Abstract
A one-dimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with time-dependent coefficients. This trial function allows for general time-dependent 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 time-dependent 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 Kepler-Coulomb problem when transformed to the model of a four-dimensional 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.
Highlights
Coherent states of the harmonic oscillator (HO) were introduced already at the beginning of wave mechanics [1]
The connection of HO and reversed harmonic oscillator (RO) with the quantum mechanics of the Kepler-Coulomb problem is briefly discussed in the context of the derivation of the classical Kepler orbits from quantum mechanics
In the following we examine two cases for the HO: type-I and type-II are characterized by a constant and time-dependent function Γ, respectively
Summary
Coherent states of the harmonic oscillator (HO) were introduced already at the beginning of wave mechanics [1]. In [14], the elliptic Kepler orbits were derived in the classical limit on the basis of coherent HO states. By means of coherent RO states, the classical limit for hyperbolic Kepler orbits was achieved in [16,17], whereby the elliptic regime could be obtained by analytic continuation from the hyperbolic side. As compared to the classical elliptic Kepler orbits, the derivation of hyperbolic orbits from quantum mechanics was accomplished quite recently [16,17]. For this achievement, it was crucial to devise a suitable time-dependent ansatz for the wave function, see (1) below, in order to construct coherent RO states. The connection of HO and RO with the quantum mechanics of the Kepler-Coulomb problem is briefly discussed in the context of the derivation of the classical Kepler orbits from quantum mechanics
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