  For almost 75 years, the general solution for the Schr\"odinger equation was
assumed to be generated by a time-ordered exponential known as the Dyson
series. We discuss under which conditions the unitarity of this solution is
broken, and additional singular dynamics emerges. Then, we provide an
alternative construction that is manifestly unitary, regardless of the choice
of the Hamiltonian, and study various aspects of the implications. The new
construction involves an additional self-adjoint operator that might evolve in
a non-gradual way. Its corresponding dynamics for gauge theories exhibit the
behavior of a collective object governed by a singular Liouville's equation
that performs transitions at a measure $0$ set. Our considerations show that
Schr\"odinger's and Liouville's equations are, in fact, two sides of the same
coin, and together they become the unified description of quantum systems.
