The harmonic oscillator as a tutorial introduction to quantum mechanics (original) (raw)

Stemming from the similar linearities of the Schrödinger equation in quantum mechanics on the one hand and of the harmonic oscillations in classical mechanics on the other hand, the idea that any N-degree-of-freedom harmonic oscillator (HON) is formally equivalent to a N-level quantum system is put forward. It is shown that the complex dynamic variables α introduced by R. J. Glauber can be regarded as the components of a state vector belonging to some N-dimension complex Hilbert space, and whose time-evolution is ruled by a Schrödinger-like equation. In case the classical HON is parametrically excited, the unitarity of the time-evolution of the associated quantum system is related to the Ehrenfest adiabaticity of the parametric excitation.