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Display information for equation id:math.2232.9 on revision:2232
* Page found: Quantentheoretischer Zugang (eq math.2232.9)
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Hash: c67ef3f3f522ac7bb7eafcefc38d50a7
TeX (original user input):
\begin{align}
& \Rightarrow {{e}^{i\vec{k}.\vec{r}}}={{e}^{i\vec{k}.\left( \vec{r}+\vec{L} \right)}},\quad \vec{L}=\left( L,L,L \right) \\
& \Rightarrow {{e}^{i\vec{k}.\vec{r}}}=1\text{ w }\!\!\ddot{\mathrm{a}}\!\!\text{ hlen} \\
& \Rightarrow {{k}_{i}}=\left( {{k}_{x}},{{k}_{y}},{{k}_{z}} \right):\,\,{{k}_{i}}=\frac{2\pi }{L}{{m}_{i}},\,\,{{m}_{i}}\in \mathbb{Z} \\
\end{align}
TeX (checked):
{\begin{aligned}&\Rightarrow {{e}^{i{\vec {k}}.{\vec {r}}}}={{e}^{i{\vec {k}}.\left({\vec {r}}+{\vec {L}}\right)}},\quad {\vec {L}}=\left(L,L,L\right)\\&\Rightarrow {{e}^{i{\vec {k}}.{\vec {r}}}}=1{\text{ w }}\!\!{\ddot {\mathrm {a} }}\!\!{\text{ hlen}}\\&\Rightarrow {{k}_{i}}=\left({{k}_{x}},{{k}_{y}},{{k}_{z}}\right):\,\,{{k}_{i}}={\frac {2\pi }{L}}{{m}_{i}},\,\,{{m}_{i}}\in \mathbb {Z} \\\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><mo>⇒</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>k</mi><mo>→</mo></mover></mrow></mrow><mo>.</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>→</mo></mover></mrow></mrow></mrow></mrow></msup><mo>=</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>k</mi><mo>→</mo></mover></mrow></mrow><mo>.</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>→</mo></mover></mrow></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>→</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow></msup><mo>,</mo><mspace width="1em"></mspace><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>→</mo></mover></mrow></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>L</mi><mo>,</mo><mi>L</mi><mo>,</mo><mi>L</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>k</mi><mo>→</mo></mover></mrow></mrow><mo>.</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>→</mo></mover></mrow></mrow></mrow></mrow></msup><mo>=</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mtext> w </mtext></mrow><mspace width="-0.167em"></mspace><mspace width="-0.167em"></mspace><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">a</mi></mrow><mo>¨</mo></mover></mrow></mrow><mspace width="-0.167em"></mspace><mspace width="-0.167em"></mspace><mrow data-mjx-texclass="ORD"><mtext> hlen</mtext></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub><mo>,</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub><mo>,</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>:</mi><mspace width="0.167em"></mspace><mspace width="0.167em"></mspace><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><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>L</mi></mrow></mfrac></mrow><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>,</mo><mspace width="0.167em"></mspace><mspace width="0.167em"></mspace><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>∈</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">ℤ</mi></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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