2010/01/05 by Gao Xianlong, Xianlong Gao, Jianmin Tao +3 · 1 citation
Mathematics · Physics and Astronomy · #Atomic orbital #Classical mechanics #Cold Atom Physics and Bose-Einstein Condensates #Density matrix #Eigenfunction #Eigenvalues and eigenvectors #Equations of motion #Ground state #Hermitian matrix #Mathematics #Physics #Quantum #Quantum mechanics #Quantum system #Quantum, superfluid, helium dynamics #Slater determinant #Spectroscopy and Quantum Chemical Studies #Wave function #cond-mat.mtrl-sci #cond-mat.other
paper · pdf · doi:10.1103/physrevb.81.195106
23 pages, 4 figures, 1 table, 6 Appendices This paper is a follow-up to PRL 103, 086401 (2009)
arxiv created 2010/01/05 · openalex publication_date 2010/05/07 · arxiv updated 2015/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We derive a closed equation of motion for the current density of an inhomogeneous quantum many-body system under the assumption that the time-dependent wave function can be described as a geometric deformation of the ground-state wave function. By describing the many-body system in terms of a single collective field we provide an alternative to traditional approaches, which emphasize one-particle orbitals. We refer to our approach as continuum mechanics for quantum many-body systems. In the linear response regime, the equation of motion for the displacement field becomes a linear fourth-order integrodifferential equation, whose only inputs are the one-particle density matrix and the pair-correlation function of the ground state. The complexity of this equation remains essentially unchanged as the number of particles increases. We show that our equation of motion is a Hermitian eigenvalue problem, which admits a complete set of orthonormal eigenfunctions under a scalar product that involves the ground-state density. Further, we show that the excitation energies derived from this approach satisfy a sum rule which guarantees the exactness of the integrated spectral strength. Our formulation becomes exact for systems consisting of a single particle and for any many-body system in the high-frequency limit. The theory is illustrated by explicit calculations for simple one- and two-particle systems.