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Physical Review A

Variational and diagrammatic evaluations of the ground-state energy of quantum liquids

E. Krotscheck, R. A. Smith, Andrew David Jackson

Published May 1, 1986• 28 citations
Publisher

Abstract

The Feynman-Hellmann theorem shows that the exact ground-state energy of a quantum system may be obtained by integrating the potential energy with respect to a coupling constant. For approximation schemes in which the energy is actually calculated from an energy functional, we show that direct and coupling-constant integration evaluations of the energy functional are identical, provided that the energy functional is optimized.

DOI
10.1103/physreva.33.3535

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Communications in Theoretical Physics•Dec 1, 2005•1 citation
The Feynman-Hellmann theorem shows that the exact ground-state energy of a quantum system may be obtained by integrating the potential energy with respect to a coupling constant. For approximation schemes in which the energy is actually calculated from an energy functional, we show that direct and coupling-constant integration evaluations of the energy functional are identical, provided that the energy functional is optimized.
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.
The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He^{4}\mathrm{He}4He, Fermi neutron gas, and liquid 3He^{3}\mathrm{He}3He are discussed. The correlation functions are parametrized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of the lowest-order contributions. The healing distance has a prescribed value in lowest-order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The many-body cluster contributions in Bose fluids are summed with successive approximations of an integral equation due to van Leeuwen et al. A simple diagrammatic cluster expansion is presented for Fermi liquids, and its direct diagrams are summed with the integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.
Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: 'completion of square' and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.
It is shown by dimensional analysis that Pauli's ground-state-energy theorem, which expresses the ground-state energy in the form of a coupling-constant integration of the potential energy, and the virial theorem are equivalent for a many-particle system interacting with a homogeneous interparticle potential ($\ensuremath{\propto}{r}^{l}$) in arbitrary (ddd) dimensions. The finite-temperature case is also discussed.