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Display information for equation id:math.1732.62 on revision:1732
* Page found: Zeitabhängige Störungsrechnung (eq math.1732.62)
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Hash: 370c9ef2f7063ef0c91830cf466a0e4f
TeX (original user input):
\begin{align}
& {{D}_{t}}(0)={{\left( \frac{t}{\hbar } \right)}^{2}} \\
& \begin{matrix}
\lim \\
t\to \infty \\
\end{matrix}\left( {{D}_{t}}(0) \right)=\infty \\
& \int_{-\infty }^{\infty }{{{D}_{t}}(E)}=\int_{-\infty }^{\infty }{{}}dE\frac{4{{\sin }^{2}}\left( \frac{Et}{2\hbar } \right)}{{{E}^{2}}}=\frac{2t}{\hbar }\int_{-\infty }^{\infty }{{}}d\xi \frac{{{\sin }^{2}}\xi }{{{\xi }^{2}}} \\
& \int_{-\infty }^{\infty }{{}}d\xi \frac{{{\sin }^{2}}\xi }{{{\xi }^{2}}}=\pi \\
& \Rightarrow \int_{-\infty }^{\infty }{{{D}_{t}}(E)}=\frac{2\pi }{\hbar }t \\
\end{align}
TeX (checked):
{\begin{aligned}&{{D}_{t}}(0)={{\left({\frac {t}{\hbar }}\right)}^{2}}\\&{\begin{matrix}\lim \\t\to \infty \\\end{matrix}}\left({{D}_{t}}(0)\right)=\infty \\&\int _{-\infty }^{\infty }{{{D}_{t}}(E)}=\int _{-\infty }^{\infty }{}dE{\frac {4{{\sin }^{2}}\left({\frac {Et}{2\hbar }}\right)}{{E}^{2}}}={\frac {2t}{\hbar }}\int _{-\infty }^{\infty }{}d\xi {\frac {{{\sin }^{2}}\xi }{{\xi }^{2}}}\\&\int _{-\infty }^{\infty }{}d\xi {\frac {{{\sin }^{2}}\xi }{{\xi }^{2}}}=\pi \\&\Rightarrow \int _{-\infty }^{\infty }{{{D}_{t}}(E)}={\frac {2\pi }{\hbar }}t\\\end{aligned}}
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<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>D</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">ℏ</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>lim</mi></mtd></mtr><mtr><mtd><mi>t</mi><mo accent="false">→</mo><mi mathvariant="normal">∞</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>D</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mtd></mtr><mtr><mtd></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msub><mi>D</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mi>d</mi><mi>E</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>4</mn><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><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>E</mi><mi>t</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi data-mjx-alternate="1">ℏ</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>E</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>t</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">ℏ</mi></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mi>d</mi><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>ξ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mi>d</mi><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>ξ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo>=</mo><mi>π</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msub><mi>D</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>π</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">ℏ</mi></mrow></mfrac></mrow><mi>t</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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