  A Brownian particle in an ideal quantum gas is considered. The mean square
displacement (MSD) is derived. The Bose-Einstein or Fermi-Dirac distribution,
other than the Maxwell-Boltzmann distribution, provides a different stochastic
force compared with the classical Brownian motion. The MSD, which depends on
the thermal wavelength and the density of medium particles, reflects the
quantum effect on the Brownian particle explicitly. The result shows that the
MSD in an ideal Bose gas is shorter than that in a Fermi gas. The behavior of
the quantum Brownian particle recovers the classical Brownian particle as the
temperature raises. At low temperatures, the quantum effect becomes obvious.
For example, there is a random motion of the Brownian particle due to the
fermionic exchange interaction even the temperature is near the absolute zero.
