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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Partialwelle</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><b>Partialwellen</b>, wörtlich: <i>Teil</i>wellen, ein Begriff der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a>, sind <a href="Station%C3%A4rer_Zustand_(Quantenmechanik)" title="Stationärer Zustand (Quantenmechanik)">stationäre Lösungen</a> eines <a href="Streuprozess" class="mw-redirect" title="Streuprozess">Streuproblems</a> und gleichzeitig <a href="Eigenfunktion" class="mw-redirect" title="Eigenfunktion">Eigenfunktionen</a> des <a href="Drehimpulsoperator" class="mw-redirect" title="Drehimpulsoperator">Drehimpulses</a>. Die <a href="Streuamplitude#Partialwellenentwicklung" title="Streuamplitude">Zerlegung einer Streuamplitude in Partialwellen</a>, d. h. eine <a href="Reihenentwicklung" title="Reihenentwicklung">Reihenentwicklung</a> nach Drehimpulsen, ist sinnvoll vor allem bei Wechselwirkungen mit kurzer Reichweite, wie z. B. der <a href="Starke_Wechselwirkung" title="Starke Wechselwirkung">starken Wechselwirkung</a>. Aufgrund der kurzen Reichweite tragen nämlich für niedrige Energien nur kleine Drehimpulse zur Streuung bei.
</p>
<div class="mw-heading mw-heading2"><h2 id="Erklärung_im_Teilchenbild"><span id="Erkl.C3.A4rung_im_Teilchenbild"></span>Erklärung im Teilchenbild</h2></div>
<p>Wird ein bewegtes <a href="Teilchen" title="Teilchen">Teilchen</a> im <a href="Feld_(Physik)" title="Feld (Physik)">Feld</a> eines <a href="Streuk%C3%B6rper" class="mw-redirect" title="Streukörper">Streuzentrums</a> – z. B. eines <a href="Atomkern" title="Atomkern">Atomkerns</a> – aus seiner Bahn abgelenkt, so gehört zu dieser Bewegung ein <a href="Bahndrehimpuls" class="mw-redirect" title="Bahndrehimpuls">Bahndrehimpuls</a>. Dieser kann nur <a href="Diskret" title="Diskret">diskrete</a>, durch eine <a href="Quantenzahl" title="Quantenzahl">Quantenzahl</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span> beschriebene Werte annehmen; im Einzelfall hängt er bei gegebener <a href="Geschwindigkeit" title="Geschwindigkeit">Geschwindigkeit</a> des Teilchens vom <a href="Sto%C3%9Fparameter" title="Stoßparameter">Stoßparameter</a> ab. Die Beiträge der Einzelprozesse mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=0,1,2,\dots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l=0,1,2,\dots }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94d80127519cc3fff1b1ca59864787dddf124155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.104ex; height:2.509ex;" alt="{\displaystyle l=0,1,2,\dots }" loading="lazy"></span> heißen im <a href="Welle-Teilchen-Dualismus" title="Welle-Teilchen-Dualismus">Wellenbild</a> Partialwellen und wirken sich jeweils charakteristisch aus, z. B. in der Verteilung der insgesamt gestreuten Teilchen auf die Streurichtungen, die <a href="Winkelverteilung" class="mw-redirect" title="Winkelverteilung">Winkelverteilung</a>.
</p><p>Die Kennbuchstaben für die Werte von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"></span> werden benutzt <a href="Atomorbital#Neben-_oder_Bahndrehimpuls-Quantenzahl_l" title="Atomorbital">wie beim gebundenen Elektron im Atom</a>, man spricht also von der s-Welle (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66485a3e3da13d226eb36a131bf1fc7e16403a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=0}" loading="lazy"></span>), p-Welle (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fc60b21200ebf2b338c4fa71b103cb697b02bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=1}" loading="lazy"></span>), d-Welle (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>=</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l=2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/568d606c605ed04ee4beb2bc2d3bed232e0b07f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=2}" loading="lazy"></span>) usw.
</p>
<div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2></div>
<p>Ziel ist es, eine Lösung der <a href="Schr%C3%B6dingergleichung" title="Schrödingergleichung">Schrödingergleichung</a> für ein sphärisch-symmetrisches <a href="Potential_(Physik)" title="Potential (Physik)">Potential</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\vec {r}})=V(r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V({\vec {r}})=V(r)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/135eae7f4ad039ed5c357d3199f7186dd6c04d98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.563ex; height:2.843ex;" alt="{\displaystyle V({\vec {r}})=V(r)}" loading="lazy"></span> wie z. B. das <a href="Coulombpotential" class="mw-redirect" title="Coulombpotential">Coulombpotential</a> zu finden.
