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Display information for equation id:math.1719.77 on revision:1719

* Page found: Identische Teilchen: Spin und Statistik (eq math.1719.77)

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TeX (original user input):

\begin{align}
& {{f}_{n}}{{^{2}}_{a}}\left\langle  {{a}_{1}},...,{{a}_{N}} \right|{{\left| {{a}_{1}},...,{{a}_{N}} \right\rangle }_{a}}={{f}_{n}}^{2}\left( _{1}\left\langle  {{a}_{1}} \right|{{...}_{N}}\left\langle  {{a}_{N}} \right| \right)\hat{A}\hat{A}\left( {{\left| {{a}_{1}} \right\rangle }_{1}}...{{\left| {{a}_{N}} \right\rangle }_{N}} \right) \\
& ={{f}_{n}}^{2}\left( _{1}\left\langle  {{a}_{1}} \right|{{...}_{N}}\left\langle  {{a}_{N}} \right| \right)\hat{A}\left( {{\left| {{a}_{1}} \right\rangle }_{1}}...{{\left| {{a}_{N}} \right\rangle }_{N}} \right)={{f}_{n}}^{2}\left( _{1}\left\langle  {{a}_{1}} \right|{{...}_{N}}\left\langle  {{a}_{N}} \right| \right)\frac{1}{N!}\left| \begin{matrix}
{{\left| {{a}_{1}} \right\rangle }_{1}} & {{\left| {{a}_{1}} \right\rangle }_{2}} & ... & {{\left| {{a}_{1}} \right\rangle }_{N}}  \\
{{\left| {{a}_{2}} \right\rangle }_{1}} & {{\left| {{a}_{2}} \right\rangle }_{2}} & ... & {{\left| {{a}_{2}} \right\rangle }_{N}}  \\
... & ... & ... & ...  \\
{{\left| {{a}_{N}} \right\rangle }_{1}} & {{\left| {{a}_{N}} \right\rangle }_{2}} & ... & {{\left| {{a}_{N}} \right\rangle }_{N}}  \\
\end{matrix} \right| \\
& ={{f}_{n}}^{2}\frac{1}{N!}\left| \begin{matrix}
_{1}\left\langle  {{a}_{1}} \right|{{\left| {{a}_{1}} \right\rangle }_{1}} & _{2}\left\langle  {{a}_{2}} \right|{{\left| {{a}_{1}} \right\rangle }_{2}} & ... & _{N}\left\langle  {{a}_{N}} \right|{{\left| {{a}_{1}} \right\rangle }_{N}}  \\
_{1}\left\langle  {{a}_{1}} \right|{{\left| {{a}_{2}} \right\rangle }_{1}} & _{2}\left\langle  {{a}_{2}} \right|{{\left| {{a}_{2}} \right\rangle }_{2}} & ... & _{N}\left\langle  {{a}_{N}} \right|{{\left| {{a}_{2}} \right\rangle }_{N}}  \\
... & ... & ... & ...  \\
_{1}\left\langle  {{a}_{1}} \right|{{\left| {{a}_{N}} \right\rangle }_{1}} & _{2}\left\langle  {{a}_{2}} \right|{{\left| {{a}_{N}} \right\rangle }_{2}} & ... & _{N}\left\langle  {{a}_{N}} \right|{{\left| {{a}_{N}} \right\rangle }_{N}}  \\
\end{matrix} \right| \\
& _{1}\left\langle  {{a}_{1}} \right|{{\left| {{a}_{1}} \right\rangle }_{1}}=1 \\
& _{1}\left\langle  {{a}_{i}} \right|{{\left| {{a}_{j}} \right\rangle }_{1}}=0\quad i\ne j \\
& \Rightarrow {{f}_{n}}^{2}\frac{1}{N!}\left| \begin{matrix}
_{1}\left\langle  {{a}_{1}} \right|{{\left| {{a}_{1}} \right\rangle }_{1}} & _{2}\left\langle  {{a}_{2}} \right|{{\left| {{a}_{1}} \right\rangle }_{2}} & ... & _{N}\left\langle  {{a}_{N}} \right|{{\left| {{a}_{1}} \right\rangle }_{N}}  \\
_{1}\left\langle  {{a}_{1}} \right|{{\left| {{a}_{2}} \right\rangle }_{1}} & _{2}\left\langle  {{a}_{2}} \right|{{\left| {{a}_{2}} \right\rangle }_{2}} & ... & _{N}\left\langle  {{a}_{N}} \right|{{\left| {{a}_{2}} \right\rangle }_{N}}  \\
... & ... & ... & ...  \\
_{1}\left\langle  {{a}_{1}} \right|{{\left| {{a}_{N}} \right\rangle }_{1}} & _{2}\left\langle  {{a}_{2}} \right|{{\left| {{a}_{N}} \right\rangle }_{2}} & ... & _{N}\left\langle  {{a}_{N}} \right|{{\left| {{a}_{N}} \right\rangle }_{N}}  \\
\end{matrix} \right|={{f}_{n}}^{2}\frac{1}{N!}\left| \begin{matrix}
1 & 0 & ... & 0  \\
0 & 1 & ... & 0  \\
... & ... & ... & ...  \\
0 & 0 & ... & 1  \\
\end{matrix} \right|={{f}_{n}}^{2}\frac{1}{N!}\equiv 1! \\
& \Rightarrow {{f}_{n}}=\sqrt{N!} \\
\end{align}

