Time Evolution of Quantum-Mechanical Harmonic Oscillator with Time-Dependent Frequency

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.)