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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Morse-Potential</span></h1>
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<p>Das <b>Morse-Potential</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 V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> ist ein Begriff aus der <a href="Molek%C3%BClphysik" title="Molekülphysik">Molekülphysik</a>. Der 1929 vom <a href="US-amerikanisch" class="mw-redirect" title="US-amerikanisch">US-amerikanischen</a> <a href="Physiker" title="Physiker">Physiker</a> <a href="Philip_McCord_Morse" class="mw-redirect" title="Philip McCord Morse">Philip McCord Morse</a><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> vorgeschlagene Zusammenhang beschreibt den Verlauf des <a href="Elektrisches_Potential" title="Elektrisches Potential">elektronischen Potentials</a> eines zweiatomigen <a href="Molek%C3%BCl" title="Molekül">Moleküls</a> in Abhängigkeit vom <a href="Atomkern" title="Atomkern">Kern</a>bindungsabstand <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> durch eine exponentielle Näherung:
</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 V(R)=D_{\text{e}}\cdot \left(1-\mathrm {e} ^{-a\cdot (R-R_{\text{e}})}\right)^{2}}">
<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>
<mo>=</mo>
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
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<mo>⋅<!-- ⋅ --></mo>
<msup>
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<mo>(</mo>
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<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
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<mtext>e</mtext>
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<mo>)</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle V(R)=D_{\text{e}}\cdot \left(1-\mathrm {e} ^{-a\cdot (R-R_{\text{e}})}\right)^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7bbc6e4e5289c8e7dc726cc08164c999b790b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.537ex; height:5.176ex;" alt="{\displaystyle V(R)=D_{\text{e}}\cdot \left(1-\mathrm {e} ^{-a\cdot (R-R_{\text{e}})}\right)^{2}}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li><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 D_{\text{e}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D_{\text{e}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e6f5ef5f3db239e27a99e3c1ddc99439cea752e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.887ex; height:2.509ex;" alt="{\displaystyle D_{\text{e}}}" loading="lazy"></span> die (spektroskopische) <a href="Dissoziationsenergie" class="mw-redirect" title="Dissoziationsenergie">Dissoziationsenergie</a></li>
<li><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_{\text{e}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{\text{e}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1544f236e6e024c50ea57b563b425b6e66a4e23e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.726ex; height:2.509ex;" alt="{\displaystyle R_{\text{e}}}" loading="lazy"></span> der Kernabstand mit der geringsten <a href="Potentielle_Energie" title="Potentielle Energie">potentiellen Energie</a> und</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> eine Konstante (manchmal als „Steifigkeit des Potentials“<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> bezeichnet)</li></ul>
<p>Diese Größen sind für das betrachtete Molekül charakteristisch.
</p><p>Da man üblicherweise das Potential im Unendlichen als null definiert:
</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 V(\infty )=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\infty )=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fdcedf5cd8095709a82912edf1b1fa3cbfc797a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.181ex; height:2.843ex;" alt="{\displaystyle V(\infty )=0}" loading="lazy"></span></dd></dl></dd></dl>
<p>wird das Morse-Potential häufig in der alternativen Form angegeben:
</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 V(R)-D_{\text{e}}=D_{\text{e}}\cdot \left(\mathrm {e} ^{-2a\cdot (R-R_{\text{e}})}-2\mathrm {e} ^{-a\cdot (R-R_{\text{e}})}\right)}">
<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>
<mo>−<!-- − --></mo>
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>2</mn>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(R)-D_{\text{e}}=D_{\text{e}}\cdot \left(\mathrm {e} ^{-2a\cdot (R-R_{\text{e}})}-2\mathrm {e} ^{-a\cdot (R-R_{\text{e}})}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f091ee97e60aa8afdd948f16843d78a8b44a1a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.711ex; height:4.843ex;" alt="{\displaystyle V(R)-D_{\text{e}}=D_{\text{e}}\cdot \left(\mathrm {e} ^{-2a\cdot (R-R_{\text{e}})}-2\mathrm {e} ^{-a\cdot (R-R_{\text{e}})}\right)}" loading="lazy"></span></dd></dl>
<p>Dadurch verschiebt sich das Nullpunktpotential um <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 -D_{\text{e}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -D_{\text{e}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1148d6a46ab53fcc457011aab871f9d258be8f1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.695ex; height:2.509ex;" alt="{\displaystyle -D_{\text{e}}}" loading="lazy"></span>. Diese Verschiebung ermöglicht die Definition eines <i>cutoff</i>-Radiuses, ab dem das Potential nicht mehr berücksichtigt wird.
