Rauh, Alexander
(2016)
Coherent States of Harmonic and Reversed Harmonic Oscillator.
Symmetry, 8(6) (46).
ISSN 20738994
Abstract
A onedimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with timedependent coefficients. This trial function allows for general timedependent solutions both of the harmonic oscillator (HO) and the reversed harmonic oscillator (RO). For the HO, apart from the standard coherent states, a further class of solutions is derived with a timedependent width parameter. The width of the corresponding probability density fluctuates, or "breathes" periodically with the oscillator frequency. In the case of the RO, one also obtains normalized wave packets which, however, show diffusion through exponential broadening with time. At the initial time, the integration constants give rise to complete sets of coherent states in the three cases considered. The results are applicable to the quantum mechanics of the KeplerCoulomb problem when transformed to the model of a fourdimensional harmonic oscillator with a constraint. In the classical limit, as was shown recently, the wave packets of the RO basis generate the hyperbolic Kepler orbits, and, by means of analytic continuation, the elliptic orbits are also obtained quantum mechanically.
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