We report a measurement of the local acceleration $g$ with ultracold neutrons based on quantum states in the gravity potential of the Earth. The new method uses resonant transitions between the states $|1> -> |3>$ and for the first time between $|1> -> |4>$. The measurements demonstrate that Newtons Inverse Square Law of Gravity is understood at micron distances at an energy level of $10^{-14}$ eV with $frac{Delta g}{g}=4times10^{-3}$. The results provide constraints on any possible gravity-like interaction at a micrometer interaction range. In particular, a dark energy candidate, the chameleon field is restricted to $beta<6.9times10^{6}$ for $n=2$ (95% C.L.).
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