the new fuels with magnecular structure - Institute for Basic Research

THE NEW FUELS WITH MAGNECULAR STRUCTURE 173
atom under the conditions here considered. For other atoms the critical value of
the magnetic field may be evidently different.
After outlining the quantum dynamics of a single electron in an external magnetic
field, Aringazin [8] turns to the consideration of the H atom under an
external static uniform magnetic field.
In the cylindrical coordinate system (r, ϕ, z), in which the external magnetic
field is B ⃗ = (0, 0, B), i.e., the magnetic field is directed along the z-axis, Schrödinger’s
equation for an electron moving around a fixed proton (Born-Oppenheimer
approximation) in the presence of the external magnetic field is given by
− 2
2m
(
∂ 2 r + 1 r ∂ r + 1 r 2 ∂2 ϕ + ∂ 2 z +
2me 2
)
2√ r 2 + z − 2 γ2 r 2 + 2iγ∂ ϕ ψ = Eψ,
(A.12)
where γ = eB/2c.
The main problem in the nonrelativistic study of the hydrogen atom in an
external magnetic field is to solve the above Schrödinger equation and find the
energy spectrum. This equation is not analytically tractable so that one is led to
use approximations.
In the approximation of a very strong magnetic field, B ≫ B 0 = 2.4·10 9 Gauss,
Coulomb interaction of the electron with the nucleus is not important, in the
transverse plane, in comparison to the interaction of the electron with external
magnetic field. Therefore, in accord to the exact solution (A.4) for a single
electron, one can look for an approximate ground state solution of Eq. (A.12) in
the form of factorized transverse and longitudinal parts,
ψ = e −γr2 /2 χ(z),
(A.13)
where χ(z) is the longitudinal wave function to be found. This is so called adiabatic
approximation. In general, the adiabatic approximation corresponds to
the case when the transverse motion of electron is totally determined by the
intense magnetic field, which makes the electron “dance” at its cyclotron frequency.
Specifically, the radius of the orbit is then much smaller than Bohr
radius, R 0 ≪ a 0 . The remaining problem is thus to find longitudinal energy
spectrum, in the z direction.
Inserting the wave function (A.13) into the Schrödinger equation (A.12), multiplying
it by ψ ∗ , and integrating over variables r and ϕ in cylindrical coordinate
system, one gets the following equation characterizing the z dependence of the