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

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210-1-2-2 -1 0 1 2Figure 4: A contour plot of the (r, ϕ) probability density for the case ofthe Landau ground state of a single electron, 2πr|ψ 000 | 2 , Eq. (2.14), in thestrong external magnetic field B = B 0 = 2.4 · 10 9 Gauss, as a function of thedistance in a.u. (1 a.u. = 0.53 · 10 −8 cm). Lighter area corresponds to higherprobability to find electron. The set of maximal values of the probabilitydensity is referred to as an ”orbit”.The condition that the Landau radius R 0 is smaller than the Bohr radius,R 0 < a 0 , which is adopted here as the condition of a considerable ”deformation”of the electron orbit of the hydrogen atom, then impliesB > B 0 = m2 ce 3¯h 3 = 2.351 · 10 9 Gauss, (2.17)where m is mass of electron. Equivalently, this deformation condition correspondsto the case when the binding energy of the hydrogen atom, |E0 Bohr | =|−me 4 /2¯h 2 | = 0.5 a.u. = 13.6 eV, is smaller than the ground Landau energyE0 ⊥ .The above critical value of the magnetic field, B 0 , is naturally taken asan atomic unit for the strength of the magnetic field, and corresponds to the9
case when the pure Coulomb interaction energy of electron with nucleus isequal to the interaction energy of a single electron with the external magneticfield, |E0 Bohr | = E0 ⊥ = 13.6 eV, or equivalently, when the Bohr radius is equalto the Landau radius, a 0 = R 0 = 0.53 · 10 −8 cm.It should be stressed here that we take the characteristic parameters,Bohr energy |E0 Bohr | and Bohr radius a 0 , of the hydrogen atom with a singlepurpose to establish criterium for the critical strength of the external magneticfield, for the hydrogen atom under consideration. For other atoms thecritical value of the magnetic field may be evidently different.3 Hydrogen atom in magnetic fieldsAfter we have considered quantum dynamics of a single electron in the externalmagnetic field, we turn to consideration of the hydrogen atom in theexternal static uniform magnetic field.We choose cylindrical coordinate system (r, ϕ, z), in which the externalmagnetic field is ⃗ B = (0, 0, B), i.e., the magnetic field is directed along thez-axis. General Schrödinger equation for the electron moving around a fixedproton (Born-Oppenheimer approximation) in the presence of the externalmagnetic field is(− ¯h2 ∂r 2 + 1 2m r ∂ r + 1 r 2 ∂2 ϕ + ∂z 2 +2me 2)¯h 2√ r 2 + z − 2 γ2 r 2 + 2iγ∂ ϕ ψ = Eψ,(3.1)where γ = eH/2¯hc. Note the presence of the term 2iγ∂ ϕ , which is linear inB, and of the term −γ 2 r 2 , which is quadratic in B. The other terms do notdepend on the magnetic field B.The main problem in the nonrelativistic study of the hydrogen atom inthe external magnetic field is to solve the above Schrödinger equation andfind the energy spectrum.In this equation, we can not directly separate all the variables, r, ϕ, andz, because of the presence of the Coulomb potential, e 2 / √ r 2 + z 2 , which doesnot allow us to make a direct separation in variables r and z.Also, for the general case of an arbitrary intensity of the magnetic field,we can not ignore the term γ 2 r 2 in Eq. (3.1), which is quadratic in B, sincethis term is not small at high intensities of the magnetic field.10
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: 0.60.50.4Landau0.30.20.100 1 2 3 4
Page 13 and 14: Gauss), and the energy spectrum in
Page 15 and 16: In atomic units (e = ¯h = m = 1),
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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