Thank you very much for your prompt assistance. Reducing the grid steps in time and space does help improve the numerical calculation of the discrete Schrödinger equation. However, my main mistake was not shifting the kinetic energy to the center. The command "ResourceFunction["FourierShift"]" resolves this issue. Unfortunately, I only realized this some time after submitting the question.
Using Fourier to get the numerical FT of a Gaussian
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