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The purpose of this thesis is to implement three numerical methods for solving and examining the time-dependent Schrödinger equation (TDSE) of the analytically solved quantum harmonic oscillator (QHO) and the, to our knowledge, analytically unsolved double well potential (DW), and to compare the numerical solutions. These methods are the Crank-Nicolson method and two split operator methods of different orders. For the QHO, the exact solution is used to determine the errors of the methods, while other methods for examining the DW had to be used (due to the lack of exact solutions). The solutions converged for the QHO, and depending the desired accuracy, either the Crank-Nicolson method or the modified split operator method were the most effective choices with regards to the global error and computation time. Applied on the DW potential, the numerical methods succeeded in displaying expected behaviours, such as locally behaving like a QHO under specific conditions, and displaying the quantum tunneling effect. For very small time-steps and fine spacial discretizations, the solutions did not deviate much from each other, indicating on convergence toward a correct solution. |
