The harmonic oscillator, described by the Schrödinger equation
is a central textbook example in quantum mechanics. Its time evolution can be easily given in closed form. More generally, the time evolution of a harmonic oscillator with a time-dependent frequency

can also be given in quadratures. This allows the efficient solution of the Schrödinger equation as a system of just three coupled nonlinear ordinary differential equations.
This Demonstration lets you see

and

for various time-dependent frequencies of the functional form
using the initial wave function (moving harmonic oscillator ground state)

.
("Calculating ..." sometimes appears when the Demonstration cannot compute a solution.)