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We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems -the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can bedoi:10.1590/s0103-97332008000100030 fatcat:2sgxxt6dnvfv7n7xsi4uemqn34