  A vortex in a quasi two-dimensional Bose-Einstein condensate is subject to
the Magnus force and can be effectively described as a planar particle in a
uniform magnetic field. Quantization of this effective particle leads to the
lowest Landau level where the most localized wave function is a gaussian. In
this gaussian state vortex position seems to fluctuate with an average
magnitude set by the magnetic width of the gaussian. We readdress this problem
using the number-conserving version of the Bogoliubov theory. We find that the
Bogoliubov mode that might be interpreted as a fluctuation of vortex position
actually does not contribute to position fluctuation at all. The only non-zero
contribution comes from phonons but it is an order of magnitude less than the
simple estimate, based on the magnetic width of the effective gaussian wave
packet.
