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Display information for equation id:math.1821.27 on revision:1821
* Page found: Der nichtrelativistische Grenzfall (eq math.1821.27)
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\begin{align}
& \left( \bar{\pi }\times \bar{\pi } \right){{\phi }_{a}}=\left( \bar{p}-e\bar{A} \right)\times \left( \bar{p}-e\bar{A} \right){{\phi }_{a}} \\
& =\bar{p}\times \left( \bar{p}{{\phi }_{a}} \right)-e\left[ \bar{p}\times \left( \bar{A}{{\phi }_{a}} \right)+\bar{A}\times \bar{p}{{\phi }_{a}} \right]+{{e}^{2}}\left( \bar{A}\times \bar{A} \right)i{{\phi }_{a}} \\
& \bar{p}\times \left( \bar{p}{{\phi }_{a}} \right)=0 \\
& {{e}^{2}}\left( \bar{A}\times \bar{A} \right)i{{\phi }_{a}}=0 \\
& e\left[ \bar{p}\times \left( \bar{A}{{\phi }_{a}} \right)+\bar{A}\times \bar{p}{{\phi }_{a}} \right]=\frac{e\hbar }{i}\bar{B}{{\phi }_{a}} \\
& \\
& \Rightarrow \left( \bar{\sigma }\bar{\pi } \right)\left( \bar{\sigma }\bar{\pi } \right)={{{\bar{\pi }}}^{2}}+i\bar{\sigma }\left( \bar{\pi }\times \bar{\pi } \right)={{\left( \bar{p}-e\bar{A} \right)}^{2}}-e\hbar \bar{\sigma }\bar{B} \\
\end{align}
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data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mo>−</mo><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></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>p</mi><mo>¯</mo></mover></mrow></mrow><mo>−</mo><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</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>p</mi><mo>¯</mo></mover></mrow></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mi>e</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</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>A</mi><mo>¯</mo></mover></mrow></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><msup><mi>e</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"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>i</mi><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</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>p</mi><mo>¯</mo></mover></mrow></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><msup><mi>e</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"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>i</mi><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mi>e</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</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>A</mi><mo>¯</mo></mover></mrow></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>e</mi><mi data-mjx-alternate="1">ℏ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>B</mi><mo>¯</mo></mover></mrow></mrow><msub><mi>ϕ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd><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>σ</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>π</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>σ</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>π</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>π</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>+</mo><mi>i</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>σ</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>π</mi><mo>¯</mo></mover></mrow></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>π</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mo>−</mo><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>−</mo><mi>e</mi><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>σ</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>B</mi><mo>¯</mo></mover></mrow></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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