Uncertainty relations for the time-dependent singular oscillator in the number state and in the coherent state are investigated. We applied our developement to the Caldirola–Kanai oscillator perturbed by a singularity. For this system, the variation (Δx) decreased exponentially while (Δp) increased exponentially with time both in the number and in the coherent states. As k → 0 and χ → 0, the number state uncertainty relation in the ground state becomes 0.583216？ which is somewhat larger than that of the standard harmonic oscillator, ？/2. On the other hand, the uncertainty relation in all excited states become smaller than that of the standard harmonic oscillator with the same quantum number n. However, as k → ∞ and χ → 0, the uncertainty relations of the system approach the uncertainty relations of the standard harmonic oscillator, (n+1/2)？.

We investigated coherent and squeezed states of light in linear media whose parameters are explicitly dependent on time by making use of the Lewis–Riesenfeld invariant operator method. Not only the field strengths but also the fluctuations of the fields both in coherent and in squeezed states are decayed with time. The relative noise of the field strengths are calculated in coherent state. Quantum statistical properties of the chaotic field are investigated. We applied our theory to a phenomenological model of the biophoton system and compared the corresponding result of the uncertainty product with that obtained from a previous report.

We showed that the idea of Schleich and Wheeler (1987, Nature 326, 574) for the semiclassical approach of the interference in phase space of harmonic oscillator squeezed states can be extended to that of general time-dependent Hamiltonian system. The quantum phase properties of squeezed states for the general time-dependent Hamiltonian system are investigated by using the quantum distribution function. The weighted overlaps A _n and phases θ _n for the system are evaluated in the semiclassical limit.