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Kontinuitätsgleichung (Quantenmechnik) Eq: math.1594.5

Hash: 9f89cbfc3263d8e238adae1758500ccc

TeX (original user input):

\begin{align}
& i\hbar \frac{\partial }{\partial t}{{\left| \Psi (\bar{r},t) \right|}^{2}}=\frac{-{{\hbar }^{2}}}{2m}\left( \Psi *\Delta \Psi -\Psi \Delta \Psi * \right)+\frac{{{e}^{2}}}{2m}\left[ \Psi *{{{\bar{A}}}^{2}}\Psi -\Psi {{{\bar{A}}}^{2}}\Psi * \right]+\Psi *V\Psi -\Psi V\Psi * \\
& \quad \quad \quad \quad \quad \quad +\frac{i\hbar e}{2m}\left( \Psi *\nabla \left( \bar{A}\Psi  \right)+\bar{A}\Psi \nabla \Psi *+\Psi \nabla \left( \bar{A}\Psi * \right)+\bar{A}\Psi *\nabla \Psi  \right) \\
& \Psi *{{{\bar{A}}}^{2}}\Psi -\Psi {{{\bar{A}}}^{2}}\Psi *=0 \\
& \Psi *V\Psi -\Psi V\Psi *=0 \\
& \Psi *\nabla \left( \bar{A}\Psi  \right)+\bar{A}\Psi \nabla \Psi *=\Psi \nabla \left( \bar{A}\Psi * \right)+\bar{A}\Psi *\nabla \Psi =\nabla \left( \Psi \bar{A}\Psi * \right) \\
& \Rightarrow i\hbar \frac{\partial }{\partial t}{{\left| \Psi (\bar{r},t) \right|}^{2}}=\frac{-{{\hbar }^{2}}}{2m}\left( \Psi *\Delta \Psi -\Psi \Delta \Psi * \right)+\frac{i\hbar e}{m}\nabla \left( \Psi \bar{A}\Psi * \right) \\
& \Psi *\Delta \Psi -\Psi \Delta \Psi *=\nabla \left( \Psi *\nabla \Psi -\Psi \nabla \Psi * \right)-\left( \nabla \Psi *\nabla \Psi -\nabla \Psi \nabla \Psi * \right) \\
& \left( \nabla \Psi *\nabla \Psi -\nabla \Psi \nabla \Psi * \right)=0 \\
& \Rightarrow i\hbar \frac{\partial }{\partial t}{{\left| \Psi (\bar{r},t) \right|}^{2}}=\frac{-{{\hbar }^{2}}}{2m}\nabla \left( \Psi *\nabla \Psi -\Psi \nabla \Psi * \right)+\frac{i\hbar e}{m}\nabla \left( \Psi \bar{A}\Psi * \right) \\
& \Rightarrow i\hbar \frac{\partial }{\partial t}{{\left| \Psi (\bar{r},t) \right|}^{2}}=\nabla \left[ \frac{-{{\hbar }^{2}}}{2m}\left( \Psi *\nabla \Psi -\Psi \nabla \Psi * \right)+\frac{i\hbar e}{m}\left( \Psi \bar{A}\Psi * \right) \right] \\
\end{align}

TeX (checked):

{\begin{aligned}&i\hbar {\frac {\partial }{\partial t}}{{\left|\Psi ({\bar {r}},t)\right|}^{2}}={\frac {-{{\hbar }^{2}}}{2m}}\left(\Psi *\Delta \Psi -\Psi \Delta \Psi *\right)+{\frac {{e}^{2}}{2m}}\left[\Psi *{{\bar {A}}^{2}}\Psi -\Psi {{\bar {A}}^{2}}\Psi *\right]+\Psi *V\Psi -\Psi V\Psi *\\&\quad \quad \quad \quad \quad \quad +{\frac {i\hbar e}{2m}}\left(\Psi *\nabla \left({\bar {A}}\Psi \right)+{\bar {A}}\Psi \nabla \Psi *+\Psi \nabla \left({\bar {A}}\Psi *\right)+{\bar {A}}\Psi *\nabla \Psi \right)\\&\Psi *{{\bar {A}}^{2}}\Psi -\Psi {{\bar {A}}^{2}}\Psi *=0\\&\Psi *V\Psi -\Psi V\Psi *=0\\&\Psi *\nabla \left({\bar {A}}\Psi \right)+{\bar {A}}\Psi \nabla \Psi *=\Psi \nabla \left({\bar {A}}\Psi *\right)+{\bar {A}}\Psi *\nabla \Psi =\nabla \left(\Psi {\bar {A}}\Psi *\right)\\&\Rightarrow i\hbar {\frac {\partial }{\partial t}}{{\left|\Psi ({\bar {r}},t)\right|}^{2}}={\frac {-{{\hbar }^{2}}}{2m}}\left(\Psi *\Delta \Psi -\Psi \Delta \Psi *\right)+{\frac {i\hbar e}{m}}\nabla \left(\Psi {\bar {A}}\Psi *\right)\\&\Psi *\Delta \Psi -\Psi \Delta \Psi *=\nabla \left(\Psi *\nabla \Psi -\Psi \nabla \Psi *\right)-\left(\nabla \Psi *\nabla \Psi -\nabla \Psi \nabla \Psi *\right)\\&\left(\nabla \Psi *\nabla \Psi -\nabla \Psi \nabla \Psi *\right)=0\\&\Rightarrow i\hbar {\frac {\partial }{\partial t}}{{\left|\Psi ({\bar {r}},t)\right|}^{2}}={\frac {-{{\hbar }^{2}}}{2m}}\nabla \left(\Psi *\nabla \Psi -\Psi \nabla \Psi *\right)+{\frac {i\hbar e}{m}}\nabla \left(\Psi {\bar {A}}\Psi *\right)\\&\Rightarrow i\hbar {\frac {\partial }{\partial t}}{{\left|\Psi ({\bar {r}},t)\right|}^{2}}=\nabla \left[{\frac {-{{\hbar }^{2}}}{2m}}\left(\Psi *\nabla \Psi -\Psi \nabla \Psi *\right)+{\frac {i\hbar e}{m}}\left(\Psi {\bar {A}}\Psi *\right)\right]\\\end{aligned}}

