Supplementary notes: Feynman-Hellmann TheoremConsider a HamiltonianH(λ) which depends on a parameterλ, such thatH(λ)|φ(λ)i=E(λ)|φ(λ)i(1)whereφ(λ) is the normalised eigenstate corresponding to the eigenenergyE(λ). Underan infinitesimal change inλso thatλ→λ+dλ, the change in the Hamiltonian is equalto∂H(λ)∂λdλ.From the first-order perturbation theory (which is exact since the change inH(λ) isinfinitesimal), the corresponding change inE(λ) is∂E(λ)∂λdλ=hφ(λ)|∂H(λ)∂λdλ|φ(λ)i.(2)Dividing throughout bydλ, we obtain the Feynman-Hellmann theorem which states that∂E(λ)/∂λis equal to the expectation value of∂H(λ)/∂λ∂E(λ)∂λ=hφ(λ)|∂H(λ)∂λ|φ(λ)i.(3)For illustration, we apply the Feynman-Hellmann theorem to evaluate
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Fundamental physics concepts, 2m, Dλ, feynman-hellmann theorem, eigenstate |nlm