</p><p>Die <a href="Schr%C3%B6dinger-Gleichung" class="mw-redirect" title="Schrödinger-Gleichung">Schrödinger-Gleichung</a> ist mit dem Morsepotential analytisch lösbar. So können die <a href="Schwingungsenergie" class="mw-redirect" title="Schwingungsenergie">Schwingungsenergien</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_{\nu }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{\nu }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5844d2c0cbe7cb43c7b2c2e58cb3d64e85fa3aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.819ex; height:2.509ex;" alt="{\displaystyle E_{\nu }}" loading="lazy"></span> berechnet werden:
</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 E_{\nu }=h\omega _{0}\cdot \left(\nu +{\frac {1}{2}}\right)-{\frac {h^{2}\omega _{0}^{2}}{4D_{\text{e}}}}\cdot \left(\nu +{\frac {1}{2}}\right)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
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</msub>
<mo>=</mo>
<mi>h</mi>
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>ν<!-- ν --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
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<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msubsup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
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<mrow>
<mn>4</mn>
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>ν<!-- ν --></mi>
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<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle E_{\nu }=h\omega _{0}\cdot \left(\nu +{\frac {1}{2}}\right)-{\frac {h^{2}\omega _{0}^{2}}{4D_{\text{e}}}}\cdot \left(\nu +{\frac {1}{2}}\right)^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4dd7ed6bad9194d3a6b027e1998b4e7d2ffe6e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.723ex; height:6.676ex;" alt="{\displaystyle E_{\nu }=h\omega _{0}\cdot \left(\nu +{\frac {1}{2}}\right)-{\frac {h^{2}\omega _{0}^{2}}{4D_{\text{e}}}}\cdot \left(\nu +{\frac {1}{2}}\right)^{2}}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li>der <a href="Planck-Konstante" title="Planck-Konstante">Planck-Konstante</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 \ h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext> </mtext>
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ h}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8eb515077d0b969895a7e62dfe0ae6a198e2426f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.92ex; height:2.176ex;" alt="{\displaystyle \ h}" loading="lazy"></span></li>
<li>der Schwingungs<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 \nu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ν<!-- ν --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"></span></li>
<li>der <a href="Frequenz" title="Frequenz">Frequenz</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 \omega _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a713d16c489051d4f515e12b1f86061c6be799b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}" loading="lazy"></span>, die über die Teilchenmasse <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 m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> mit der Konstante <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> des Morse-Potentials verknüpft ist</li></ul>
<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 \omega _{0}={\frac {a}{2\pi }}{\sqrt {\frac {2D_{\text{e}}}{m}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>a</mi>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mn>2</mn>
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>e</mtext>
</mrow>
</msub>
</mrow>
<mi>m</mi>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega _{0}={\frac {a}{2\pi }}{\sqrt {\frac {2D_{\text{e}}}{m}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8a2d579ebbacf4a235ba4de3797cce6f338f236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.138ex; height:6.176ex;" alt="{\displaystyle \omega _{0}={\frac {a}{2\pi }}{\sqrt {\frac {2D_{\text{e}}}{m}}}}" loading="lazy"></span></dd></dl></dd></dl>
