In this article, we present the mathematical analysis of the convergence of the linearized Crank-Nicolson Galerkin method for a nonlinear Schrodinger problem related to a domain with a moving boundary. The convergence analysis of the numerical method is carried out for both semi-discrete and fully discrete problems. An optimal error estimate in the $L^2$-norm with order ${O}(τ^2+ h^s),~ 2\leq s\leq r$, where $h$ is the finite element mesh size parameter, $τ$ is the time step, and $r-1$ represents the degree of the finite element polynomial basis. Numerical simulations are provided to confirm the consistency between theoretical and numerical results, validating the method and the order of convergence for different degrees $p\geq 1$ of the Lagrange polynomials and also for Hermite polynomials (degree $p=3$), which form the basis of the approximate solution.