The non-degeneracy perturbation theory was investigated in the Torres-Vega and Frederick(T-F) phase space.On the basis of one perturbation,the second and third perturbation formulas of energies and function were deduced.Contrast the probability density of coordinate and momentum with phase space by harmonic oscillator,show that the quantum phase space can get more information than coordinate representation's or momentum representation's when study the perturbation of system in quantum phase space.The non-degeneracy perturbation theory was investigated in the Torres-Vega and Frederick(T-F) phase space.On the basis of one perturbation,the second and third perturbation formulas of energies and function were deduced.Contrast the probability density of coordinate and momentum with phase space by harmonic oscillator,show that the quantum phase space can get more information than coordinate representation's or momentum representation's when study the perturbation of system in quantum phase space.

Journal of Jinggangshan UniversityJan 1, 20080 citations

Perturbation Theory Based on Currents for Interacting Bosons at Absolute Zero

B. Tsu-Shen Chang

As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in $k$ if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in $k$ . However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in $k$ if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in $k$. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.

Physical Review AMar 1, 19723 citations

Perturbation Theory

Pedro Pereyra Padilla

Undergraduate lecture notes in physicsJan 1, 20121 citation

Collective bath coordinate mapping of “hierarchy” in hierarchical equations of motion

Tatsushi Ikeda, Akira Nakayama

The theory of hierarchical equations of motion (HEOM) is one of the standard methods to give exact evaluations of the dynamics as coupled to harmonic oscillator environments. However, the theory is numerically demanding due to its hierarchy, which is the set of auxiliary elements introduced to capture the non-Markovian and non-perturbative effects of environments. When system-bath coupling becomes relatively strong, the required computational resources and precision move beyond the regime that can be currently handled. This article presents a new representation of HEOM theory in which the hierarchy is mapped into a continuous space of a collective bath coordinate and several auxiliary coordinates as the form of the quantum Fokker-Planck equation. This representation gives a rigorous time evolution of the bath coordinate distribution and is more stable and efficient than the original HEOM theory, particularly when there is a strong system-bath coupling. We demonstrate the suitability of this approach to treat vibronic system models coupled to environments.The theory of hierarchical equations of motion (HEOM) is one of the standard methods to give exact evaluations of the dynamics as coupled to harmonic oscillator environments. However, the theory is numerically demanding due to its hierarchy, which is the set of auxiliary elements introduced to capture the non-Markovian and non-perturbative effects of environments. When system-bath coupling becomes relatively strong, the required computational resources and precision move beyond the regime that can be currently handled. This article presents a new representation of HEOM theory in which the hierarchy is mapped into a continuous space of a collective bath coordinate and several auxiliary coordinates as the form of the quantum Fokker-Planck equation. This representation gives a rigorous time evolution of the bath coordinate distribution and is more stable and efficient than the original HEOM theory, particularly when there is a strong system-bath coupling. We demonstrate the suitability of this approach to treat vibronic system models coupled to environments.

arXiv:2112.09861The Journal of Chemical PhysicsFeb 9, 202214 citations

Perturbation Theory

Michele Cini

UNITEXT for physicsJan 1, 20180 citations

k

if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in

k

. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.