Toroidal configuration of the orbit of the electron of the hydrogen ...

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0-50-100C, eV-150-200-250-30002·10 -9 4·10 -9 6·10 -9 8·10 -9 1·10 -8z, cm0-500Coloumb, eV-1000-1500-2000-2500-300002·10 -9 4·10 -9 6·10 -9 8·10 -9 1·10 -8z, cmFigure 5: The exact effective potential C(z) (top panel), at B = 2.4 · 10 11Gauss = 100B 0 , C(0) = −337 eV, and (for a comparison) Coulomb potential−e 2 /|z| (bottom panel).Then, one can perform the simplified integral and obtain the resultC(z) ≃ V (z) = − e2|z| , at γ〈z2 〉 ≫ 1, (3.7)which appears to be a pure Coulomb interaction of electron with the nucleus,in the z direction. Therefore, Eq. (3.4) reduces to one-dimensionalSchrödinger equation for Coulomb potential,( ¯h2d 2)2m dz + e22 |z| + ¯h2 γm + E χ(z) = 0. (3.8)13
In atomic units (e = ¯h = m = 1), using the notationE ′ = ¯h2 γm + E, n2 = 1−2E ′ , (3.9)and introducing new variablex = 2z n , (3.10)we rewrite the above equation in the following form:[ d(−dx + 1 2 4 + n ) ] χ(x) = 0, (3.11)xwhere we assume, to simplify representation, x > 0. Introducing new functionv(x),χ(x) = xe −x/2 v(x), (3.12)we get the final form of the equation,xv ′′ + (2 − x)v ′ − (1 − n)v = 0. (3.13)Note that this equation is identical to that obtained for the radial part of thethree-dimensional Schrödinger equation for the hydrogen atom, at orbitalquantum number zero. We note that it is a particular case of Cummer’sequation,xv ′′ + (b − x)v ′ − av = 0, (3.14)the general solution of which is given bywhereand1F 1 (a, b, x) =v(x) = C 1 1 F 1 (a, b, x) + C 2 U(a, b, x) (3.15)∫Γ(b) 1e xt t a−1 (1 − t) b−a−1 dt (3.16)Γ(b − a)Γ(a) 0U(a, b, x) = 1 ∫ ∞e −xt t a−1 (1 + t) b−a−1 dt (3.17)Γ(a) 0are the confluent hypergeometric functions, and C 1,2 are constants. In ourcase, a = 1 − n and b = 2.14
Page 1 and 2: Toroidal configuration of the orbit
Page 3 and 4: problem of atoms and molecules expo
Page 5 and 6: formally compare the Bohr radius of
Page 7 and 8: Q n−ss is Laguerre polynomial, L
Page 9 and 10: 0.60.50.4Landau0.30.20.100 1 2 3 4
Page 11 and 12: case when the pure Coulomb interact
Page 13: Gauss), and the energy spectrum in
Page 17 and 18: One can see that the problem is in
Page 19 and 20: yxzFigure 6: Schematic view on the
Page 21 and 22: where x 0 > 0 (we remind that x = 2
Page 23 and 24: zyxFigure 8: Schematic view of the
Page 25 and 26: Transverse size L, a.u.0.10.090.080
Page 27 and 28: functions made by Heyl and Hernquis
Page 29 and 30: Magnetic field B(Gauss)Binding ener
Page 31 and 32: Figure 16: Contour plots of the (r,
Page 33 and 34: Figure 17: The ground and first-exc
Page 35 and 36: Appendix2000201500151F11000U1050050
Page 37 and 38: References[1] A.A. Sokolov, I.M. Te
Page 39: M.V. Ivanov and P. Schmelcher, Grou

Toroidal configuration of the orbit of the electron of the hydrogen ...

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