Introduction to Quantum Mechanics · Problem 7.38

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · feynman_hellmann_theorem

Problem

Suppose the Hamiltonian HH, for a particular quantum system, is a function of some parameter λ\lambda; let En(λ)E_n(\lambda) and ψn(λ)\psi_n(\lambda) be the eigenvalues and eigenfunctions of H(λ)H(\lambda). The Feynman–Hellmann theorem states that ∂En∂λ=⟨ψn∣∂H∂λ∣ψn⟩\frac{\partial E_n}{\partial \lambda} = \left\langle \psi_n \left| \frac{\partial H}{\partial \lambda} \right| \psi_n \right\rangle (assuming either that EnE_n is nondegenerate, or—if degenerate—that the ψn\psi_ns are the "good" linear combinations of the degenerate eigenfunctions). (a) Prove the Feynman–Hellmann theorem. Hint: Use Equation 7.9. (b) Apply it to the one-dimensional harmonic oscillator, (i) using λ=ω\lambda = \omega (this yields a formula for the expectation value of VV), (ii) using λ=ℏ\lambda = \hbar (this yields ⟨T⟩\langle T \rangle), and (iii) using λ=m\lambda = m (this yields a relation between ⟨T⟩\langle T \rangle and ⟨V⟩\langle V \rangle). Compare your answers to Problem 2.12, and the virial theorem predictions (Problem 3.37).

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