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Title: Amplitude flux, probability flux, and gauge invariance in the finite volume scheme for the Schrödinger equation

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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 280; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-07-04 09:23:33; Journal ID: ISSN 0021-9991
Country of Publication:
United States

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Gordon, D. F., Hafizi, B., and Landsman, A. S.. Amplitude flux, probability flux, and gauge invariance in the finite volume scheme for the Schrödinger equation. United States: N. p., 2015. Web. doi:10.1016/
Gordon, D. F., Hafizi, B., & Landsman, A. S.. Amplitude flux, probability flux, and gauge invariance in the finite volume scheme for the Schrödinger equation. United States. doi:10.1016/
Gordon, D. F., Hafizi, B., and Landsman, A. S.. 2015. "Amplitude flux, probability flux, and gauge invariance in the finite volume scheme for the Schrödinger equation". United States. doi:10.1016/
title = {Amplitude flux, probability flux, and gauge invariance in the finite volume scheme for the Schrödinger equation},
author = {Gordon, D. F. and Hafizi, B. and Landsman, A. S.},
abstractNote = {},
doi = {10.1016/},
journal = {Journal of Computational Physics},
number = C,
volume = 280,
place = {United States},
year = 2015,
month = 1

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  • The time-dependent Schrödinger equation can be put in a probability conserving, gauge invariant form, on arbitrary structured grids via finite volume discretization. The gauge terms in the discrete system cancel with a portion of the amplitude flux to produce abbreviated flux functions. The resulting time translation operator is strictly unitary, and is compatible with an efficient operator splitting scheme that allows for multi-dimensional simulation with complex grid geometries. Moreover, the abbreviated amplitude flux is necessary to the construction of a conservative probability current. This construction turns out to be important when computing Bohmian trajectories in multi-dimensions. Bohmian trajectories are usefulmore » in the interpretation of quantum mechanical phenomena such as tunneling ionization, and provide a bridge between quantum and classical regimes.« less
  • A certain class of geometric objects is considered against the background of a classical gauge field associated with an arbitrary structural Lie group. It is assumed that the components of these objects depend on the gauge potentials and their first derivatives, and also on certain gauge-dependent parameters whose properties are suggested by the interaction of an isotopic spin particle with a classical Yang-Mills field. It is shown that the necessary and sufficient conditions for the invariance of the given objects under a finite gauge transformation are embodied in a set of three relations involving the derivatives of their components. Asmore » a special case these so-called invariance identities indicate that there cannot exist a gauge-invariant Lagrangian that depends on the gauge potentials, the interaction parameters, and the 4-velocity components of a test particle. However, the requirement that the equations of motion that result from such a Lagrangian be gauge-invariant, uniquely determines the structure of these equations.« less
  • A scheme, the so-called {open_quote}{open_quote}projection,{close_quote}{close_quote} for handling singularities in processes such as {ital e}{sup +}{ital e}{sup {minus}}{r_arrow}{ital t{bar b}e{sup {minus}}}{bar {nu}} (or {ital e}{sup +}{ital e}{sup {minus}}{r_arrow}{ital u{bar d}e{sup {minus}}}{bar {nu}}) is proposed. In the scheme, with the help of gauge invariance, the large power quantities ({ital s}/{ital m}{sup 2}{sub {ital e}}){sup {ital n}} ({ital n}{ge}1;{ital s}{r_arrow}{infinity}) are removed from the calculation totally, while in the usual schemes the large quantities appear and only will be canceled at last. The advantages of the scheme in numerical calculations are obvious; thus, we focus our discussions mainly on the advantages of themore » scheme in the special case where the absorptive part for some propagators relevant to the process could not be ignored, and a not satisfactory but widely adopted approximation is made; i.e., a finite constant {open_quote}{open_quote}width{close_quote}{close_quote} is introduced to approximate the absorptive part of the propagators phenomenologically even though QED gauge invariance is violated. {copyright} {ital 1996 The American Physical Society.}« less
  • The free-complement (FC) method is a general method for solving the Schrödinger equation (SE): The produced wave function has the potentially exact structure as the solution of the Schrödinger equation. The variables included are determined either by using the variational principle (FC-VP) or by imposing the local Schrödinger equations (FC-LSE) at the chosen set of the sampling points. The latter method, referred to as the local Schrödinger equation (LSE) method, is integral-free and therefore applicable to any atom and molecule. The purpose of this paper is to formulate the basic theories of the LSE method and explain their basic features.more » First, we formulate three variants of the LSE method, the AB, HS, and H{sup T}Q methods, and explain their properties. Then, the natures of the LSE methods are clarified in some detail using the simple examples of the hydrogen atom and the Hooke’s atom. Finally, the ideas obtained in this study are applied to solving the SE of the helium atom highly accurately with the FC-LSE method. The results are very encouraging: we could get the world’s most accurate energy of the helium atom within the sampling-type methodologies, which is comparable to those obtained with the FC-VP method. Thus, the FC-LSE method is an easy and yet a powerful integral-free method for solving the Schrödinger equation of general atoms and molecules.« less