Why the square root stabilizes Poisson counts

You estimate a Poisson rate $\lambda$ by the sample mean $\bar X$ of $n$ counts, so $\operatorname{Var}(\bar X) = \lambda / n$ (for a Poisson, the variance equals the mean). The problem: this variance depends on $\lambda$, so high-rate and low-rate cells have different noise.

Use the delta method to show that $g(\bar X) = \sqrt{\bar X}$ has a variance that does not depend on $\lambda$.