Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

By modifying ψ as ψ + δψ, the expectation value becomes
〈H〉 + δ〈H〉 =
=
=
∫ (ψ ∗ + δψ ∗ )H(ψ + δψ) dv
∫ (ψ ∗ + δψ ∗ )(ψ + δψ) dv
∫ ψ ∗ Hψ dv + ∫ δψ ∗ Hψ dv + ∫ ψ ∗ Hδψ dv
∫ ψ ∗ ψ dv + ∫ δψ ∗ ψ dv + ∫ ψ ∗ δψ dv
(∫
∫
ψ ∗ Hψ dv +
(
1
∫ ψ ∗ 1 −
ψ dv
∫
δψ ∗ Hψ dv +
)
ψ ∗ Hδψ dv
∫ δψ ∗ ∫
ψ dv ψ ∗ )
∫ ψ ∗ ψ dv − δψ dv
∫ ψ ∗ ψ dv
×
(4.3)
where second-order terms in δψ have been omitted and we have used the approximation
1/(1+x) ≃ 1−x, valid for x