  The motion of a free quantum particle in a thermal environment is usually
described by the quantum Langevin equation, where the effect of the bath is
encoded through a dissipative and a noise term, related to each other via the
fluctuation dissipation theorem. The quantum Langevin equation can be derived
starting from a microscopic model of the thermal bath as an infinite collection
of harmonic oscillators prepared in an initial equilibrium state. The spectral
properties of the bath oscillators and their coupling to the particle determine
the specific form of the dissipation and noise. Here we investigate in detail
the well-known Rubin bath model, which consists of a one-dimensional harmonic
chain with the boundary bath particle coupled to the Brownian particle. We show
how in the limit of infinite bath bandwidth, we get the Drude model and a
second limit of infinite system-bath coupling gives the Ohmic model. A detailed
analysis of relevant correlation functions, such as the mean squared
displacement, velocity auto-correlation functions, and the response function
are presented, with the aim of understanding of the various temporal regimes.
In particular, we discuss the quantum to classical crossover time scales where
the mean square displacement changes from a $\sim \ln t$ to a $\sim t$
dependence. We relate our study to recent work using linear response theory to
understand quantum Brownian motion.
