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Journal of Shenyang University

A Calculating Method of Average Value of Dynamical Quantity

Xiao Hongfei

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

The Hellmann-Feynman theorem and it's applicable range in quantum mechanics are discussed. The calculating method of average value of dynamical quantity by using the theorem is stressed. The movement of particle in the centre force field is discussed.

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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 $\ensuremath{\partial}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}]}_{C}∕\ensuremath{\partial}\ensuremath{\lambda}$ 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.
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.