</p><p>Die <a href="Wellenfunktion" title="Wellenfunktion">Wellenfunktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\vec {k}}({\vec {r}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>k</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}({\vec {r}})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2fbfcd6784c9dbdad9d00eed0b3bb935e751a8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.634ex; height:3.176ex;" alt="{\displaystyle \psi _{\vec {k}}({\vec {r}})}" loading="lazy"></span> wird für <a href="Asymptotisch" class="mw-redirect" title="Asymptotisch">asymptotische</a> Abstände <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 r\rightarrow \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r\rightarrow \infty }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/170e3065b668161cd003c9fd57119e51d3424ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\rightarrow \infty }" loading="lazy"></span> als Überlagerung einer einlaufenden <a href="Ebene_Welle" title="Ebene Welle">ebenen Welle</a> und einer durch die <a href="Streuamplitude" title="Streuamplitude">Streuamplitude</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta ,\phi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(\theta ,\phi )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b09584135fea5049be8213c6f87e96e0313e7085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle f(\theta ,\phi )}" loading="lazy"></span> modifizierten <a href="Kugelwelle" title="Kugelwelle">Kugelwelle</a> angesetzt:
</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 \psi _{\vec {k}}({\vec {r}}){\xrightarrow {r\rightarrow \infty }}[e^{i{\vec {k}}{\vec {r}}}+f(\theta ,\phi ){\frac {e^{ikr}}{r}}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>k</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mo>→</mo>
<mpadded width="+0.611em" lspace="0.278em" voffset=".15em">
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mpadded>
</mover>
</mrow>
<mo stretchy="false">[</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>k</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mrow>
</msup>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
<mi>r</mi>
</mrow>
</msup>
<mi>r</mi>
</mfrac>
</mrow>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}({\vec {r}}){\xrightarrow {r\rightarrow \infty }}[e^{i{\vec {k}}{\vec {r}}}+f(\theta ,\phi ){\frac {e^{ikr}}{r}}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae2e55b7c55ba284ba51fe8ad0b83732f6a8867b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.677ex; height:5.676ex;" alt="{\displaystyle \psi _{\vec {k}}({\vec {r}})\xrightarrow {r\rightarrow \infty } [e^{i{\vec {k}}{\vec {r}}}+f(\theta ,\phi ){\frac {e^{ikr}}{r}}]}" loading="lazy"></span></dd></dl>
<p>In diesem Fall ist die Streuamplitude aufgrund der <a href="Kugelsymmetrie" class="mw-redirect" title="Kugelsymmetrie">Kugelsymmetrie</a> <i>unabhängig</i> vom Winkel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span>:
</p>
<dl><dd><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 f(\theta ,\phi )=f(\theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(\theta ,\phi )=f(\theta )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f83fd2b54287f981a56fbb82ac88a85fe717ca7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.875ex; height:2.843ex;" alt="{\displaystyle f(\theta ,\phi )=f(\theta )}" loading="lazy"></span></dd></dl></dd></dl>
<p>Nach einigen Umformungen ergibt sich die Lösungswellenfunktion des Streuproblems für asymptotische Distanzen zu:
</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 \psi _{\vec {k}}({\vec {r}})=\sum _{l=0}^{\infty }i^{l}(2l+1)\;R_{lk}(r)\;P_{l}(\cos \theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>k</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msup>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>l</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}({\vec {r}})=\sum _{l=0}^{\infty }i^{l}(2l+1)\;R_{lk}(r)\;P_{l}(\cos \theta )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46e9e83150a7700f169c1b2e7242029f3ce9af2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.772ex; height:7.009ex;" alt="{\displaystyle \psi _{\vec {k}}({\vec {r}})=\sum _{l=0}^{\infty }i^{l}(2l+1)\;R_{lk}(r)\;P_{l}(\cos \theta )}" loading="lazy"></span></dd></dl>
<p>dabei sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{l}(\cos \theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{l}(\cos \theta )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb0c2979e5a1be871c49bad2304ad7bf58af484c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.613ex; height:2.843ex;" alt="{\displaystyle P_{l}(\cos \theta )}" loading="lazy"></span> die <a href="Legendre-Polynome" class="mw-redirect" title="Legendre-Polynome">Legendre-Polynome</a>.
