Motion of a Gaussian wave packet in a periodically pulsed harmonic oscillator

We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.

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QUANTUM PARAMETRIC RESONANCE: ANALYTICAL STUDIES

Journal of Theoretical and Computational Chemistry ◽

10.1142/s0219633608004015 ◽

2008 ◽

Vol 07 (04) ◽

pp. 629-637 ◽

Cited By ~ 1

Author(s):

JIUSHU SHAO

Keyword(s):

Wave Packet ◽

Harmonic Oscillator ◽

Parametric Resonance ◽

Gaussian Wave Packet ◽

Analytical Studies ◽

Propagator Method ◽

Quantum Version ◽

Floquet Operator ◽

Discrete Map

The quantum version of parametric resonance of the harmonic oscillator is studied in terms of Feynman's propagator method and a discrete map. The eigenfunctions and eigenvalues of the Floquet operator are derived explicitly. The solutions follow the general laws: In the regions of parametric resonance the spectrum of the Floquet operator becomes continuous, while it is pointlike in the stable regimes. The Gaussian wave packet undergoes spreading exponentially and linearly in the two regimes, respectively.

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GEOMETRIC PHASES, SQUEEZED QUANTUM STATES AND GAUSSIAN WAVE PACKET STATES OF RELIC GRAVITONS

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194512008136 ◽

2012 ◽

Vol 18 ◽

pp. 13-17

Author(s):

K. BAKKE ◽

I. A. PEDROSA ◽

C. FURTADO

Keyword(s):

Wave Packet ◽

Linear Time ◽

Invariant Operator ◽

Quantum States ◽

Time Dependent ◽

Gaussian Wave Packet ◽

Geometric Phases ◽

Uncertainty Product ◽

Quantized Field ◽

Generalized Time

In this contribution, we discuss quantum effects on relic gravitons described by the Friedmann-Robertson-Walker (FRW) spacetime background by reducing the problem to that of a generalized time-dependent harmonic oscillator, and find the corresponding Schrödinger states with the help of the dynamical invariant method. Then, by considering a quadratic time-dependent invariant operator, we show that we can obtain the geometric phases and squeezed quantum states for this system. Furthermore, we also show that we can construct Gaussian wave packet states by considering a linear time-dependent invariant operator. In both cases, we also discuss the uncertainty product for each mode of the quantized field.

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