the new fuels with magnecular structure - Institute for Basic Research

THE NEW FUELS WITH MAGNECULAR STRUCTURE 179
y
x
z
Figure A5. The axial wavefunctions of hydrogen in an intense magnetic field (analytic calculation)
for B = 4.7 · 10 12 Gauss. The first four even states with axial excitations, |000〉 (ground
state), |002〉, |004〉, and |006〉 (left panel), and odd states |001〉 and |003〉 (right panel) are
depicted; n = 1/ √ 15.58, ζ = 2z/n corresponds to x in the used notation; z in a.u., 1 a.u. =
0.53 · 10 −8 cm (reproduction of Figure 3 by Heyl and Hernquist [14]).
Heyl and Hernquist calculated the first-order perturbative corrections to the
above energies and obtained the values, which are in a good agreement with
the results by Ruder et al. [9] and Lai [12].
The associated probability density of the above excited states is evidently of a
cylindrical (axial) symmetry and can be described as two Landau orbits of radius
R 0 in different (r, ϕ) planes, one at the level z = −L z , and the other at the
level z = +L z , with the nucleus at z = 0, as schematically depicted in Fig. A6.
Presence of two Landau orbits occurs in accord to the excited wave functions,
which is symmetrical with respect to the inversion, z → −z, and the largest peaks
of which are away from the center x = 0. The electron moves simultaneously on
these two Landau orbits.
A review of approximate, variational, and numerical solutions can be found
in the paper by Lai [12]. The accuracy of numerical solutions is about 3%,
for the external magnetic field in the range from 10 11 to 10 15 Gauss. Particularly,
due to the variational results [12], the z-size of the hydrogen atom in the
ground state is well approximated by the formula L z ≃ [ln(B/B 0 )] −1 a.u.; the