This is related to the probability density of the quantum harmonic oscillator. For large $k$ the density
$$P_k(x)=\frac{1}{\sqrt{2\pi}k!} h_k^2 \, e^{-x^2/2},$$
normalized to unity, has the approximation
$$p_k(x)=\frac{1}{\pi}(4k-x^2)^{-1/2},\;\;|x|<2\sqrt{k}.$$
The boundaries $\pm 2\sqrt{k}$ are the classical turning points of the harmonic oscillator.

The plot compares $P_k$ and $p_k$ for $k=100$. The function $p_k$ is a smoothed version of $P_k$, suitable for averaging quantities that vary little on the scale of $1/\sqrt{k}$.

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