  We consider a quantum system weakly coupled to a large heat bath of harmonic
oscillators. It is well known that such a boson bath initially at thermal
equilibrium thermalises the system. We show that assuming a priori an
equilibrium state is not necessary to obtain the thermalisation of the system.
We determine the complete Schr\"odinger time evolution of the subsystem of
interest for an initial pure product state of the composite system consisting
of the considered system and the bath. We find that the system relaxes into
canonical equilibrium for almost all initial pure bath states of
macroscopically well-defined energy. The temperature of the system asymptotic
thermal state is determined by the energy of the initial bath state as the
corresponding microcanonical temperature. Moreover, the time evolution of the
system is identical to the one obtained assuming that the boson bath is
initially at thermal equilibrium with this temperature. A significant part of
our approach is applicable to other baths and we identify the bath features
which are requisite for the thermalisation studied.
