  Let $\beta>0$ and consider an $n$-point process $\lambda_1, \lambda_2,...,
\lambda_n$ from Hermite $\beta$ ensemble on the real line $\mathbb{R}$.
Dumitriu and Edelman discovered a tri-diagonal matrix model and established the
global Wigner semicircle law for normalized empirical measures. In this paper
we prove that the average number of states in a small interval in the bulk
converges in probability when the length of the interval is larger than $\sqrt
{\log n}$, i.e., local semicircle law holds. And the number of positive states
in $(0,\infty)$ is proved to fluctuate normally around its mean $n/2$ with
variance like $\log n/\pi^2\beta$. The proofs rely largely on the way invented
by Valk$\acute{o}$ and Vir$\acute{a}$g of counting states in any interval and
the classical martingale argument.
