The Hydrogen-Atom Problem and Coulomb Sturmian Functions in Spheroidal Coordinates

Kereselidze, Tamaz; Ogilvie, J. F.

doi:10.1016/bs.aiq.2018.02.002

Cited by 3 publications

(3 citation statements)

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“…The numbers λ ∈ C for which (2) has a nontrivial bounded solution w(x) on (−1, 1) are the eigenvalues (or characteristic values) of the CSWE, and the corresponding eigenfunctions w(x) are called Coulomb spheroidal functions. They appear in astrophysics and molecular physics and provide, for example, exact wave functions for a one-electron diatomic molecule with fixed nuclei (see [1,Chapter 9] or [2]). In our subsequent considerations, it does not make much difference whether we include the term βx or not.…”

Section: Introductionmentioning

confidence: 99%

Computation of the eigenvalues for the angular and Coulomb spheroidal wave equation

Schmid¹

2023

Applied Numerical Mathematics

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Section: Introductionmentioning

confidence: 99%

Computation of the eigenvalues for the angular and Coulomb spheroidal wave equation

Schmid¹

2023

Applied Numerical Mathematics

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“…There are also two special cases for t ¹ 0. One is the case s=0, m ≠ 0 and t = -c This equation has extremely wide applications in many fields such as electromagnetic field theory [17][18][19][20], and atomic and molecular physics [21][22][23][24][25]. Up to now, many authors have used different methods to approximately calculate the eigenvalues and the eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions to the angular Teukolsky equation with s ≠ 0

Chen¹,

Wang²,

You³

et al. 2022

Commun. Theor. Phys.

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We first convert the angular Teukolsky equation under the special condition of τ ≠ 0, s ≠ 0, m = 0 into a confluent Heun differential equation (CHDE) by taking different function transformation and variable substitution. And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function (CHF), we find two linearly dependent solutions corresponding to the same eigenstate, from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant. After that, we are able to localize the positions of the eigenvalues on the real axis or on the complex plane when τ is a real number, a pure imaginary number, and a complex number, respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l = |s|+ n, n = 0, 1, 2 ∙∙∙. The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple. The features of the angular probability distribution (APD) and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed. We find that for a real number τ, the eigenvalue is a real number and the eigenfunction is real function, and the eigenfunction system is an orthogonal complete system, and the APD is asymmetric in the northern and southern hemispheres. For a pure imaginary number τ, the eigenvalue is still a real number and the eigenfunction is a complex function, but the APD is symmetric in the northern and southern hemispheres. When τ is a complex number, the eigenvalue is a complex number, the eigenfunction is still a complex function, and the APD in the northern and southern hemispheres is also asymmetric.

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“…This equation was originally from the study by Coulson and Robinson who studied the Schrödinger equation for the hydrogen atom in the spheroidal coordinates [1]. Up to now, the solutions of Equation ( 1) have extremely wide applications in many fields such as electromagnetic field theory [2][3][4], astrophysics [5][6][7][8][9][10][11][12], atomic and molecular physics [13][14][15][16][17][18][19][20]. In particular, when c 2 > 0 its solutions have important applications in communication theory, too [21][22][23][24][25][26].…”

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confidence: 99%

The visualization of the angular probability distribution for the angularTeukolsky equationwithm ≠ 0

Chen

Sun

et al. 2020

Int J of Quantum Chemistry

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We present the exact solutions of the angular Teukolsky equation with m ≠ 0 given by a confluent Heun function. This equation is first transformed to a confluent Heun differential equation through some variable transformations. The Wronskian determinant, which is constructed by two linearly dependent solutions, is used to calculate the eigenvalues precisely. The normalized eigenfunctions can be obtained by substituting the calculated eigenvalues into the unnormalized eigenfunctions. The relations among the linearly dependent eigenfunctions are also discussed. When c2=cR2+i0.5emcI2, the eigenvalues are approximately expressed as Alm≈l()l+1+()cR2+i0.12emcI20.12em[]1−m2/l()l+1/2 for small |c|2 but large l. The isosurface and contour visualizations of the angular probability distribution (APD) are presented for the cases of the real and complex values c2. It is found that the APD has obvious directionality, but the northern and southern hemispheres are always symmetrical regardless of the value of the parameter c2, which is real or imaginary.

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The Hydrogen-Atom Problem and Coulomb Sturmian Functions in Spheroidal Coordinates

Cited by 3 publications

References 36 publications

Computation of the eigenvalues for the angular and Coulomb spheroidal wave equation

Computation of the eigenvalues for the angular and Coulomb spheroidal wave equation

Exact solutions to the angular Teukolsky equation with s ≠ 0

The visualization of the angular probability distribution for the angularTeukolsky equationwithm ≠ 0

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