab859d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.459ex; height:2.509ex;" alt="{\displaystyle T_{a}}" loading="lazy"> wie folgt aus dem Noether-Theorem ablesen:

Man rekonstruiert zunächst aus der Schrödingergleichung die zugehörige <a href="Lagrange-Dichte" title="Lagrange-Dichte">Lagrange-Dichte</a> und bestimmt dann explizit den bei einer infinitesimalen Verschiebung der Wellenfunktion erhaltenen Erwartungswert.

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

<ul><li><a href="Torsten_Flie%C3%9Fbach" title="Torsten Fließbach">Torsten Fließbach</a>: Quantenmechanik: Lehrbuch zur Theoretischen Physik III. Spektrum Akademischer Verlag, 2008, ISBN 978-3-8274-2020-6.</li>

<li><a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a>: Feynman Vorlesungen über Physik, Bd. 3, Quantenmechanik. Oldenbourg, 2007, ISBN 978-3-486-58109-6.</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> siehe z. B. die Originalarbeit von <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> (1931): <a rel="nofollow" class="external text" href="https://eudml.org/doc/159483">Die Eindeutigkeit der Schrödingerschen Operatoren.</a> In: eudml.org. Abgerufen am 9. April 2023.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AImpulsoperator&rft.title=Die+Eindeutigkeit+der+Schr%C3%B6dingerschen+Operatoren&rft.description=Die+Eindeutigkeit+der+Schr%C3%B6dingerschen+Operatoren&rft.identifier=https%3A%2F%2Feudml.org%2Fdoc%2F159483">

</li>

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

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