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

## Question:

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).

## Step by Step Answer:

**Related Book For**

## Introduction To Quantum Mechanics

**ISBN:** 9781107189638

3rd Edition

**Authors:** David J. Griffiths, Darrell F. Schroeter