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* Page found: Variationsverfahren (eq math.1765.4)

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Variationsverfahren Eq: math.1767.4

Hash: 9cfd1232529ca24b55b3294ee3b05c73

TeX (original user input):

\begin{align}

& \left\langle  \Psi  \right|\hat{H}\left| \Psi  \right\rangle =\sum\limits_{n}^{{}}{{}}\left\langle  \Psi  \right|\hat{H}\left| {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle =\sum\limits_{n}^{{}}{{}}{{E}_{n}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle  \\

& {{E}_{n}}\ge {{E}_{0}} \\

& \Rightarrow \sum\limits_{n}^{{}}{{}}{{E}_{n}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle \ge {{E}_{0}}\sum\limits_{n}^{{}}{{}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle ={{E}_{0}}\left\langle  \Psi   |  \Psi  \right\rangle  \\

\end{align}

TeX (checked):

{\begin{aligned}&\left\langle \Psi \right|{\hat {H}}\left|\Psi \right\rangle =\sum \limits _{n}^{}{}\left\langle \Psi \right|{\hat {H}}\left|{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle =\sum \limits _{n}^{}{}{{E}_{n}}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle \\&{{E}_{n}}\geq {{E}_{0}}\\&\Rightarrow \sum \limits _{n}^{}{}{{E}_{n}}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle \geq {{E}_{0}}\sum \limits _{n}^{}{}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle ={{E}_{0}}\left\langle \Psi |\Psi \right\rangle \\\end{aligned}}

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MathML (4.137 KB / 453 B) :

\begin{matrix} ⟨ Ψ | \hat{H} | Ψ ⟩ = \sum_{n}^{} ⟨ Ψ | \hat{H} | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ = \sum_{n}^{} E_{n} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ \\ E_{n} \geq E_{0} \\ \Rightarrow \sum_{n}^{} E_{n} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ \geq E_{0} \sum_{n}^{} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ = E_{0} ⟨ Ψ | Ψ ⟩ \end{matrix}

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Identifiers

$Ψ$

$\hat{H}$

$Ψ$

$n$

$Ψ$

$\hat{H}$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$n$

$E_{n}$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$E_{n}$

$E_{0}$

$n$

$E_{n}$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$E_{0}$

$n$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$E_{0}$

$Ψ$

$Ψ$

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