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The one dimensional Schrödinger equation: symmetries, solutions and Feynman propagators

A simple method to find the symmetries of the Schrödinger equation in one dimension with arbitrary potentials is presented. The method hereby used can be of interest to students in quantum mechanics at the undergraduate level. Several physical questions arising from this symmetry analysis are also discussed. Finally, some attention is paid to how the result leads to the explicit description of the exact propagators in the linear and quadratic case without using the more popular method of describing the classical action in the stationary phase approximation.