We will look at the axioms of quantum mechanics as expressed, for example, in the book by M. A. Nielsen and I. L. Chung ("Quantum Computation and Quantum Information"). We then take a critical look at these axioms, raising several questions as we go. In particular, we will look at the possible informational completeness property of the family of operators that we measure. We will propose physical solutions based on the results of quantum mechanics on phase space and the measurement of quantum particles by quantum mechanical means. We illustrate this with both momentum-position measurements and spin measurements.