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Physical Review C

Diagrammatic analysis of the Hellmann-Feynman theorem: Numerical applications

R. Sartor

Published Oct 4, 2001• 6 citations
Publisher

Abstract

Various versions of the Hellmann-Feynman theorem that were studied in a previous paper are applied to the actual computation of certain contributions to the expectation value of the kinetic and potential energies of nuclear matter. The numerical importance of the contributions which violate the Bethe-Brandow-Petscheck theorem in some versions is examined. We also provide a discussion of the so-called conserving approximation which directly derives the expectation value of the kinetic energy from that of the potential energy.

DOI
10.1103/physrevc.64.054303

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Various versions of the Hellmann-Feynman theorem that were studied in a previous paper are applied to the actual computation of certain contributions to the expectation value of the kinetic and potential energies of nuclear matter. The numerical importance of the contributions which violate the Bethe-Brandow-Petscheck theorem in some versions is examined. We also provide a discussion of the so-called conserving approximation which directly derives the expectation value of the kinetic energy from that of the potential energy.
We present a simple derivation of the Hellmann–Feynman theorem at finite temperature. We illustrate its validity by considering three relevant examples, which can be used in quantum mechanics lectures: the one-dimensional harmonic oscillator, the one-dimensional Ising model, and the Lipkin model. We show that the Hellmann–Feynman theorem allows one to calculate expectation values of operators that appear in the Hamiltonian. This is particularly useful when the total free energy is available, but there is no direct access to the thermal average of the operators themselves.
In this work, the Hellmann-Feynman theorem is extended within the doubly occupied configuration interaction space to enable practical calculations of reduced density matrices and expected values. This approach is straightforward, employing finite energy differences, yet remains reliable and accurate even with approximate energies from successive approximation methods. The method's validity is rigorously tested against the Richardson-Gaudin-Kitaev and reduced Bardeen-Cooper-Schrieffer models using approximate excitation energies procured from the Hermitian operator method within the same space, effectively proving the approach's reliability with median error rates for reduced density matrix calculations around 0.1%. These results highlight the procedure's potential as a practical tool for computing reduced density matrices and expected values, particularly valuable as an ad hoc method in scenarios where only system energies are easily available.
The Hellmann-Feynman theorem in a pure ensemble is extended to a quasidegenerate mixed ensemble (canonical) as an ensemble average value. The theorem is also extended to a grand canonical ensemble wherein average particle number is conserved. Connections with the thermodynamic observables are made. An approximate relation between $\ensuremath{\partial}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}]}_{C}∕\ensuremath{\partial}\ensuremath{\lambda}$ and $\ensuremath{\partial}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}]}_{G}∕\ensuremath{\partial}\ensuremath{\lambda}$ is established at the leading order of the saddle-point approximation.
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.