Invariant operators for quadratic Hamiltonians

The motion of wave packets can be easily determined for any Hamiltonian tha t is quadratic in position and momentum, even if the coefficients of the te rms in the Hamiltonian vary with timer The method is based on the existence of an invariant operator, linear in position and momentum, the coefficient s of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple Wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy e igenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite-Gaussia n). This set provides a complete basis for finding the evolution of any sta te and the exact propagator is readily determined. For the harmonic oscilla tor, this set includes coherent states, squeezed states, displaced number s tates, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time depen dence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2 ) Every detail of the evolution of the quantum system can be calculated fro m the Solutions of the corresponding classical system. (3) Wave functions o f Gaussian (or Hermite-Gaussian) form will retain that form as they evolve. (C) 1999 American Association of Physics Teachers.