All separable Hilbert spaces of given dimension being isomorphic implies exactly six basic generalized coordinate operators and seven coordinate-momentum pairs via self-adjoint or Neumark extensions.
Self-adjoint extensions of operators and the teaching of quantum mechanics
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abstract
For the example of the infinitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self-adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different physical situations. Some consequences are worked out, which could lead to experimental checks.
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2025 1verdicts
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All Hilbert spaces are the same: consequences for generalized coordinates and momenta
All separable Hilbert spaces of given dimension being isomorphic implies exactly six basic generalized coordinate operators and seven coordinate-momentum pairs via self-adjoint or Neumark extensions.