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* Page found: Operatoren im Hilbertraum (eq math.1626.19)

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\begin{align}

& \Phi :=\hat{\bar{r}}\left| \Psi  \right\rangle  \\

& \Phi (\bar{p})\equiv \left\langle  {\bar{p}} | \Phi  \right\rangle =\left\langle  {\bar{p}} \right|\hat{\bar{r}}\left| \Psi  \right\rangle  \\

& \Phi (\bar{p})=\int_{{}}^{{}}{{{d}^{3}}r}\left\langle  {\bar{p}} | {\bar{r}} \right\rangle \left\langle  {\bar{r}} \right|\hat{\bar{r}}\left| \Psi  \right\rangle  \\

& \left\langle  {\bar{p}} | {\bar{r}} \right\rangle =\frac{1}{{{\left( 2\pi \hbar  \right)}^{\tfrac{3}{2}}}}{{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}} \\

& \left\langle  {\bar{r}} \right|\hat{\bar{r}}\left| \Psi  \right\rangle =\bar{r}\left\langle  {\bar{r}} | \Psi  \right\rangle  \\

& \Rightarrow \Phi (\bar{p})=\int_{{}}^{{}}{{{d}^{3}}r}\left\langle  {\bar{p}} | {\bar{r}} \right\rangle \left\langle  {\bar{r}} \right|\hat{\bar{r}}\left| \Psi  \right\rangle =\int_{{}}^{{}}{{{d}^{3}}r}\frac{1}{{{\left( 2\pi \hbar  \right)}^{\tfrac{3}{2}}}}{{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}}\bar{r}\Psi (\bar{r})=\frac{1}{{{\left( 2\pi \hbar  \right)}^{\tfrac{3}{2}}}}\int_{{}}^{{}}{{{d}^{3}}r}\bar{r}{{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}}\Psi (\bar{r}) \\

& \bar{r}{{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}}=-\frac{\hbar }{i}{{\nabla }_{p}}\left( {{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}} \right) \\

& {{\nabla }_{p}}:=\left( \begin{matrix}

\frac{\partial }{\partial {{p}_{x}}}, & \frac{\partial }{\partial {{p}_{y}}}, & \frac{\partial }{\partial {{p}_{z}}}  \\

\end{matrix} \right) \\

& \Rightarrow \Phi (\bar{p})=\frac{1}{{{\left( 2\pi \hbar  \right)}^{\tfrac{3}{2}}}}\int_{{}}^{{}}{{{d}^{3}}r}\bar{r}{{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}}\Psi (\bar{r})=-\frac{\hbar }{i}{{\nabla }_{p}}\left[ \int_{{}}^{{}}{{{d}^{3}}r\frac{1}{{{\left( 2\pi \hbar  \right)}^{\tfrac{3}{2}}}}}{{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}}\Psi (\bar{r}) \right] \\

& \frac{1}{{{\left( 2\pi \hbar  \right)}^{\tfrac{3}{2}}}}{{e}^{-i\frac{\bar{p}\bar{r}}{\hbar }}}=\left\langle  {\bar{p}} | {\bar{r}} \right\rangle  \\

& \Rightarrow \Phi (\bar{p})=-\frac{\hbar }{i}{{\nabla }_{p}}\left[ \int_{{}}^{{}}{{{d}^{3}}r\left\langle  {\bar{p}} | {\bar{r}} \right\rangle \left\langle  {\bar{r}} | \Psi  \right\rangle } \right]=-\frac{\hbar }{i}{{\nabla }_{p}}\tilde{\Psi }(\bar{p}) \\

\end{align}

TeX (checked): 

