  By analogy with the fundamental quantum units of electrical conductance
$G_0^e=\frac{2 e^2}{h}$ and thermal conductance $K_0^t=\frac{2 K_B^2 T}{h}$ we
define a fundamental quantum unit of conductance, $G_0^m$, and diffusion of a
massive gas of atomic particles, respectively given by $$ G_0^m=\frac{m^2}{h} \
, \ D_0=\frac{h}{m}$$ with $h$ the Planck constant, $K_B$ the Boltzmann
constant, $T$ the absolute temperature, $e$ the unit charge and $m$ the mass of
the atomic gas particle that move balistically in a one dimensional medium of
length $L$. The effect of scattering can be accounted for by introducing an
appropriate transmission probability in analogy with the quantum electrical
conductance model introduced by Landauer in 1957. For an electron gas
$G_0^m=1.25 \times 10^{-27} \ Kg^2/(J s)$ and $D_0 = 7.3 \times 10^{-3} \
m^2/s$, and we found a quantum expression for the generalized Einstein relation
that writes $$G_0^e = \frac{2e^2m}{h^2} D_0 $$
