Hellmann-Feynman Theorems in Classical and Quantum Mechanics

Abstract

Theorems giving exact expressions for the derivative of the energy with respect to a parameter in the Hamiltonian for cyclic classical mechanical problems in both the relativistic and nonrelativistic cases are derived. Comparison is made with similar well-known theorems in the quantum mechanical case. Applications of both classical and quantum derivative relations are made to deduction of parametric dependence of the energy in simple cases, deduction of a meaning to the expectation value of the Dirac operator β, and construction of a test of consistency for a well posed problem in the theory of a Dirac particle in a bound state.