THE FEYNMAN–HELLMANN THEOREM
المؤلف:
Thomas Banks
المصدر:
Quantum Mechanics
الجزء والصفحة:
287
28-3-2021
2708
THE FEYNMAN–HELLMANN THEOREM
Feynman and Hellmann proved a simple theorem about derivatives of energy levels with respect to parameters, which is often helpful in evaluating the expectation values required for computing first-order perturbation theory. Let H(λ) be a one parameter set of Hamiltonians, En(λ) one of its discrete eigenvalues and |Ψn(λ)〉 the corresponding normalized eigenstate.
Assume that we are working at a value of λ where the eigenstate is nondegenerate. The norm of | n(λ)〉 is independent of λ so
If we write
and use the previous equation, we find the Feynman–Hellmann result
This can be viewed as a generalization of the first-order perturbation theory formula to Hamiltonians that depend on nonlinear functions of λ.
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