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FeynmanHellmann

# FeynmanHellmann - Supplementary notes Feynman-Hellmann...

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Supplementary notes: Feynman-Hellmann Theorem Consider a Hamiltonian H ( λ ) which depends on a parameter λ , such that H ( λ ) | φ ( λ ) i = E ( λ ) | φ ( λ ) i (1) where φ ( λ ) is the normalised eigenstate corresponding to the eigenenergy E ( λ ). Under an infinitesimal change in λ so that λ λ + , the change in the Hamiltonian is equal to ∂H ( λ ) ∂λ . From the first-order perturbation theory (which is exact since the change in H ( λ ) is infinitesimal), the corresponding change in E ( λ ) is ∂E ( λ ) ∂λ = h φ ( λ ) | ∂H ( λ ) ∂λ | φ ( λ ) i . (2) Dividing throughout by , we obtain the Feynman-Hellmann theorem which states that ∂E ( λ ) /∂λ is equal to the expectation value of ∂H ( λ ) /∂λ ∂E ( λ ) ∂λ = h φ ( λ ) | ∂H ( λ ) ∂λ | φ ( λ ) i . (3) For illustration, we apply the Feynman-Hellmann theorem to evaluate
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