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For the example of the infinitely deep well potential, we point out someparadoxes which are solved by a careful analysis of what is a trulyself-adjoint operator. We then describe the self-adjoint extensions and theirspectra for the momentum and the Hamiltonian operators in different physicalsituations. Some consequences are worked out, which could lead to experimentalchecks.Comment: 25 pages, Latex file, extended version of Am. J. Phys. 69 (2001) 32

Self-adjoint extensions of operators and the teaching of quantum mechanics