Modern Physics Letters A

VARIATIONAL RANDOM PHASE APPROXIMATION FOR THE ANHARMONIC OSCILLATOR

J. Dukelsky, Peter Schuck

Published Aug 30, 1991• 17 citations

Abstract

The recently derived Variational Random Phase Approximation is examined using the anharmonoic oscillator model. Special attention is paid to the ground state RPA wave function and the convergence of the proposed truncation scheme to obtain the diagonal density matrix.

DOI: 10.1142/s0217732391002852

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Modern Physics Letters A

VARIATIONAL RANDOM PHASE APPROXIMATION FOR THE ANHARMONIC OSCILLATOR

J. Dukelsky, Peter Schuck

Published Aug 30, 1991• 17 citations

Publisher

Abstract

DOI: 10.1142/s0217732391002852

Related papers

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arXiv:quant-ph/0105095WORLD SCIENTIFIC eBooksJun 1, 20013 citations

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Yu Zhou, Jay D. Mancini, P. F. Meier +1 more

We present here a systematic scheme for improving the variational wave functions and corresponding energy levels for quantum systems. By expanding the wave function around a variational parameter value, a family of independent functions can be systematically generated. The eigenstates are then obtained by diagonalizing the Hamiltonian matrix within the basis and optimized with respect to the variational parameter. As a test, the ground state of the quartic anharmonic oscillator has been investigated, and it is found that this scheme converges more rapidly than the conventional Lanczos method and yields better approximations of the energy levels than other variational methods. The effectiveness of this scheme for larger systems remains to be seen.We present here a systematic scheme for improving the variational wave functions and corresponding energy levels for quantum systems. By expanding the wave function around a variational parameter value, a family of independent functions can be systematically generated. The eigenstates are then obtained by diagonalizing the Hamiltonian matrix within the basis and optimized with respect to the variational parameter. As a test, the ground state of the quartic anharmonic oscillator has been investigated, and it is found that this scheme converges more rapidly than the conventional Lanczos method and yields better approximations of the energy levels than other variational methods. The effectiveness of this scheme for larger systems remains to be seen.

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The random phase approximation (RPA) builds in correlations left out by mean-field theory. In full $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces we calculate the Hartree-Fock

(\mathrm{HF})+\mathrm{RPA}

binding energy, and compare it to exact diagonalization. We find that often, but not always, the

\mathrm{HF}+\mathrm{RPA}

gives a good approximation to the ``exact'' ground state energy. In those cases where the RPA is less satisfactory, however, there is no obvious correlation with properties of the HF state, such as deformation or overlap with the exact ground state wave function. This weakens the reliability of the RPA for computing binding energies.

A connection between the random-phase-approximation (RPA) total energy

{E}_{\text{RPA}}

and the

GW

quasiparticle energy

{ϵ}_{GW}

has been pointed out. More specifically we show that: $\ensuremath{\delta}{E}_{\text{RPA}}(N)/\ensuremath{\delta}N={ϵ}_{GW}$, where

N

is the total number of electron. The variational property of the RPA total energy is also discussed and the variational equations for the orbital wave functions are derived. We argue that the RPA formalism is a method which can provide both accurate ground-state energies and single quasiparticle energies.