</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 R_{lk}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{lk}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06ad574071424fc7f852a2e82fb24eb353daf2a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle R_{lk}}" loading="lazy"></span> ist die Lösung der radialen Schrödingergleichung, welche aus einer <a href="Linearkombination" title="Linearkombination">Linearkombination</a> der sphärischen <a href="Besselsche_Differentialgleichung" class="mw-redirect" title="Besselsche Differentialgleichung">Bessel-Funktionen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{l}(\rho )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>j</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j_{l}(\rho )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5fa79b3fb3ec070912ae2d47759ec719163bcdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:4.719ex; height:2.843ex;" alt="{\displaystyle j_{l}(\rho )}" loading="lazy"></span> und der Von-Neumann-Funktion <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 n_{l}(\rho )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n_{l}(\rho )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9066782843a4a3fc7d758c62065b75198f92b85d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.128ex; height:2.843ex;" alt="{\displaystyle n_{l}(\rho )}" loading="lazy"></span> besteht:
</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 R_{lk}{\stackrel {\mathrm {r\rightarrow \infty } }{=}}A_{l}\;j_{l}(kr)+B_{l}\;n_{l}(kr)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mi>k</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-REL">
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo>=</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">r</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</mrow>
</mover>
</mrow>
</mrow>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>j</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{lk}{\stackrel {\mathrm {r\rightarrow \infty } }{=}}A_{l}\;j_{l}(kr)+B_{l}\;n_{l}(kr)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23726b118e04d3bf0c0dbaf8ba5940737b769917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.293ex; height:3.509ex;" alt="{\displaystyle R_{lk}{\stackrel {\mathrm {r\rightarrow \infty } }{=}}A_{l}\;j_{l}(kr)+B_{l}\;n_{l}(kr)}" loading="lazy"></span></dd></dl>
<p>Im nächsten Schritt wird die <b>Streuphase</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95c6f3002cd5f9ce093642c80a15ad13b8e46429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.755ex; height:2.676ex;" alt="{\displaystyle \delta _{l}}" loading="lazy"></span> folgendermaßen definiert:
</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 A_{l}=+a_{l}\cos \delta _{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mi>cos</mi>
<mo><!-- --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{l}=+a_{l}\cos \delta _{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a9b5cda87c9cee25769ad32203c8d84f5f272bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.965ex; height:2.676ex;" alt="{\displaystyle A_{l}=+a_{l}\cos \delta _{l}}" 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 B_{l}=-a_{l}\sin \delta _{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mi>sin</mi>
<mo><!-- --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B_{l}=-a_{l}\sin \delta _{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02f3a67e543f61f934380d33e42c5000a1b5ac25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.73ex; height:2.676ex;" alt="{\displaystyle B_{l}=-a_{l}\sin \delta _{l}}" loading="lazy"></span></dd></dl>
<p>Die Phase der auslaufenden Kugelwelle wird also durch das Potential <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 V(r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(r)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/114fdc48547ee60d02d7a2f4765d52a6ee3507d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle V(r)}" loading="lazy"></span> verschoben: bei <a href="Elastischer_Sto%C3%9F" class="mw-redirect" title="Elastischer Stoß">elastischer Streuung</a> unterscheidet sich die gestreute Welle von der ungestörten Welle des <a href="Freies_Teilchen" title="Freies Teilchen">freien Teilchens</a> nur durch einen <a href="Zustand_(Quantenmechanik)#Phasenfaktor_und_Superposition" title="Zustand (Quantenmechanik)">Phasenfaktor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\cdot \delta _{l}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{i\cdot \delta _{l}}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/014ddc33b7d7fef3a810d5343af930e0a788c12c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.28ex; height:2.676ex;" alt="{\displaystyle e^{i\cdot \delta _{l}}.}" loading="lazy"></span>
</p><p>Durch Einsetzen der sphärischen Bessel- und Von-Neumann-Funktionen und Vergleich mit dem Ansatz für die Wellenfunktion für asymptotische Distanzen kommt man nach einigen Umformungen auf den folgenden Zusammenhang zwischen Streuamplitude <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 f(\theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(\theta )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/205329be9269c44e4efaeb7001fc60bbe5188eaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.178ex; height:2.843ex;" alt="{\displaystyle f(\theta )}" loading="lazy"></span> und der Streuphase <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95c6f3002cd5f9ce093642c80a15ad13b8e46429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.755ex; height:2.676ex;" alt="{\displaystyle \delta _{l}}" 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 f(\theta )=\sum _{l=0}^{\infty }(2l+1)\;f_{l}(\delta _{l})\;P_{l}(\cos \theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>l</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(\theta )=\sum _{l=0}^{\infty }(2l+1)\;f_{l}(\delta _{l})\;P_{l}(\cos \theta )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c2ba82fa85780e208639151a9909fbf800e55bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -3.171ex; width:33.629ex; height:7.009ex;" aria-hidden="true" alt="{\displaystyle f(\theta )=\sum _{l=0}^{\infty }(2l+1)\;f_{l}(\delta _{l})\;P_{l}(\cos \theta )}" loading="lazy"></span></dd></dl>