<p>Heutzutage wird für die Berechnung von Schwingungsenergien eher das RKR-Potential (RKR steht hierbei für Ragnar Rydberg, <a href="Oskar_Benjamin_Klein" class="mw-redirect" title="Oskar Benjamin Klein">Oskar Klein</a> und Lloyd Rees) oder das <a href="Lennard-Jones-Potential" title="Lennard-Jones-Potential">Lennard-Jones-Potential</a> angewendet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Wolfgang_Demtr%C3%B6der" title="Wolfgang Demtröder">Wolfgang Demtröder</a>: <cite style="font-style:italic">Molekülphysik: Theoretische Grundlagen und experimentelle Methoden</cite>. Oldenbourg Wissenschaftsverlag, 2003, ISBN 978-3-486-24974-3, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>93–94</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:Morse-Potential&rft.au=Wolfgang+Demtr%C3%B6der&rft.btitle=Molek%C3%BClphysik%3A+Theoretische+Grundlagen+und+experimentelle+Methoden&rft.date=2003&rft.genre=book&rft.isbn=9783486249743&rft.pages=93-94&rft.pub=Oldenbourg+Wissenschaftsverlag" style="display:none"> </span></li>
<li>Ludwig Bergmann, Clemens Schaefer, Wilhelm Raith, Mit Beitragen Von <a href="Hans_Kleinpoppen" title="Hans Kleinpoppen">H. Kleinpoppen</a>, M. Fink, N. Risch: <cite style="font-style:italic">Bestandteile der Materie: Atome, Moleküle, Atomkerne, Elementarteilchen</cite>. Walter de Gruyter, 2003, ISBN 978-3-11-016800-6, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>460–462</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:Morse-Potential&rft.au=Ludwig+Bergmann%2C+Clemens+Schaefer%2C+Wilhelm+Raith%2C+...&rft.btitle=Bestandteile+der+Materie%3A+Atome%2C+Molek%C3%BCle%2C+Atomkerne%2C+Elementarteilchen&rft.date=2003&rft.genre=book&rft.isbn=9783110168006&rft.pages=460-462&rft.pub=Walter+de+Gruyter" style="display:none"> </span></li>
<li>Gerd Otter, Raimund Honecker: <cite style="font-style:italic">Atome – Moleküle – Kerne: Molekül- und Kernphysik</cite>. Vieweg +Teubner, 1996, ISBN 978-3-519-03220-5, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>152–154</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:Morse-Potential&rft.au=Gerd+Otter%2C+Raimund+Honecker&rft.btitle=Atome+-+Molek%C3%BCle+-+Kerne%3A+Molek%C3%BCl-+und+Kernphysik&rft.date=1996&rft.genre=book&rft.isbn=9783519032205&rft.pages=152-154&rft.pub=Vieweg+%2BTeubner" style="display:none"> </span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Philip M. Morse: <cite style="font-style:italic">Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels</cite>. In: <cite style="font-style:italic">Physical Review</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>34</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, 1. Juni 1929, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>57</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/10.1103/PhysRev.34.57">10.1103/PhysRev.34.57</a></span>.<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:Morse-Potential&rft.atitle=Diatomic+Molecules+According+to+the+Wave+Mechanics.+II.+Vibrational+Levels&rft.au=Philip+M.+Morse&rft.date=1929-06-01&rft.doi=10.1103%2FPhysRev.34.57&rft.genre=journal&rft.issue=1&rft.jtitle=Physical+Review&rft.pages=57&rft.volume=34" 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">Ingolf V. Hertel, C.-P. Schulz: <cite style="font-style:italic">Atome, Moleküle und Optische Physik 2: Moleküle und Photonen-Spektroskopie und Streuphysik</cite>. Springer, 2011, ISBN 978-3-642-11972-9, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>13</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:Morse-Potential&rft.au=Ingolf+V.+Hertel%2C+C.-P.+Schulz&rft.btitle=Atome%2C+Molek%C3%BCle+und+Optische+Physik+2%3A+Molek%C3%BCle+und+Photonen-Spektroskopie+und+Streuphysik&rft.date=2011&rft.genre=book&rft.isbn=9783642119729&rft.pages=13&rft.pub=Springer" style="display:none"> </span></span>
</li>
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