  In this paper, we investigate the convergence in probability of a stochastic
symplectic scheme for stochastic nonlinear Schr\"{o}dinger equation with
quadratic potential and an additive noise. Theoretical analysis shows that our
symplectic semi-discretization is of order one in probability under appropriate
regularity conditions for the initial value and noise. Numerical experiments
are given to simulate the long time behavior of the discrete average charge and
energy as well as the influence of the external potential and noise, and to
test the convergence order.
