  The longitudinal friction acting on a vortex line in superfluid $^4$He is
investigated within a simple model based on the analogy between such vortex
dynamics and that of the quantal Brownian motion of a charged point particle in
a uniform magnetic field. The scattering of superfluid quasiparticle
excitations by the vortex stems from a translationally invariant interaction
potential which, expanded to first order in the vortex velocity operator, gives
rise to vortex transitions between nearest Landau levels. The corresponding
friction coefficient is shown to be, in the limit of elastic scattering
(vanishing cyclotron frequency), equivalent to that arising from the Iordanskii
formula. Proposing a simple functional form for the scattering amplitude, with
only one adjustable parameter whose value is set in order to get agreement to
the Iordanskii result for phonons, an excellent agreement is also found with
the values derived from experimental data up to temperatures about 1.5 K.
Finite values of the cyclotron frequency arising from recent theories are shown
to yield similar results. The incidence of vortex-induced quasiparticle
transitions on the friction process is estimated to be, in the roton dominated
regime, about 50 % of the value of the friction coefficient, $\sim$8 % of which
corresponds to roton-phonon transitions and $\sim$42 % to roton
$R^+\leftrightarrow R^-$ ones.
