Closed-form solutions of the Schrödinger equation for a class of smoothed Coulomb potentials

Schrödinger's equation with the attractive potential V (r) = −Z/(r q + β q ) 1 q , Z > 0, β > 0, q ≥ 1, is shown, for general values of the parameters Z and β, to be reducible to the confluent Heun equation in the case q = 1, and to the generalized Heun equation in case q = 2. In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases this factor becomes a polynomial. The Asymptotic Iteration Method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations for them. Detail solutions for both cases are provided.

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“…as given earlier by Liu & Clark [17] for −2E = k 2 . For the purpose of applying AIM, we may note for ν = l + 1 and…”

Section: Soft-core Coulomb Potentials Vq(r)mentioning

confidence: 65%

Soft-core Coulomb potentials and Heun’s differential equation

Hall

Saad

Sen

2010

Journal of Mathematical Physics

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“…Furthermore, tunneling in the fully three-dimensional Coulomb problem can be described by an effective one-dimensional tunneling barrier via introducing parabolic coordinates [28]. Thus, we restrict ourselves to one-dimensional systems and consider an electron bound to the soft-core potential −Z/ x 2 + α(Z) [29][30][31][32] to model the essential features of an electron in a threedimensional Coulomb potential. Here, Z is the atomic number, and the softening parameter α(Z) = 2/Z 2 is chosen such that the ground state energy of the soft-core potential is −I p = −Z 2 /2, which equals the ground state energy of the Coulomb potential…”

Section: Considered Systemmentioning

confidence: 99%

Ionization Time and Exit Momentum in Strong-Field Tunnel Ionization

et al. 2016

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Tunnel ionization belongs to the fundamental processes of atomic physics. The so-called two-step model, which describes the ionization as instantaneous tunneling at the electric field maximum and classical motion afterwards with zero exit momentum, is commonly employed to describe tunnel ionization in adiabatic regimes. In this contribution, we show by solving numerically the time-dependent Schrödinger equation in one dimension and employing a virtual detector at the tunnel exit that there is a nonvanishing positive time delay between the electric field maximum and the instant of ionization. Moreover, we find a nonzero exit momentum in the direction of the electric field. To extract proper tunneling times from asymptotic momentum distributions of ionized electrons, it is essential to incorporate the electron's initial momentum in the direction of the external electric field.

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“…The magnetic field is measured in units of 1 au = 2.350 52 × 10 5 T. The soft-core parameters in (19) are chosen such that the ground-state energy of this potential yields in case of the nonrelativistic Schrödinger equation −Z 2 /2, which is the same value as for the three-dimensional Coulomb potential with the same atomic number Z. The nonrelativistic normalized ground state of the soft-core potential (19) is [44]…”

Section: Soft-core Potential In Two Dimensionsmentioning

confidence: 99%

Krylov subspace methods for the Dirac equation

Beerwerth

Bauke

2015

Computer Physics Communications

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The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac Lanczos propagator utilizing the Message Passing Interface standard.

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Closed-form solutions of the Schrödinger equation for a class of smoothed Coulomb potentials

Abstract: Abstract. An infinite family of closed-form solutions is exhibited for the Schrödinger equation for the potential V (r) = −Z/ |r| 2 + a 2 . Evidence is presented for an approximate dynamical symmetry for large values of the angular momentum l.

Cited by 11 publications

References 13 publications

Soft-core Coulomb potentials and Heun’s differential equation

Soft-core Coulomb potentials and Heun’s differential equation

Ionization Time and Exit Momentum in Strong-Field Tunnel Ionization

Krylov subspace methods for the Dirac equation

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