∑ λ 1 , λ 2 ∈ S p e c ( H ) ⟨ ψ λ 1 ∣ ψ λ 2 ⟩ = δ λ 1 λ 2 {\displaystyle \sum _{\lambda _{1},\,\lambda _{2}\in \mathrm {Spec} (H)}\langle \psi _{\lambda _{1}}\mid \psi _{\lambda _{2}}\rangle =\delta _{\lambda _{1}\lambda _{2}}}
H ^ | ψ λ ⟩ = λ | ψ λ ⟩ . {\displaystyle {\hat {H}}\left|\psi _{\lambda }\right\rangle =\lambda \left|\psi _{\lambda }\right\rangle .}
⟨ ψ ∣ H ∣ ψ ⟩ = ∑ λ 1 , λ 2 ∈ S p e c ( H ) ⟨ ψ | ψ λ 1 ⟩ ⟨ ψ λ 1 | H | ψ λ 2 ⟩ ⟨ ψ λ 2 | ψ ⟩ = ∑ λ ∈ S p e c ( H ) λ | ⟨ ψ λ ∣ ψ ⟩ | 2 ≥ ∑ λ ∈ S p e c ( H ) E 0 | ⟨ ψ λ ∣ ψ ⟩ | 2 = E 0 {\displaystyle {\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}}}
⟨ ψ ( α i ) ∣ ψ ( α i ) ⟩ = 1 {\displaystyle \left\langle \psi (\alpha _{i})\mid \psi (\alpha _{i})\right\rangle =1}
ε ( α i ) = ⟨ ψ ( α i ) | H | ψ ( α i ) ⟩ . {\displaystyle \varepsilon (\alpha _{i})=\left\langle \psi (\alpha _{i})|H|\psi (\alpha _{i})\right\rangle .}
| ψ ⟩ = | ψ test ⟩ − ⟨ ψ g r ∣ ψ test ⟩ | ψ gr ⟩ {\displaystyle \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 ground ≤ ⟨ ϕ | H | ϕ ⟩ . {\displaystyle E_{\text{ground}}\leq \left\langle \phi |H|\phi \right\rangle .}
ϕ = ∑ n c n ψ n . {\displaystyle \phi =\sum _{n}c_{n}\psi _{n}.\,}
⟨ ϕ | H | ϕ ⟩ = ⟨ ∑ n c n ψ n | H | ∑ m c m ψ m ⟩ = ∑ n ∑ m ⟨ c n ψ n | E m | c m ψ m ⟩ = ∑ n ∑ m c n ∗ c m E m ⟨ ψ n ∣ ψ m ⟩ = ∑ n | c n | 2 E n . {\displaystyle {\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}}}
⟨ ϕ | H | ϕ ⟩ ≥ E g ∑ n | c n | 2 = E g . {\displaystyle \left\langle \phi |H|\phi \right\rangle \geq E_{g}\sum _{n}|c_{n}|^{2}=E_{g}.\,}