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\begin{matrix} i ℏ \frac{\partial}{\partial t} {| Ψ (\bar{r}, t) |}^{2} = \frac{- ℏ^{2}}{2 m} (Ψ * Δ Ψ - Ψ Δ Ψ *) + \frac{e^{2}}{2 m} [Ψ * {\bar{A}}^{2} Ψ - Ψ {\bar{A}}^{2} Ψ *] + Ψ * V Ψ - Ψ V Ψ * \\ + \frac{i ℏ e}{2 m} (Ψ * \nabla (\bar{A} Ψ) + \bar{A} Ψ \nabla Ψ * + Ψ \nabla (\bar{A} Ψ *) + \bar{A} Ψ * \nabla Ψ) \\ Ψ * {\bar{A}}^{2} Ψ - Ψ {\bar{A}}^{2} Ψ * = 0 \\ Ψ * V Ψ - Ψ V Ψ * = 0 \\ Ψ * \nabla (\bar{A} Ψ) + \bar{A} Ψ \nabla Ψ * = Ψ \nabla (\bar{A} Ψ *) + \bar{A} Ψ * \nabla Ψ = \nabla (Ψ \bar{A} Ψ *) \\ \Rightarrow i ℏ \frac{\partial}{\partial t} {| Ψ (\bar{r}, t) |}^{2} = \frac{- ℏ^{2}}{2 m} (Ψ * Δ Ψ - Ψ Δ Ψ *) + \frac{i ℏ e}{m} \nabla (Ψ \bar{A} Ψ *) \\ Ψ * Δ Ψ - Ψ Δ Ψ * = \nabla (Ψ * \nabla Ψ - Ψ \nabla Ψ *) - (\nabla Ψ * \nabla Ψ - \nabla Ψ \nabla Ψ *) \\ (\nabla Ψ * \nabla Ψ - \nabla Ψ \nabla Ψ *) = 0 \\ \Rightarrow i ℏ \frac{\partial}{\partial t} {| Ψ (\bar{r}, t) |}^{2} = \frac{- ℏ^{2}}{2 m} \nabla (Ψ * \nabla Ψ - Ψ \nabla Ψ *) + \frac{i ℏ e}{m} \nabla (Ψ \bar{A} Ψ *) \\ \Rightarrow i ℏ \frac{\partial}{\partial t} {| Ψ (\bar{r}, t) |}^{2} = \nabla [\frac{- ℏ^{2}}{2 m} (Ψ * \nabla Ψ - Ψ \nabla Ψ *) + \frac{i ℏ e}{m} (Ψ \bar{A} Ψ *)] \end{matrix}

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data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">−</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">−</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mi>i</mi><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Ψ</mi><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><msup><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Ψ</mi><mo stretchy="false">∗</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">−</mo><mi>Ψ</mi><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi alternate="1">ℏ</mi><mi>e</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></mfrac></mrow><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Ψ</mi><mover><mi>A</mi><mo>¯</mo></mover><mi>Ψ</mi><mo stretchy="false">∗</mo><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><mi>i</mi><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Ψ</mi><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">=</mo><mi mathvariant="normal">∇</mi><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"><mo stretchy="false">−</mo><msup><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Ψ</mi><mo stretchy="false">∗</mo><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">−</mo><mi>Ψ</mi><mi mathvariant="normal">∇</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi alternate="1">ℏ</mi><mi>e</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Ψ</mi><mover><mi>A</mi><mo>¯</mo></mover><mi>Ψ</mi><mo stretchy="false">∗</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$i$

$t$

$Ψ$

$\bar{r}$

$t$

$m$

$Ψ$

$Δ$

$Ψ$

$Ψ$

$Δ$

$Ψ$

$e$

$m$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$V$

$Ψ$

$Ψ$

$V$

$Ψ$

$i$

$e$

$m$

$Ψ$

$\bar{A}$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$Ψ$

$\bar{A}$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$V$

$Ψ$

$Ψ$

$V$

$Ψ$

$Ψ$

$\bar{A}$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$Ψ$

$\bar{A}$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$Ψ$

$\bar{A}$

$Ψ$

$i$

$t$

$Ψ$

$\bar{r}$

$t$

$m$

$Ψ$

$Δ$

$Ψ$

$Ψ$

$Δ$

$Ψ$

$i$

$e$

$m$

$Ψ$

$\bar{A}$

$Ψ$

$Ψ$

$Δ$

$Ψ$

$Ψ$

$Δ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$i$

$t$

$Ψ$

$\bar{r}$

$t$

$m$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$i$

$e$

$m$

$Ψ$

$\bar{A}$

$Ψ$

$i$

$t$

$Ψ$

$\bar{r}$

$t$

$m$

$Ψ$

$Ψ$

$Ψ$

$Ψ$

$i$

$e$

$m$

$Ψ$

$\bar{A}$

$Ψ$

MathML observations

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General

LaTeXML (experimentell; verwendet MathML) rendering

MathML (experimentell; keine Bilder) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigationsmenü

General

LaTeXML (experimentell; verwendet MathML) rendering

MathML (experimentell; keine Bilder) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigationsmenü

Suche