Narrow Results

It is proven that the energy of a quantum mechanical harmonic oscillator with a generically time-dependent but cyclic frequency, ω(t₀) = ω(0), cannot decrease on an average if the system is originally in a stationary state, after the system goes through a full cycle. The energy exchange always takes place in the direction from the macroscopic system (environment) to the quantum microscopic system.

We study the consequences of the generalized Heisenberg uncertainty relation which admits a minimal uncertainty in length such as the case in a theory of quantum gravity. In particular, the theory of quantum harmonic oscillators arising from such a generalized uncertainty relation is examined. We demonstrate that all the standard properties of the quantum harmonic oscillators prevail when we employ a generalized momentum. We also show that quantum electrodynamics and coherent photon states can be described in the familiar standard manner despite the generalized uncertainty principle.

This contribution summarizes work on finite, non-cyclic Hamiltonian systems —in particular the one-dimensional finite oscillator—, in conjunction with a Lie algebraic definition of the (meta-) phase space of finite systems, and a corresponding Wigner distribution function for the state vectors. The consistency of this approach is important for the strategy of fractionalization of a finite Fourier transform, and the contraction of finite unitary to continuous symplectic transformations of Hamiltonian systems.

The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically, solving the fractional Schrödinger equation based on the Riemann and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

Analytical expressions for the PM noise in FET oscillators are derived in terms of the FET equivalent circuit elements and the passive circuitry. Efficient methods to reduce the PM noise in fundamental and in harmonic modes are suggested and implemented. The effects of the different FET equivalent circuit parameters on large-signal, small-signal, and noise behavior of FET oscillators are investigated. Finally, the effects of harmonic signal on the output noise power and on the phase noise at both fundamental and harmonic frequencies are determined.

In the wake of a preceding article [31] introducing the Schrödinger–Virasoro group, we study its affine action on a space of (1+1)-dimensional Schrödinger operators with time- and space-dependent potential V periodic in time. We focus on the subspace corresponding to potentials that are at most quadratic in the space coordinate, which is in some sense the natural quantization of the space of Hill (Sturm–Liouville) operators on the one-dimensional torus. The orbits in this subspace have finite codimension, and their classification by studying the stabilizers can be obtained by extending Kirillov's results on the orbits of the space of Hill operators under the Virasoro group. We then explain the connection to the theory of Ermakov–Lewis invariants for time-dependent harmonic oscillators. These exact adiabatic invariants behave covariantly under the action of the Schrödinger–Virasoro group, which allows a natural classification of the orbits in terms of a monodromy operator on L²(ℝ) which is closely related to the monodromy matrix for the corresponding Hill operator.