  We assume that $\mathcal{E}$ is a rank $r$ Ulrich bundle for $(P^n,
\mathcal{O}(d))$. The main result of this paper is that $\mathcal{E}(i)\otimes
\Omega^{j}(j)$ has natural cohomology for any integers $i \in \mathbb{Z}$ and
$0 \leq j \leq n$, and every Ulrich bundle $\mathcal{E}$ has a resolution in
terms of $n$ of the trivial bundle over $P^n$. As a corollary, we can give a
necessary and sufficient condition for Ulrich bundles if $n \leq 3$, which can
be used to find some new examples, i.e., rank $2$ bundles for $(P^3,
\mathcal{O}(2))$ and rank $3$ bundles for $(P^2, \mathcal{O}(3))$.
