Extending the stationary quantum mechanics of being to a nonstationary quantum theory of becoming and decaying

It is shown that the foundations of quantum mechanics allow, besides the conventional mathematical theory in the Hilbert space, another mathematical idealization which uses the Gelʹfand triplet or Rigged Hilbert Space (RHS). The mathematics of the RHS is introduced, and the similarities and distinctions of the von Neumann formulation and the RHS formulations are discussed. It is shown that in the RHS, one can obtain a more encompassing quantum theory with intrinsic (microphysical) irreversibility, for which the conventional von Neumann quantum mechanics is the restriction to the physics of stationary states and reversible time evolution.