Weyl's theory applied to the Stark effect in the hydrogen atom

Hehenberger, Michael; McIntosh, Harold V.; Brändas, Erkki

doi:10.1103/physreva.10.1494

Cited by 150 publications

(50 citation statements)

References 29 publications

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“…(A3) and (A4), can be found in Eqs. (6) and (7) of [45] (B-field, alternating series, coupling parameter g B = e 2 B 2 /m 4 e ) and in Eqs. (8) and (9) of [46] (E-field, nonalternating series, coupling parameter g E = e 2 E 2 /m 4 e ).…”

Section: Discussionmentioning

confidence: 99%

“…In comparison to a recent investigation [55], where the additional Padé-improvement in the conformal variable is also used, we perform here the analytic continuation by a mapping whose structure reflects the double cuts suggested by the asymptotic properties of the perturbative coefficients given in Eqs. (2), (3) and (6) [cf. Eq.…”

Section: Doubly-cut Borel Plane and Resummation By Conformal Mappingmentioning

confidence: 99%

“…(9), (13)(14)(15), (28)(29)(30)(31)(32)(33), (59)(60)(61)(62)(63)(64)(65)(66)(67) and (73) in [1]. The atomic unit system is used in the sequel, as is customary for this type of calculation [1,6,9,11]. The unit of energy is α 2 m e c 2 = 27.211 eV where α is the fine structure constant, and the unit of the electric field is the field strength felt by an electron at a distance of one Bohr radius a Bohr to a nucleus of elementary charge, which is 1/(4 π ǫ 0 ) (e/a 2 Bohr ) = 5.142 × 10 3 MV/cm (here, ǫ 0 is the permittivity of the vacuum).…”

Section: Numerical Calculationsmentioning

confidence: 99%

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Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Jentschura

2001

Phys. Rev. A

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We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescriptions are compared: (i) a variant of the Borel-Padé resummation method, with an additional improvement due to utilization of the leading renormalon poles (for a comprehensive discussion of renormalons see [M. Beneke, Phys. Rep. 317, 1 (1999)]), and (ii) a contour-improved combination of the Borel method with an analytic continuation by conformal mapping, and Padé approximations in the conformal variable. The singularity structure in the case of the LoSurdo-Stark effect in the complex Borel plane is shown to be similar to (divergent) perturbative expansions in quantum chromodynamics.11.15. Bt, 11.10.Jj, 32.60.+i, 32.70.Jz

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Section: Discussionmentioning

confidence: 99%

Section: Doubly-cut Borel Plane and Resummation By Conformal Mappingmentioning

confidence: 99%

Section: Numerical Calculationsmentioning

confidence: 99%

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Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Jentschura

2001

Phys. Rev. A

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“…This spectral theorem expresses the completeness of the wavefunctions of the Schrödinger equation in a general way that applies not just to the familiar discrete spectrum models (such as the infinite square well or the harmonic oscillator), but also to systems with discrete and continuum spectra (such as the finite square well or the hydrogen atom), and even to systems with a purely continuous spectrum, such as for ionization problems where there are no true bound states. The WTK approach is well suited for numerical implementation and has been applied long ago to the Stark effect in atomic hydrogen [26]. An interesting soluble one-dimensional atomic model consisting of a single finite square well on the half-line is solved using the WTK method in [27].…”

Section: Introductionmentioning

confidence: 99%

Simple soluble molecular ionization model

Dunne

Gauthier

2004

Phys. Rev. A

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We present a simple exact analytical solution, using the Weyl-Titchmarsh-Kodaira spectral theorem, for the spectral function of the one-dimensional diatomic molecule model consisting of two attractive delta function wells in the presence of a static external electric field. For sufficiently deep and far apart wells, this molecule supports both an even and an odd state, and the introduction of a static electric field turns these bound states into quasi-bound states which are Stark shifted and broadened. The continuum spectrum also inherits an intricate pattern of resonances which reflect the competition between resonant scattering between the two atomic wells and between the linear potential and one or both atomic well(s). All results are analytic and can be easily plotted. The relation between the large orders of the divergent perturbative Stark shift series and the non-perturbative widths of quasi-bound levels is studied.

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“…[9,20,32,[45][46][47][48] for more details. The nomenclature nonHermitian or non-self-adjoint operators hide an important aspect, viz.…”

Section: Non-hermitian Quantum Theorymentioning

confidence: 99%

The Origin and Evolution of Complex Enough Systems in Biology

Brändas

2017

Quantum Systems in Physics, Chemistry, and Biology

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Recent criticisms of Neo-Darwinism are considered and disputed within the setting of recent advances in chemical physics. A related query, viz., the ontological thesis, that everything is physical, confronts a crucial test on the validity of reductionism as a fundamental approach to science. While traditional 'physicalism' interprets evolution as a sequence of physical accidents governed by the second law of thermodynamics, the concepts of biology concern processes that owe their goal-directedness to the influence of an evolved program. This disagreement is met by unifying basic aspects of chemistry and physics, formulating the Correlated Dissipative Ensemble, CDE, as a characterization of a 'complex enough systems', CES, in biology. The latter entreats dissipative dynamics; non-Hermitian quantum mechanics together with modern quantum statistics thereby establish a precise spatio-temporal order of significance for living systems. The CDE grants a unitary transformation structure that comprises communication protocols of embedded Poisson statistics for molecular recognition and cellular differentiation, providing cell-hierarchies in the organism. The present conception of evolution, founded on communication with a built-in self-referential order, offers a valid argument in favour of Neo-Darwinism, providing an altogether solid response and answer to the criticisms voiced above. Introductory Remarks Reductionism in Natural ScienceA frequent understanding of natural sciences contends that biology reduces to chemistry and chemistry to physics. Even if the traditional analytic interpretation of mathematics as a language or tool to convert knowledge about nature, i.e. a formal science not being incorporated amongst the branches of physical or life sciences, some recent propositions impart that also physics can be reduced to mathematics [1]. Within such views, based on strict reductionnism, see more below, one argues that the present laws of physics are not only commensurate to biology, but also contain the necessary natural laws for expressing all the known biological facts.There are numerous differences of opinion regarding the scheme outlined above, e.g. the impossibility to derive the Coulomb Hamiltonian corresponding to a molecule from first principles and the general problem of quantum chemistry to treat nuclei and electrons on a more or less equivalent basis, see Löwdin [2]. Other arguments concern the concepts of biology as the laws of physics, at present, exclude a fundamental understanding of biological processes governed by an evolved program [3,4].Observation and deliberate experiments instigate rational, deductive theory, formulated in the language of mathematics, with the so accumulated knowledge discussed and contained by the methods of philosophy in general and the concepts of biology in particular. Recognizing the acquired demarcation between the philosophy of science, including biology, and the enactment of reductionism as embodied in the laws of physics, the endwise connections between them ...

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Weyl's theory applied to the Stark effect in the hydrogen atom

Cited by 150 publications

References 29 publications

Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Resummation of the divergent perturbation series for a hydrogen atom in an electric field

Simple soluble molecular ionization model

The Origin and Evolution of Complex Enough Systems in Biology

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