structural:expansion, quality:correction
Derivation

⟨ ψ | H | ψ ⟩ = ∫ d x (− ℏ 2 2 m ψ ∗ d 2 ψ d x 2 + V (x) | ψ (x) | 2), {\displaystyle \langle \psi |H|\psi \rangle =\int dx\,\left(-{\frac {\hbar ^{2}}{2m}}\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}+V(x)|\psi (x)|^{2}\right),}