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Display information for equation id:math.1767.4 on revision:1767
* Page found: Variationsverfahren (eq math.1767.4)
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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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