  We compute the braided groups and braided matrices $B(R)$ for the solution
$R$ of the Yang-Baxter equation associated to the quantum Heisenberg group. We
also show that a particular extension of the quantum Heisenberg group is dual
to the Heisenberg universal enveloping algebra $U_{q}(h)$, and use this result
to derive an action of $U_{q}(h)$ on the braided groups. We then demonstrate
the various covariance properties using the braided Heisenberg group as an
explicit example. In addition, the braided Heisenberg group is found to be
self-dual. Finally, we discuss a physical application to a system of n braided
harmonic oscillators. An isomorphism is found between the n-fold braided and
unbraided tensor products, and the usual `free' time evolution is shown to be
equivalent to an action of a primitive generator of $U_{q}(h)$ on the braided
tensor product.
