The description of waves in a random medium propagating mainly in one direction leads to the random Schrödinger equation \(i \partial V/\partial t+\Delta V+\mu V=0,\) \(x\in (x_ 1,x_ 2)\in {\mathbb{R}}^ 2,\) \(V(0,x_ 1,x_ 2)=V_ 0(x_ 1,x_ 2),\) \(\Delta =\partial^ 2/\partial x^ 2_ 1+\partial^ 2/\partial x^ 2_ 2,\) \(\mu (t,x_ 1,x_ 2)\) a given real-valued process. Under the assumption that \(\mu\) is Gaussian white noise in t a unique mild solution of the corresponding Fourier-transformed stochastic partial differential equation is constructed in \(L^ 2({\mathbb{R}}^ 2)\) by a Wiener-Itô expansion. This solution turns out to be a Markov diffusion process on the unit sphere of \(L^ 2({\mathbb{R}}^ 2)\) (conservation of energy). A limiting regime (narrow beam spot-dancing) is investigated.