Zur Navigation springen
Zur Suche springen
General
Display information for equation id:math.1737.27 on revision:1737
* Page found: Induzierte Emission und Absorption von Lichtquanten in Atomen (eq math.1737.27)
(force rerendering)Occurrences on the following pages:
Hash: 544042cc8bf04bea2ce07d7553ac69cc
TeX (original user input):
\begin{align}
& {{\Psi }_{nlm}}(\bar{r})=\frac{{{u}_{nl}}(r)}{r}{{Y}_{l}}^{m}\left( \vartheta ,\phi \right)\tilde{\ }{{P}_{l}}^{m}(\cos \vartheta ){{e}^{im\phi }} \\
& \left\langle n\acute{\ }l\acute{\ }m\acute{\ } \right|\hat{\bar{\xi }}\left| nlm \right\rangle \tilde{\ }\int_{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left( \vartheta \right){{P}_{l\acute{\ }}}^{m\acute{\ }}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta )\int_{0}^{2\pi }{d}\phi {{e}^{i\left( m-m\acute{\ }+1 \right)\phi }} \\
& \int_{0}^{2\pi }{d}\phi {{e}^{i\left( m-m\acute{\ }+1 \right)\phi }}\tilde{\ }{{\delta }_{m\acute{\ },m+1}} \\
& \Rightarrow \left\langle n\acute{\ }l\acute{\ }m\acute{\ } \right|\hat{\bar{\xi }}\left| nlm \right\rangle \tilde{\ }\int_{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left( \vartheta \right){{P}_{l\acute{\ }}}^{m+1}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta ) \\
& \int_{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left( \vartheta \right){{P}_{l\acute{\ }}}^{m+1}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta )\tilde{\ }{{\delta }_{l\acute{\ },l\pm 1}} \\
& \Rightarrow \left\langle n\acute{\ }l\acute{\ }m\acute{\ } \right|\hat{\bar{\xi }}\left| nlm \right\rangle \tilde{\ }{{\delta }_{m\acute{\ },m+1}}{{\delta }_{l\acute{\ },l\pm 1}} \\
\end{align}
LaTeXML (experimentell; verwendet MathML) rendering
MathML (0 B / 8 B) :
SVG image empty. Force Re-Rendering
SVG (0 B / 8 B) :
MathML (experimentell; keine Bilder) rendering
MathML (11.589 KB / 845 B) :
<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><msub><mi mathvariant="normal">Ψ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>n</mi><mi>l</mi><mi>m</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msub><mi>u</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>n</mi><mi>l</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></mfrac></mrow><msup><msub><mi>Y</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ϑ</mi><mo>,</mo><mi>ϕ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo>~</mo></mover></mrow></mrow><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>ϑ</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>m</mi><mi>ϕ</mi></mrow></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>ξ</mi><mo>¯</mo></mover></mrow></mrow><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>n</mi><mi>l</mi><mi>m</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo>~</mo></mover></mrow></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>π</mi></mrow></munderover></mstyle><mi>d</mi><mi>ϑ</mi><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ϑ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></mrow></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></mrow></mrow></msup><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>ϑ</mi><mo stretchy="false">)</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>π</mi></mrow></mrow></munderover></mstyle><mi>d</mi><mi>ϕ</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>m</mi><mo>−</mo><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ϕ</mi></mrow></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>π</mi></mrow></mrow></munderover></mstyle><mi>d</mi><mi>ϕ</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>m</mi><mo>−</mo><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ϕ</mi></mrow></mrow></msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo>~</mo></mover></mrow></mrow><msub><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>,</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>ξ</mi><mo>¯</mo></mover></mrow></mrow><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>n</mi><mi>l</mi><mi>m</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo>~</mo></mover></mrow></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>π</mi></mrow></munderover></mstyle><mi>d</mi><mi>ϑ</mi><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ϑ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></mrow></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msup><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>ϑ</mi><mo stretchy="false">)</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>ϑ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>π</mi></mrow></munderover></mstyle><mi>d</mi><mi>ϑ</mi><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ϑ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></mrow></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msup><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>ϑ</mi><mo stretchy="false">)</mo><msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo>~</mo></mover></mrow></mrow><msub><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>,</mo><mi>l</mi><mo>±</mo><mn>1</mn></mrow></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>ξ</mi><mo>¯</mo></mover></mrow></mrow><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>n</mi><mi>l</mi><mi>m</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo>~</mo></mover></mrow></mrow><msub><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>,</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub><msub><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>,</mo><mi>l</mi><mo>±</mo><mn>1</mn></mrow></mrow></msub></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
Translations to Computer Algebra Systems
Translation to Maple
In Maple:
Translation to Mathematica
In Mathematica:
Similar pages
Calculated based on the variables occurring on the entire Induzierte Emission und Absorption von Lichtquanten in Atomen page
Identifiers
MathML observations
0results
0results
no statistics present please run the maintenance script ExtractFeatures.php
0 results
0 results