  We consider the problem of determining the energy distribution of quantum
states that satisfy exponential decay of correlation and product states, with
respect to a quantum local hamiltonian on a spin lattice. For a quantum state
on a $D$-dimensional lattice that has correlation length $\sigma$ and has
average energy $e$ with respect to a given local hamiltonian (with $n$ local
terms, each of which has norm at most $1$), we show that the overlap of this
state with eigenspace of energy $f$ is at most
$exp(-((e-f)^2\sigma)^{\frac{1}{D+1}}/n^{\frac{1}{D+1}}D\sigma)$. This bound
holds whenever $|e-f|>2^{D}\sqrt{n\sigma}$. Thus, on a one dimensional lattice,
the tail of the energy distribution decays exponentially with the energy.
  For product states, we improve above result to obtain a Gaussian decay in
energy, even for quantum spin systems without an underlying lattice structure.
Given a product state on a collection of spins which has average energy $e$
with respect to a local hamiltonian (with $n$ local terms and each local term
overlapping with at most $m$ other local terms), we show that the overlap of
this state with eigenspace of energy $f$ is at most $exp(-(e-f)^2/nm^2)$. This
bound holds whenever $|e-f|>m\sqrt{n}$.