{\begin{aligned}&\Phi :={\hat {\bar {r}}}\left|\Psi \right\rangle \\&\Phi ({\bar {p}})\equiv \left\langle {\bar {p}}|\Phi \right\rangle =\left\langle {\bar {p}}\right|{\hat {\bar {r}}}\left|\Psi \right\rangle \\&\Phi ({\bar {p}})=\int _{}^{}{{{d}^{3}}r}\left\langle {\bar {p}}|{\bar {r}}\right\rangle \left\langle {\bar {r}}\right|{\hat {\bar {r}}}\left|\Psi \right\rangle \\&\left\langle {\bar {p}}|{\bar {r}}\right\rangle ={\frac {1}{{\left(2\pi \hbar \right)}^{\tfrac {3}{2}}}}{{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}\\&\left\langle {\bar {r}}\right|{\hat {\bar {r}}}\left|\Psi \right\rangle ={\bar {r}}\left\langle {\bar {r}}|\Psi \right\rangle \\&\Rightarrow \Phi ({\bar {p}})=\int _{}^{}{{{d}^{3}}r}\left\langle {\bar {p}}|{\bar {r}}\right\rangle \left\langle {\bar {r}}\right|{\hat {\bar {r}}}\left|\Psi \right\rangle =\int _{}^{}{{{d}^{3}}r}{\frac {1}{{\left(2\pi \hbar \right)}^{\tfrac {3}{2}}}}{{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}{\bar {r}}\Psi ({\bar {r}})={\frac {1}{{\left(2\pi \hbar \right)}^{\tfrac {3}{2}}}}\int _{}^{}{{{d}^{3}}r}{\bar {r}}{{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}\Psi ({\bar {r}})\\&{\bar {r}}{{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}=-{\frac {\hbar }{i}}{{\nabla }_{p}}\left({{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}\right)\\&{{\nabla }_{p}}:=\left({\begin{matrix}{\frac {\partial }{\partial {{p}_{x}}}},&{\frac {\partial }{\partial {{p}_{y}}}},&{\frac {\partial }{\partial {{p}_{z}}}}\\\end{matrix}}\right)\\&\Rightarrow \Phi ({\bar {p}})={\frac {1}{{\left(2\pi \hbar \right)}^{\tfrac {3}{2}}}}\int _{}^{}{{{d}^{3}}r}{\bar {r}}{{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}\Psi ({\bar {r}})=-{\frac {\hbar }{i}}{{\nabla }_{p}}\left[\int _{}^{}{{{d}^{3}}r{\frac {1}{{\left(2\pi \hbar \right)}^{\tfrac {3}{2}}}}}{{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}\Psi ({\bar {r}})\right]\\&{\frac {1}{{\left(2\pi \hbar \right)}^{\tfrac {3}{2}}}}{{e}^{-i{\frac {{\bar {p}}{\bar {r}}}{\hbar }}}}=\left\langle {\bar {p}}|{\bar {r}}\right\rangle \\&\Rightarrow \Phi ({\bar {p}})=-{\frac {\hbar }{i}}{{\nabla }_{p}}\left[\int _{}^{}{{{d}^{3}}r\left\langle {\bar {p}}|{\bar {r}}\right\rangle \left\langle {\bar {r}}|\Psi \right\rangle }\right]=-{\frac {\hbar }{i}}{{\nabla }_{p}}{\tilde {\Psi }}({\bar {p}})\\\end{aligned}}

