  This paper is primarily devoted to a class of interpolation inequalities of
Hardy and Rellich types on the Heisenberg group $\mathbb{H}^n$. Consequently,
several weighted Hardy type, Heisenberg-Pauli-Weyl uncertainty principle and
Hardy-Rellich type inqualities are established on $\mathbb{H}^n$. Moreover, new
weighted Sobolev type embeddings are derived. Finally, an integral inequality
for vector fields in a domain of the Heisenberg group is obtained, leading to
several specific weighted Hardy type inequalities by making careful choices of
vector fields.
