where nmax is the maximum Landau number for a given e. The latter number is an integral part of the variable Y

nwc 2me

Instead of Eq. (4.4), one can use any other basis of the type

It is sufficient to assume that 0 < ( < n/2; ( may depend on n but should be zero for n = 0.

In particular, by choosing ( = arcsin ^/(1 — pz/po)/2 we obtain the basis of states with fixed electron helicities (i.e., spin projections on the canonical momentum). In this case, s defines the helicity sign.

The spin magnetic moment of the electron contains a small anomalous part whose relative magnitude is determined by the difference of the electron gyro-magnetic factor ge = 1.00116 (the ratio of the actual magnetic moment to the Bohr magneton) from 1. The anomalous magnetic moment splits the energy levels n > 1 and, strictly speaking, removes the spin degeneracy. In neutron star envelopes, however, this splitting is typically negligible, because 5e is smaller than either the thermal width — kBT of the Fermi level or the collisional width of the Landau levels (see, e.g., Kaminker & Yakovlev 1981).

Non-relativistic limit. In the non-relativistic limit, the basis of bispinors (4.4) is often most convenient, because it corresponds to fixed spin projections (sh/2) on the z-axis (two lower components of bispinors ^ns are negligible in this case). Then the coordinate part of the wave function is formally given by Eq. (4.3) with

Let us also mention that in the cylindrical gauge of the vector potential, A = (—By/2,Bx/2, 0), px is not a good quantum number; the magnetic quantum number m (i.e., the z-projection of the angular momentum in units of h) takes its place. At any n, one has m = n, n — 1, n — 2 ... In the non-relativistic limit, the coordinate parts of the basic wave functions do not depend on s (but one should not forget different statistical weights of the Landau levels with n = 0 and n > 0). These coordinate parts are

Lz ipz z/h

gim^

v 2n am is a Landau function, Lz is the normalization length, and Inn' (u) = (-l)n -nIn 'n (u) is a Laguerre function (Sokolov & Ternov, 1974). Assuming n' > n, one has nnW^-W-** g(-l)k kMn - ¿ifi- n + k, uk <4->3>

A wave function of the relativistic electron in a magnetic field in the cylindrical gauge can be also expressed in terms of §n—m(r±) (e.g., Sokolov & Ternov 1974).

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