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Φ:=r¯^|Ψ⟩Φ(p¯)≡⟨p¯|Φ⟩=⟨p¯|r¯^|Ψ⟩Φ(p¯)=∫d3r⟨p¯|r¯⟩⟨r¯|r¯^|Ψ⟩⟨p¯|r¯⟩=1(2πℏ)32e−ip¯r¯ℏ⟨r¯|r¯^|Ψ⟩=r¯⟨r¯|Ψ⟩⇒Φ(p¯)=∫d3r⟨p¯|r¯⟩⟨r¯|r¯^|Ψ⟩=∫d3r1(2πℏ)32e−ip¯r¯ℏr¯Ψ(r¯)=1(2πℏ)32∫d3rr¯e−ip¯r¯ℏΨ(r¯)r¯e−ip¯r¯ℏ=−ℏi∇p(e−ip¯r¯ℏ)∇p:=(∂∂px,∂∂py,∂∂pz)⇒Φ(p¯)=1(2πℏ)32∫d3rr¯e−ip¯r¯ℏΨ(r¯)=−ℏi∇p[∫d3r1(2πℏ)32e−ip¯r¯ℏΨ(r¯)]1(2πℏ)32e−ip¯r¯ℏ=⟨p¯|r¯⟩⇒Φ(p¯)=−ℏi∇p[∫d3r⟨p¯|r¯⟩⟨r¯|Ψ⟩]=−ℏi∇pΨ~(p¯)
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data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>,</mo></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x21D2;</mo><mi mathvariant="normal">&#x03A6;</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi data-mjx-alternate="1">&#x210F;</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow></mstyle></mrow></mrow></msup></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>&#x2212;</mo><mi>i</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow></mfrac></mrow></mrow></mrow></msup><mi mathvariant="normal">&#x03A8;</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></mfrac></mrow><msub><mi mathvariant="normal">&#x2207;</mi><mrow data-mjx-texclass="ORD"><mi>p</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi data-mjx-alternate="1">&#x210F;</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow></mstyle></mrow></mrow></msup></mrow></mfrac></mrow></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>&#x2212;</mo><mi>i</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow></mfrac></mrow></mrow></mrow></msup><mi mathvariant="normal">&#x03A8;</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi data-mjx-alternate="1">&#x210F;</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow></mstyle></mrow></mrow></msup></mrow></mfrac></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>&#x2212;</mo><mi>i</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow></mfrac></mrow></mrow></mrow></msup><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</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="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x21D2;</mo><mi mathvariant="normal">&#x03A6;</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></mfrac></mrow><msub><mi mathvariant="normal">&#x2207;</mi><mrow data-mjx-texclass="ORD"><mi>p</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</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="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">&#x27E8;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>|</mo><mi mathvariant="normal">&#x03A8;</mi><mo data-mjx-texclass="CLOSE">&#x27E9;</mo></mrow></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></mfrac></mrow><msub><mi mathvariant="normal">&#x2207;</mi><mrow data-mjx-texclass="ORD"><mi>p</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="normal">&#x03A8;</mi><mo>~</mo></mover></mrow></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

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Calculated based on the variables occurring on the entire Operatoren im Hilbertraum page

Identifiers

  • Φ
  • r¯^
  • Ψ
  • Φ
  • p¯
  • p¯
  • Φ
  • p¯
  • r¯^
  • Ψ
  • Φ
  • p¯
  • r
  • p¯
  • r¯
  • r¯
  • r¯^
  • Ψ
  • p¯
  • r¯
  • π
  • e
  • i
  • p¯
  • r¯
  • r¯
  • r¯^
  • Ψ
  • r¯
  • r¯
  • Ψ
  • Φ
  • p¯
  • r
  • p¯
  • r¯
  • r¯
  • r¯^
  • Ψ
  • r
  • π
  • e
  • i
  • p¯
  • r¯
  • r¯
  • Ψ
  • r¯
  • π
  • r
  • r¯
  • e
  • i
  • p¯
  • r¯
  • Ψ
  • r¯
  • r¯
  • e
  • i
  • p¯
  • r¯
  • i
  • p
  • e
  • i
  • p¯
  • r¯
  • p
  • px
  • py
  • pz
  • Φ
  • p¯
  • π
  • r
  • r¯
  • e
  • i
  • p¯
  • r¯
  • Ψ
  • r¯
  • i
  • p
  • r
  • π
  • e
  • i
  • p¯
  • r¯
  • Ψ
  • r¯
  • π
  • e
  • i
  • p¯
  • r¯
  • p¯
  • r¯
  • Φ
  • p¯
  • i
  • p
  • r
  • p¯
  • r¯
  • r¯
  • Ψ
  • i
  • p
  • Ψ~
  • p¯

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