We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on $\mathbb{R}$, with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő-type central limit theorem (CLT) for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalization to a Gaussian limit with a $H^{1/2}$-type variance, which depends on the potential. In the one-well (one-cut) case, using the quantum action-angle theorem and additional micro-local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi-cut case, we show that for generic potentials, a similar result holds and the contributions of each wells are independent in the limit.