Matrix Numerov method for solving Schrödinger’s equation

Pillai, Mohandas; Goglio, Joshua; Walker, Thad

doi:10.1119/1.4748813

Cited by 89 publications

(58 citation statements)

References 8 publications

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“…These techniques are well established e.g., [29,30] and could be improved upon by using higher-order finite-difference elements, or using the Numerov method in matrix form.…”

Section: Finite Difference Matrix To Solve the Schrödinger Equation Fmentioning

confidence: 99%

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Gomez

Nagayama

Fontes

et al. 2018

Atoms

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Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numerical complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. This technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.

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“…These techniques are well established e.g., [29,30] and could be improved upon by using higher-order finite-difference elements, or using the Numerov method in matrix form.…”

Section: Finite Difference Matrix To Solve the Schrödinger Equation Fmentioning

confidence: 99%

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Gomez

Nagayama

Fontes

et al. 2018

Atoms

View full text Add to dashboard Cite

show abstract

“…Numerical simulations were carried out on a personal computer using Python and the scientific libraries Numpy and Scipy for matrix diagonalizations and optimization operations [24,25]. When necessary, the exact solution results in 1D were obtained by building an Hamiltonian based on a matrix Numerov method [26]. This approach was then expanded to higher dimensionality; the details are explained in Appendix C. Since this method uses a direct space basis set, all potentials are treated effectively as if they were enclosed in an infinite well.…”

Section: Simulation Detailsmentioning

confidence: 99%

Computational applications of the many-interacting-worlds interpretation of quantum mechanics

Sturniolo

2018

Phys. Rev. E

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While historically many quantum-mechanical simulations of molecular dynamics have relied on the Born-Oppenheimer approximation to separate electronic and nuclear behavior, recently a great deal of interest has arisen in quantum effects in nuclear dynamics as well. Due to the computational difficulty of solving the Schrödinger equation in full, these effects are often treated with approximate methods. In this paper, we present an algorithm to tackle these problems using an extension to the many-interacting-worlds approach to quantum mechanics. This technique uses a kernel function to rebuild the probability density, and therefore, in contrast with the approximation presented in the original paper, it can be naturally extended to n-dimensional systems. This opens up the possibility of performing quantum ground-state searches with steepest-descent methods, and it could potentially lead to real-time quantum molecular-dynamics simulations. The behavior of the algorithm is studied in different potentials and numbers of dimensions and compared both to the original approach and to exact Schrödinger equation solutions whenever possible.

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“…Extending ideas of Ref. 68, we can convert this system of equations into a generalized eigenvalue problem. We introduce a matrix A having c j /a 2 on the j-th diagonal, where j > 0 refers to the upper diagonals and j < 0 refers to the lower diagonals, a diagonal matrix V = diag(V j ) representing the potential, and a matrix B having −(r!)…”

Section: (B1)mentioning

confidence: 99%

Dual approach to circuit quantization using loop charges

Ulrich

Hassler

2016

Phys. Rev. B

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The conventional approach to circuit quantization is based on node fluxes and traces the motion of node charges on the islands of the circuit. However, for some devices, the relevant physics can be best described by the motion of polarization charges over the branches of the circuit that are in general related to the node charges in a highly nonlocal way. Here, we present a method, dual to the conventional approach, for quantizing planar circuits in terms of loop charges. In this way, the polarization charges are directly obtained as the differences of the two loop charges on the neighboring loops. The loop charges trace the motion of fluxes through the circuit loops. We show that loop charges yield a simple description of the flux transport across phase-slip junctions. We outline a concrete construction of circuits based on phase-slip junctions that are electromagnetically dual to arbitrary planar Josephson junction circuits. We argue that loop charges also yield a simple description of the flux transport in conventional Josephson junctions shunted by large impedances. We show that a mixed circuit description in terms of node fluxes and loop charges yields an insight into the flux decompactification of a Josephson junction shunted by an inductor. As an application, we show that the fluxonium qubit is well approximated as a phase-slip junction for the experimentally relevant parameters. Moreover, we argue that the 0-π qubit is effectively the dual of a Majorana Josephson junction.

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Matrix Numerov method for solving Schrödinger’s equation

Cited by 89 publications

References 8 publications

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Computational applications of the many-interacting-worlds interpretation of quantum mechanics

Dual approach to circuit quantization using loop charges

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