Suppose the Hamiltonian H, for a particular quantum system, is a function of some parameter ; let

Suppose the Hamiltonian H, for a particular quantum system, is a function of some parameter λ; let E_n(λ) and Ψ_n(λ) be the eigenvalues and eigenfunctions of H (λ). The Feynman Hellmann theorem states that

(assuming either that  E_n is nondegenerate, or—if degenerate—that the Ψn^s are the “good” linear combinations of the degenerate eigenfunctions).
(a) Prove the Feynman–Hellmann theorem. Use Equation 7.9.
(b) Apply it to the one-dimensional harmonic oscillator, (i) using λ = ω (this yields a formula for the expectation value of V), (ii) using λ = ћ (this yields (T)), and (iii) using λ = m  (this yields a relation between (T) and (V)) . Compare your answers to Problem 2.12, and the virial theorem predictions (Problem 3.37).

Transcribed Image Text:

а En Ye = шп не re 31 (7.110)