<p>wobei
</p>
<dl><dd><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 f_{l}(\delta _{l})={\frac {1}{k}}e^{i\cdot \delta _{l}}\sin \delta _{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>k</mi>
</mfrac>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mrow>
</msup>
<mi>sin</mi>
<mo><!-- --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{l}(\delta _{l})={\frac {1}{k}}e^{i\cdot \delta _{l}}\sin \delta _{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ab7bec44ee2953a4b5830902da280c9fe2590a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.589ex; height:5.343ex;" alt="{\displaystyle f_{l}(\delta _{l})={\frac {1}{k}}e^{i\cdot \delta _{l}}\sin \delta _{l}}" loading="lazy"></span></dd></dl></dd></dl>
<p>den Beitrag der <i>l</i>-ten Partialwelle darstellt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Streulänge"><span id="Streul.C3.A4nge"></span>Streulänge</h2></div>
<p>Eine weitere wichtige Größe für die Analyse von Streuproblemen, die sich aus der Streuamplitude ableiten lässt, ist die <b>Streulänge</b> <i>a</i>. Sie ergibt sich aus dem <a href="Wirkungsquerschnitt" title="Wirkungsquerschnitt">totalen Streuquerschnitt</a>, wenn die Energie des gestreuten Teilchens gegen 0 geht:
</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}\lim _{k\to 0}\sigma _{\text{total}}&=4\pi \cdot a^{2}\\\Leftrightarrow a&=\pm {\sqrt {\frac {\lim _{k\to 0}\sigma _{\text{total}}}{4\pi }}}\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>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>total</mtext>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>4</mn>
<mi>π<!-- π --></mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⇔<!-- ⇔ --></mo>
<mi>a</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>total</mtext>
</mrow>
</msub>
</mrow>
<mrow>
<mn>4</mn>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lim _{k\to 0}\sigma _{\text{total}}&=4\pi \cdot a^{2}\\\Leftrightarrow a&=\pm {\sqrt {\frac {\lim _{k\to 0}\sigma _{\text{total}}}{4\pi }}}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7b41eea80bbaae0a900e6805274a337cae027eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.554ex; margin-bottom: -0.283ex; width:29.578ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\lim _{k\to 0}\sigma _{\text{total}}&=4\pi \cdot a^{2}\\\Leftrightarrow a&=\pm {\sqrt {\frac {\lim _{k\to 0}\sigma _{\text{total}}}{4\pi }}}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Die Streulänge entspricht also einer effektiven Querschnittsfläche, welche sowohl die Stärke als auch die Art eines Potentials anzeigt.
</p><p>Mit folgender Definition für den totalen Querschnitt:
</p>
<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\text{total}}=\int _{4\pi }{\frac {d\sigma }{d\Omega }}\cdot d\Omega =\int _{4\pi }|f(\theta )|^{2}\cdot d\Omega ={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(2l+1)\sin ^{2}\delta _{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>total</mtext>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>π<!-- π --></mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>σ<!-- σ --></mi>
</mrow>
<mrow>
<mi>d</mi>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mrow>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>d</mi>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>π<!-- π --></mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>d</mi>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>4</mn>
<mi>π<!-- π --></mi>
</mrow>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>l</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{\text{total}}=\int _{4\pi }{\frac {d\sigma }{d\Omega }}\cdot d\Omega =\int _{4\pi }|f(\theta )|^{2}\cdot d\Omega ={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(2l+1)\sin ^{2}\delta _{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83a9f3f3e1cf6b4717a8003fb32d77e1e85ec361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.169ex; height:7.009ex;" alt="{\displaystyle \sigma _{\text{total}}=\int _{4\pi }{\frac {d\sigma }{d\Omega }}\cdot d\Omega =\int _{4\pi }|f(\theta )|^{2}\cdot d\Omega ={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(2l+1)\sin ^{2}\delta _{l}}" loading="lazy"></span></dd></dl></dd></dl>
<p>wird die Streulänge für s-Wellen (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66485a3e3da13d226eb36a131bf1fc7e16403a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=0}" loading="lazy"></span>) zu:
</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 a=\pm \lim _{k\to 0}{\frac {\sin \delta _{0}}{k}}\;.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mo>±<!-- ± --></mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mi>k</mi>
</mfrac>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=\pm \lim _{k\to 0}{\frac {\sin \delta _{0}}{k}}\;.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/317961ba33a7972142dc0415834cbebd577bbd10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.69ex; height:5.509ex;" alt="{\displaystyle a=\pm \lim _{k\to 0}{\frac {\sin \delta _{0}}{k}}\;.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Cohen-Tannoudji: Quantum Mechanics - Vol 2, Wiley-Interscience, 2006.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li>TU München: <style data-mw-deduplicate="TemplateStyles:r261891140">
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