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* Page found: Variationsverfahren (eq math.1767.30)

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Variationsverfahren Eq: math.1767.30

Hash: c67a9891a339a1df4d78b4466538788a

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=0}^{\infty }{{}}{{E}_{n}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle  \\

& \left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle =0,f\ddot{u}r\quad n=0 \\

& \Rightarrow \left\langle  \Psi  \right|\hat{H}\left| \Psi  \right\rangle =\sum\limits_{n=1}^{\infty }{{}}{{E}_{n}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle  \\

& \Rightarrow {{E}_{n}}\ge {{E}_{1}} \\

& \Rightarrow \sum\limits_{n=1}^{\infty }{{}}{{E}_{n}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle \ge {{E}_{1}}\sum\limits_{n=1}^{\infty }{{}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle  \\

& \Rightarrow \sum\limits_{n=1}^{\infty }{{}}{{E}_{n}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle \ge {{E}_{1}}\left\langle  \Psi   |  \Psi  \right\rangle \Rightarrow {{E}_{1}}\le \frac{\sum\limits_{n=1}^{\infty }{{}}{{E}_{n}}\left\langle  \Psi   |  {{\Psi }_{n}} \right\rangle \left\langle  {{\Psi }_{n}}  |  \Psi  \right\rangle }{\left\langle  \Psi   |  \Psi  \right\rangle } \\

& \Rightarrow {{E}_{1}}\le \frac{\left\langle  \Psi  \right|\hat{H}\left| \Psi  \right\rangle }{\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=0}^{\infty }{}{{E}_{n}}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle \\&\left\langle \Psi |{{\Psi }_{n}}\right\rangle =0,f{\ddot {u}}r\quad n=0\\&\Rightarrow \left\langle \Psi \right|{\hat {H}}\left|\Psi \right\rangle =\sum \limits _{n=1}^{\infty }{}{{E}_{n}}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle \\&\Rightarrow {{E}_{n}}\geq {{E}_{1}}\\&\Rightarrow \sum \limits _{n=1}^{\infty }{}{{E}_{n}}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle \geq {{E}_{1}}\sum \limits _{n=1}^{\infty }{}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle \\&\Rightarrow \sum \limits _{n=1}^{\infty }{}{{E}_{n}}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle \geq {{E}_{1}}\left\langle \Psi |\Psi \right\rangle \Rightarrow {{E}_{1}}\leq {\frac {\sum \limits _{n=1}^{\infty }{}{{E}_{n}}\left\langle \Psi |{{\Psi }_{n}}\right\rangle \left\langle {{\Psi }_{n}}|\Psi \right\rangle }{\left\langle \Psi |\Psi \right\rangle }}\\&\Rightarrow {{E}_{1}}\leq {\frac {\left\langle \Psi \right|{\hat {H}}\left|\Psi \right\rangle }{\left\langle \Psi |\Psi \right\rangle }}\\\end{aligned}}

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\begin{aligned} ⟨ Ψ | \hat{H} | Ψ ⟩ = \sum_{n}^{} ⟨ Ψ | \hat{H} | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ = \sum_{n = 0}^{\infty} E_{n} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ \\ ⟨ Ψ | Ψ_{n} ⟩ = 0, f \ddot{u} r n = 0 \\ \Rightarrow ⟨ Ψ | \hat{H} | Ψ ⟩ = \sum_{n = 1}^{\infty} E_{n} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ \\ \Rightarrow E_{n} \geq E_{1} \\ \Rightarrow \sum_{n = 1}^{\infty} E_{n} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ \geq E_{1} \sum_{n = 1}^{\infty} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ \\ \Rightarrow \sum_{n = 1}^{\infty} E_{n} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩ \geq E_{1} ⟨ Ψ | Ψ ⟩ \Rightarrow E_{1} \leq \frac{\sum_{n = 1}^{\infty} E_{n} ⟨ Ψ | Ψ_{n} ⟩ ⟨ Ψ_{n} | Ψ ⟩}{⟨ Ψ | Ψ ⟩} \\ \Rightarrow E_{1} \leq \frac{⟨ Ψ | \hat{H} | Ψ ⟩}{⟨ Ψ | Ψ ⟩} \end{aligned}

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Identifiers

$Ψ$

$\hat{H}$

$Ψ$

$n$

$Ψ$

$\hat{H}$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$n$

$E_{n}$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$Ψ$

$Ψ_{n}$

$f$

$\ddot{u}$

$r$

$n$

$Ψ$

$\hat{H}$

$Ψ$

$n$

$E_{n}$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$E_{n}$

$E_{1}$

$n$

$E_{n}$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$E_{1}$

$n$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$n$

$E_{n}$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$E_{1}$

$Ψ$

$Ψ$

$E_{1}$

$n$

$E_{n}$

$Ψ$

$Ψ_{n}$

$Ψ_{n}$

$Ψ$

$Ψ$

$Ψ$

$E_{1}$

$Ψ$

$\hat{H}$

$Ψ$

$Ψ$

$Ψ$

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