Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chinese Physics Letters

Decay Rate of Energy Eigenfunctions in Classically Energetically Inaccessible Regions

王文阁

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Classically energetically inaccessible parts of energy eigenfunctions in configuration space are studied by making use of a generalization of Brillouin-Wigner perturbation theory. Approximate formulas are proposed for describing local decaying rate of this part of energy eigenfunctions, which are useful in the study of quantum phenomena,such as tunnelling effect, and are tested in an anharmonic oscillator.

Related papers

Decay Rate of Energy Eigenfunctions in Classically Energetically Inaccessible Regions

Wen-ge Wang

Classically energetically inaccessible parts of energy eigenfunctions in configuration space are studied by making use of a generalization of Brillouin–Wigner perturbation theory. Approximate formulas are proposed for describing local decaying rate of this part of energy eigenfunctions, which are useful in the study of quantum phenomena, such as tunnelling effect, and are tested in an anharmonic oscillator.

Chinese Physics Letters•Sep 30, 2004•4 citations

Decay Rate of Energy Eigenfunctions in Classically Energetically Inaccessible Regions in more than One-Dimensional Configuration Spaces

Wen-ge Wang

An approximate formula is proposed for the decay rate of energy eigenfunctions in classically energetically inaccessible regions in more than one-dimensional configuration spaces. This is achieved by generalizing an approach proposed recently for energy eigenfunctions in the one-dimensional configuration space. The formula is tested numerically in the Pullen–Edmonds model.

Chinese Physics Letters•Nov 28, 2005•3 citations

Decay Rate of Energy Eigenfunctions in Classically Energetically Inaccessible Regions in more than One-Dimensional Configuration Spaces

王文阁

Chinese Physics Letters•Jan 1, 2005•0 citations

A Renormalized-Hamiltonian-Flow Approach to Eigenenergies and Eigenfunctions*

Wen-ge Wang

Abstract We introduce a decimation scheme of constructing renormalized Hamiltonian flows, which is useful in the study of properties of energy eigenfunctions, such as localization, as well as in approximate calculation of eigenenergies. The method is based on a generalized Brillouin-Wigner perturbation theory. Each flow is specific for a given energy and, at each step of the flow, a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step. Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows. Numerical illustration of the method is given in the Wigner-band random-matrix model .

Communications in Theoretical Physics•Jul 1, 2019•2 citations

Quantum theory of anharmonic oscillators-a variational and systematic general approximation method

K Yamazaki

This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2 q 2 + lambda q 4 ). As a lowest-order approximation an extremely simple formula for energy levels, E i (0) = (i + 1/2)1/4(3/alpha i + alpha i ), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha i is the real positive root of a cubic equation: y i alpha i 3 + alpha i 2 - 1 = 0, with y i = 6lambda(2i 2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda 4 terms, as it should.

Journal of Physics A Mathematical and General•Feb 1, 1984•26 citations

Weisskopf-Wigner decay theory for the energy-driven stochastic Schrödinger equation

Stephen L. Adler

We generalize the Weisskopf-Wigner theory for the line shape and transition rates of decaying states to the case of the energy-driven stochastic Schr\"odinger equation that has been used as a phenomenology for state vector reduction. Within the standard approximations used in the Weisskopf-Wigner analysis, and assuming that the perturbing potential inducing the decay has vanishing matrix elements within the degenerate manifold containing the decaying state, the stochastic Schr\"odinger equation linearizes. Solving the linearized equations, we find no change from the standard analysis in the line shape or the transition rate per unit time. The only effect of the stochastic terms is to alter the early time transient behavior of the decay, in a way that eliminates the quantum Zeno effect. We apply our results to estimate experimental bounds on the parameter governing the stochastic effects. In addition, elegant stochastic-theoretic methods suggested by Di\'osi are used to rederive the principal results, without the assumptions needed to linearize the stochastic equation, and to give analogous results for the Rabi oscillations of a two-level system.

arXiv:quant-ph/0208123•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jan 17, 2003•33 citations
Classically energetically inaccessible parts of energy eigenfunctions in configuration space are studied by making use of a generalization of Brillouin-Wigner perturbation theory. Approximate formulas are proposed for describing local decaying rate of this part of energy eigenfunctions, which are useful in the study of quantum phenomena,such as tunnelling effect, and are tested in an anharmonic oscillator.
Abstract We introduce a decimation scheme of constructing renormalized Hamiltonian flows, which is useful in the study of properties of energy eigenfunctions, such as localization, as well as in approximate calculation of eigenenergies. The method is based on a generalized Brillouin-Wigner perturbation theory. Each flow is specific for a given energy and, at each step of the flow, a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step. Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows. Numerical illustration of the method is given in the Wigner-band random-matrix model .
This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2 q 2 + lambda q 4 ). As a lowest-order approximation an extremely simple formula for energy levels, E i (0) = (i + 1/2)1/4(3/alpha i + alpha i ), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha i is the real positive root of a cubic equation: y i alpha i 3 + alpha i 2 - 1 = 0, with y i = 6lambda(2i 2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda 4 terms, as it should.
We generalize the Weisskopf-Wigner theory for the line shape and transition rates of decaying states to the case of the energy-driven stochastic Schr\"odinger equation that has been used as a phenomenology for state vector reduction. Within the standard approximations used in the Weisskopf-Wigner analysis, and assuming that the perturbing potential inducing the decay has vanishing matrix elements within the degenerate manifold containing the decaying state, the stochastic Schr\"odinger equation linearizes. Solving the linearized equations, we find no change from the standard analysis in the line shape or the transition rate per unit time. The only effect of the stochastic terms is to alter the early time transient behavior of the decay, in a way that eliminates the quantum Zeno effect. We apply our results to estimate experimental bounds on the parameter governing the stochastic effects. In addition, elegant stochastic-theoretic methods suggested by Di\'osi are used to rederive the principal results, without the assumptions needed to linearize the stochastic equation, and to give analogous results for the Rabi oscillations of a two-level system.