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In this work, we study the semi-classical limit of the Schr\"odinger equationwith random inputs, and show that the semi-classical Schr\"odinger equationproduces $O(\varepsilon)$ oscillations in the random variable space. With theGaussian wave packet transform, the original Schr\"odinger equation is mappedto an ODE system for the wave packet parameters coupled with a PDE for thequantity $w$ in rescaled variables. Further, we show that the $w$ equation doesnot produce $\varepsilon$ dependent oscillations, and thus it is more amenablefor numerical simulations. We propose multi-level sampling strategy inimplementing the Gaussian wave packet transform, where in the most costly part,i.e. simulating the $w$ equation, it is sufficient to use $\varepsilon$independent samples. We also provide extensive numerical tests as well asmeaningful numerical experiments to justify the properties of the numericalalgorithm, and hopefully shed light on possible future directions.

Gaussian wave packet transform based numerical scheme for the semi-classical Schr\"odinger equation with random inputs