TeX (checked):

{\begin{aligned}&{{f}_{n}}{{^{2}}_{a}}\left\langle {{a}_{1}},...,{{a}_{N}}\right|{{\left|{{a}_{1}},...,{{a}_{N}}\right\rangle }_{a}}={{f}_{n}}^{2}\left(_{1}\left\langle {{a}_{1}}\right|{{...}_{N}}\left\langle {{a}_{N}}\right|\right){\hat {A}}{\hat {A}}\left({{\left|{{a}_{1}}\right\rangle }_{1}}...{{\left|{{a}_{N}}\right\rangle }_{N}}\right)\\&={{f}_{n}}^{2}\left(_{1}\left\langle {{a}_{1}}\right|{{...}_{N}}\left\langle {{a}_{N}}\right|\right){\hat {A}}\left({{\left|{{a}_{1}}\right\rangle }_{1}}...{{\left|{{a}_{N}}\right\rangle }_{N}}\right)={{f}_{n}}^{2}\left(_{1}\left\langle {{a}_{1}}\right|{{...}_{N}}\left\langle {{a}_{N}}\right|\right){\frac {1}{N!}}\left|{\begin{matrix}{{\left|{{a}_{1}}\right\rangle }_{1}}&{{\left|{{a}_{1}}\right\rangle }_{2}}&...&{{\left|{{a}_{1}}\right\rangle }_{N}}\\{{\left|{{a}_{2}}\right\rangle }_{1}}&{{\left|{{a}_{2}}\right\rangle }_{2}}&...&{{\left|{{a}_{2}}\right\rangle }_{N}}\\...&...&...&...\\{{\left|{{a}_{N}}\right\rangle }_{1}}&{{\left|{{a}_{N}}\right\rangle }_{2}}&...&{{\left|{{a}_{N}}\right\rangle }_{N}}\\\end{matrix}}\right|\\&={{f}_{n}}^{2}{\frac {1}{N!}}\left|{\begin{matrix}_{1}\left\langle {{a}_{1}}\right|{{\left|{{a}_{1}}\right\rangle }_{1}}&_{2}\left\langle {{a}_{2}}\right|{{\left|{{a}_{1}}\right\rangle }_{2}}&...&_{N}\left\langle {{a}_{N}}\right|{{\left|{{a}_{1}}\right\rangle }_{N}}\\_{1}\left\langle {{a}_{1}}\right|{{\left|{{a}_{2}}\right\rangle }_{1}}&_{2}\left\langle {{a}_{2}}\right|{{\left|{{a}_{2}}\right\rangle }_{2}}&...&_{N}\left\langle {{a}_{N}}\right|{{\left|{{a}_{2}}\right\rangle }_{N}}\\...&...&...&...\\_{1}\left\langle {{a}_{1}}\right|{{\left|{{a}_{N}}\right\rangle }_{1}}&_{2}\left\langle {{a}_{2}}\right|{{\left|{{a}_{N}}\right\rangle }_{2}}&...&_{N}\left\langle {{a}_{N}}\right|{{\left|{{a}_{N}}\right\rangle }_{N}}\\\end{matrix}}\right|\\&_{1}\left\langle {{a}_{1}}\right|{{\left|{{a}_{1}}\right\rangle }_{1}}=1\\&_{1}\left\langle {{a}_{i}}\right|{{\left|{{a}_{j}}\right\rangle }_{1}}=0\quad i\neq j\\&\Rightarrow {{f}_{n}}^{2}{\frac {1}{N!}}\left|{\begin{matrix}_{1}\left\langle {{a}_{1}}\right|{{\left|{{a}_{1}}\right\rangle }_{1}}&_{2}\left\langle {{a}_{2}}\right|{{\left|{{a}_{1}}\right\rangle }_{2}}&...&_{N}\left\langle {{a}_{N}}\right|{{\left|{{a}_{1}}\right\rangle }_{N}}\\_{1}\left\langle {{a}_{1}}\right|{{\left|{{a}_{2}}\right\rangle }_{1}}&_{2}\left\langle {{a}_{2}}\right|{{\left|{{a}_{2}}\right\rangle }_{2}}&...&_{N}\left\langle {{a}_{N}}\right|{{\left|{{a}_{2}}\right\rangle }_{N}}\\...&...&...&...\\_{1}\left\langle {{a}_{1}}\right|{{\left|{{a}_{N}}\right\rangle }_{1}}&_{2}\left\langle {{a}_{2}}\right|{{\left|{{a}_{N}}\right\rangle }_{2}}&...&_{N}\left\langle {{a}_{N}}\right|{{\left|{{a}_{N}}\right\rangle }_{N}}\\\end{matrix}}\right|={{f}_{n}}^{2}{\frac {1}{N!}}\left|{\begin{matrix}1&0&...&0\\0&1&...&0\\...&...&...&...\\0&0&...&1\\\end{matrix}}\right|={{f}_{n}}^{2}{\frac {1}{N!}}\equiv 1!\\&\Rightarrow {{f}_{n}}={\sqrt {N!}}\\\end{aligned}}

