Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential

摘要

The unitary operator which transforms a harmonic oscillator system oftime-dependent frequency into that of a simple harmonic oscillator of differenttime-scale is found, with and without an inverse-square potential. It is shownthat for both cases, this operator can be used in finding complete sets of wavefunctions of a generalized harmonic oscillator system from the well-known setsof the simple harmonic oscillator. Exact invariants of the time-dependentsystems can also be obtained from the constant Hamiltonians of unit mass andfrequency by making use of this unitary transformation. The geometric phasesfor the wave functions of a generalized harmonic oscillator with aninverse-square potential are given.