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Physical Review A

Hellmann-Feynman theorem in a mixed ensemble

Dhurba Rai

Published Mar 21, 2007• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.75.032514

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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∂[H\ifmmode^\elseˆ\fi]C∕\ensuremath∂\ensuremathλ\ensuremath{\partial}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}]}_{C}∕\ensuremath{\partial}\ensuremath{\lambda}\ensuremath∂[H\ifmmode^\elseˆ\fi​]C​∕\ensuremath∂\ensuremathλ and \ensuremath∂[H\ifmmode^\elseˆ\fi]G∕\ensuremath∂\ensuremathλ\ensuremath{\partial}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}]}_{G}∕\ensuremath{\partial}\ensuremath{\lambda}\ensuremath∂[H\ifmmode^\elseˆ\fi​]G​∕\ensuremath∂\ensuremathλ is established at the leading order of the saddle-point approximation.
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.