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Display information for equation id:math.1656.79 on revision:1656
* Page found: Der harmonische Oszillator (eq math.1656.79)
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TeX (original user input):
\begin{align}
& {{\phi }_{n}}(\xi )=\frac{{{\left( {{a}^{+}} \right)}^{n}}}{\sqrt{n!}}{{\phi }_{0}}(\xi )=\frac{1}{{{i}^{n}}\sqrt{{{2}^{n}}n!}}{{\left( \xi -\frac{d}{d\xi } \right)}^{n}}{{\phi }_{0}}(\xi )=\frac{1}{{{i}^{n}}}\frac{{{A}_{0}}}{\sqrt{{{2}^{n}}n!}}{{\left( -1 \right)}^{n}}{{e}^{\left( \frac{{{\xi }^{2}}}{2} \right)}}\frac{{{d}^{n}}}{{{\left( d\xi \right)}^{n}}}{{e}^{-{{\xi }^{2}}}} \\
& \frac{{{A}_{0}}}{\sqrt{{{2}^{n}}n!}}:={{A}_{n}} \\
& {{A}_{0}}={{\left( \frac{m\omega }{\hbar \pi } \right)}^{\frac{1}{4}}} \\
& {{\left( -1 \right)}^{n}}{{e}^{\left( \frac{{{\xi }^{2}}}{2} \right)}}\frac{{{d}^{n}}}{{{\left( d\xi \right)}^{n}}}{{e}^{-{{\xi }^{2}}}}:={{H}_{n}}(\xi ){{e}^{-\frac{{{\xi }^{2}}}{2}}} \\
\end{align}
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data-mjx-texclass="ORD"><mi>n</mi><mi>!</mi></mrow></msqrt></mrow></mrow></mfrac></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>i</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mrow data-mjx-texclass="ORD"><msqrt><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mi>n</mi><mi>!</mi></mrow></msqrt></mrow></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ξ</mi><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>ξ</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mi>i</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msqrt><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mi>n</mi><mi>!</mi></mrow></msqrt></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>d</mi><mi>ξ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup></mrow></mfrac></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msqrt><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><mi>n</mi><mi>!</mi></mrow></msqrt></mrow></mrow></mfrac></mrow><mi>:</mi><mo>=</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><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"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>ω</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">ℏ</mi><mi>π</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>4</mn></mrow></mfrac></mrow></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>d</mi><mi>ξ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msup></mrow></mfrac></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow></msup><mi>:</mi><mo>=</mo><msub><mi>H</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ξ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow></mrow></mrow></msup></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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