Numerical Simulation of Time Dependent Schrodinger Equation using Crank–Nicolson Method

The time-dependent Schrodinger equation for a one-dimensional infinite potential well was numerically studied using the Crank-Nicolson implicit finite difference method. A quantum model with an electron confined within a potential well of length L=0.1 nm was employed. The initial wavefunction was set to be a Gaussian wave packet with a spatial width of L/20 and wave number of k = 5*1010 m^(-1). The time evolution was in steps of dt=1*10^-18 s and the spatial domain is discretized into N = 1000 points with spacing a = L/N. To form the Hamiltonian matrix, , a tridiagonal finite difference Laplacian and the boundary conditions ψ(0) = ψ(L) = 0 were used, consistent with an infinite well. When using iterative solutions, the Crank Nicolsen technique guarantees numerical stability and accuracy. After the time evolution, the probability density and the wavefunction's real and imaginary components were plotted separately to visualize their evolution behavior. The results illustrate the stable behavior within the potential well, highlighting the effectiveness of the method for modeling confined quantum systems.