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\begin{matrix} f_{n} {^{2}}_{a} ⟨ a_{1}, ..., a_{N} | {| a_{1}, ..., a_{N} ⟩}_{a} = {f_{n}}^{2} (_{1} ⟨ a_{1} | {...}_{N} ⟨ a_{N} |) \hat{A} \hat{A} ({| a_{1} ⟩}_{1} ... {| a_{N} ⟩}_{N}) \\ = {f_{n}}^{2} (_{1} ⟨ a_{1} | {...}_{N} ⟨ a_{N} |) \hat{A} ({| a_{1} ⟩}_{1} ... {| a_{N} ⟩}_{N}) = {f_{n}}^{2} (_{1} ⟨ a_{1} | {...}_{N} ⟨ a_{N} |) \frac{1}{N!} | \begin{matrix} {| a_{1} ⟩}_{1} & {| a_{1} ⟩}_{2} & ... & {| a_{1} ⟩}_{N} \\ {| a_{2} ⟩}_{1} & {| a_{2} ⟩}_{2} & ... & {| a_{2} ⟩}_{N} \\ ... & ... & ... & ... \\ {| a_{N} ⟩}_{1} & {| a_{N} ⟩}_{2} & ... & {| a_{N} ⟩}_{N} \end{matrix} | \\ = {f_{n}}^{2} \frac{1}{N!} | \begin{matrix} _{1} ⟨ a_{1} | {| a_{1} ⟩}_{1} & _{2} ⟨ a_{2} | {| a_{1} ⟩}_{2} & ... & _{N} ⟨ a_{N} | {| a_{1} ⟩}_{N} \\ _{1} ⟨ a_{1} | {| a_{2} ⟩}_{1} & _{2} ⟨ a_{2} | {| a_{2} ⟩}_{2} & ... & _{N} ⟨ a_{N} | {| a_{2} ⟩}_{N} \\ ... & ... & ... & ... \\ _{1} ⟨ a_{1} | {| a_{N} ⟩}_{1} & _{2} ⟨ a_{2} | {| a_{N} ⟩}_{2} & ... & _{N} ⟨ a_{N} | {| a_{N} ⟩}_{N} \end{matrix} | \\ _{1} ⟨ a_{1} | {| a_{1} ⟩}_{1} = 1 \\ _{1} ⟨ a_{i} | {| a_{j} ⟩}_{1} = 0 i \neq j \\ \Rightarrow {f_{n}}^{2} \frac{1}{N!} | \begin{matrix} _{1} ⟨ a_{1} | {| a_{1} ⟩}_{1} & _{2} ⟨ a_{2} | {| a_{1} ⟩}_{2} & ... & _{N} ⟨ a_{N} | {| a_{1} ⟩}_{N} \\ _{1} ⟨ a_{1} | {| a_{2} ⟩}_{1} & _{2} ⟨ a_{2} | {| a_{2} ⟩}_{2} & ... & _{N} ⟨ a_{N} | {| a_{2} ⟩}_{N} \\ ... & ... & ... & ... \\ _{1} ⟨ a_{1} | {| a_{N} ⟩}_{1} & _{2} ⟨ a_{2} | {| a_{N} ⟩}_{2} & ... & _{N} ⟨ a_{N} | {| a_{N} ⟩}_{N} \end{matrix} | = {f_{n}}^{2} \frac{1}{N!} | \begin{matrix} 1 & 0 & ... & 0 \\ 0 & 1 & ... & 0 \\ ... & ... & ... & ... \\ 0 & 0 & ... & 1 \end{matrix} | = {f_{n}}^{2} \frac{1}{N!} \equiv 1! \\ \Rightarrow f_{n} = \sqrt{N!} \end{matrix}

