Lankide:Juan Ignacio Vazquez/Proba orria

Deskribapena

$\sum _{\lambda _{1},\,\lambda _{2}\in \mathrm {Spec} (H)}\langle \psi _{\lambda _{1}}\mid \psi _{\lambda _{2}}\rangle =\delta _{\lambda _{1}\lambda _{2}}$

${\hat {H}}\left|\psi _{\lambda }\right\rangle =\lambda \left|\psi _{\lambda }\right\rangle .$

${\begin{aligned}\left\langle \psi \mid H\mid \psi \right\rangle &=\sum _{\lambda _{1},\lambda _{2}\in \mathrm {Spec} (H)}\left\langle \psi |\psi _{\lambda _{1}}\right\rangle \left\langle \psi _{\lambda _{1}}|H|\psi _{\lambda _{2}}\right\rangle \left\langle \psi _{\lambda _{2}}|\psi \right\rangle \\&=\sum _{\lambda \in \mathrm {Spec} (H)}\lambda \left|\left\langle \psi _{\lambda }\mid \psi \right\rangle \right|^{2}\geq \sum _{\lambda \in \mathrm {Spec} (H)}E_{0}\left|\left\langle \psi _{\lambda }\mid \psi \right\rangle \right|^{2}=E_{0}\end{aligned}}$

$\left\langle \psi (\alpha _{i})\mid \psi (\alpha _{i})\right\rangle =1$

$\varepsilon (\alpha _{i})=\left\langle \psi (\alpha _{i})|H|\psi (\alpha _{i})\right\rangle .$

$\left|\psi \right\rangle =\left|\psi _{\text{test}}\right\rangle -\left\langle \psi _{\mathrm {gr} }\mid \psi _{\text{test}}\right\rangle \left|\psi _{\text{gr}}\right\rangle$

$E_{\text{ground}}\leq \left\langle \phi |H|\phi \right\rangle .$

$\phi =\sum _{n}c_{n}\psi _{n}.\,$

${\begin{aligned}&\left\langle \phi |H|\phi \right\rangle \\={}&\left\langle \sum _{n}c_{n}\psi _{n}|H|\sum _{m}c_{m}\psi _{m}\right\rangle \\={}&\sum _{n}\sum _{m}\left\langle c_{n}\psi _{n}|E_{m}|c_{m}\psi _{m}\right\rangle \\={}&\sum _{n}\sum _{m}c_{n}^{*}c_{m}E_{m}\left\langle \psi _{n}\mid \psi _{m}\right\rangle \\={}&\sum _{n}|c_{n}|^{2}E_{n}.\end{aligned}}$

$\left\langle \phi |H|\phi \right\rangle \geq E_{g}\sum _{n}|c_{n}|^{2}=E_{g}.\,$