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symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">=</mo><msup><msub><mi>f</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><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"><mi>N</mi><mo stretchy="false">!</mo></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN"></mo><mtable><mtr><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow 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data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub></mtd><mtd><mi>...</mi></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd></mtr><mtr><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub></mtd><mtd><mi>...</mi></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">=</mo><mn>1</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">=</mo><mn>0</mn><mspace width="1em"></mspace><mi>i</mi><mo stretchy="false">≠</mo><mi>j</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msup><msub><mi>f</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><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"><mi>N</mi><mo stretchy="false">!</mo></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN"></mo><mtable><mtr><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub></mtd><mtd><mi>...</mi></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub></mtd><mtd><mi>...</mi></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd></mtr><mtr><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub></mtd><mtd><mi>...</mi></mtd><mtd><msub><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo stretchy="false">=</mo><msup><msub><mi>f</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><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"><mi>N</mi><mo stretchy="false">!</mo></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN"></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>...</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>...</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd><mtd><mi>...</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>...</mi></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo stretchy="false">=</mo><msup><msub><mi>f</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><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"><mi>N</mi><mo stretchy="false">!</mo></mrow></mrow></mfrac></mrow><mo stretchy="false">≡</mo><mn>1</mn><mo stretchy="false">!</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msub><mi>f</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><msqrt><mrow data-mjx-texclass="ORD"><mi>N</mi><mo stretchy="false">!</mo></mrow></msqrt></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Identifiers

$f_{n}$

$a$

$a_{1}$

$a_{N}$

$a_{1}$

$a_{N}$

$a$

$f_{n}$

$a_{1}$

$N$

$a_{N}$

$\hat{A}$

$\hat{A}$

$a_{1}_{1}$

$a_{N}_{N}$

$f_{n}$

$a_{1}$

$N$

$a_{N}$

$\hat{A}$

$a_{1}_{1}$

$a_{N}_{N}$

$f_{n}$

$a_{1}$

$N$

$a_{N}$

$N$

$a_{1}_{1}$

$a_{1}_{2}$

$a_{1}_{N}$

$a_{2}_{1}$

$a_{2}_{2}$

$a_{2}_{N}$

$a_{N}_{1}$

$a_{N}_{2}$

$a_{N}_{N}$

$f_{n}$

$N$

$a_{1}$

$a_{1}_{1}$

$a_{2}$

$a_{1}_{2}$

$N$

$a_{N}$

$a_{1}_{N}$

$a_{1}$

$a_{2}_{1}$

$a_{2}$

$a_{2}_{2}$

$N$

$a_{N}$

$a_{2}_{N}$

$a_{1}$

$a_{N}_{1}$

$a_{2}$

$a_{N}_{2}$

$N$

$a_{N}$

$a_{N}_{N}$

$a_{1}$

$a_{1}_{1}$

$a_{i}$

$a_{j}_{1}$

$i$

$j$

$f_{n}$

$N$

$a_{1}$

$a_{1}_{1}$

$a_{2}$

$a_{1}_{2}$

$N$

$a_{N}$

$a_{1}_{N}$

$a_{1}$

$a_{2}_{1}$

$a_{2}$

$a_{2}_{2}$

$N$

$a_{N}$

$a_{2}_{N}$

$a_{1}$

$a_{N}_{1}$

$a_{2}$

$a_{N}_{2}$

$N$

$a_{N}$

$a_{N}_{N}$

$f_{n}$

$N$

$f_{n}$

$N$

$f_{n}$

$N$

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