Quantum and Classical Exact Solutions of the Time-Dependent Driven Generalized Harmonic Oscillator

2003 ◽  
Vol 68 (1) ◽  
pp. 41-44 ◽  
Author(s):  
Mai-Lin Liang ◽  
Hai-Bo Wu
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Time Dependent ◽  
Generalized Harmonic Oscillator

COMPLETE EXACT QUANTUM STATES OF THE GENERALIZED TIME-DEPENDENT HARMONIC OSCILLATOR

2004 ◽  
Vol 18 (24) ◽  
pp. 1267-1274 ◽  
Author(s):  
I. A. PEDROSA
Keyword(s):  
General Solution ◽  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Continuous Spectrum ◽  
Invariant Operator ◽  
Quantum States ◽  
Time Dependent ◽  
Quadratic Invariants ◽  
Generalized Time ◽  
Discrete And Continuous Spectrum

By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.

INVARIANCE OF THE GENERALIZED HANNAY ANGLE UNDER GAUGE TRANSFORMATIONS: APPLICATION TO THE TIME-DEPENDENT GENERALIZED HARMONIC OSCILLATOR

1993 ◽  
Vol 07 (11) ◽  
pp. 2147-2162
Author(s):  
DONALD H. KOBE
Keyword(s):  
Harmonic Oscillator ◽  
Time Derivative ◽  
Classical Mechanics ◽  
Rate Of Change ◽  
Time Dependent ◽  
Gauge Transformations ◽  
Correct Energy ◽  
The One ◽  
Time Dependent Problems ◽  
Generalized Harmonic Oscillator

The Hannay angle of classical mechanics is generalized so that it is invariant under gauge transformations, which are a restricted class of canonical transformations. A distinction between the Hamiltonian and the energy is essential to make in time-dependent problems. A time-dependent generalized harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. The energy, the time derivative of which is the power, is obtained from the equation of motion and related to the action variable. Hamilton’s equations give the time rate of change of the angle and action variables. The generalized Hannay angle is shown to be zero, and remains invariant under gauge transformations. On the other hand, if the original Hamiltonian is chosen as the energy, a nonzero generalized Hannay angle is obtained, but the power is given incorrectly. Nevertheless in the adiabatic limit, the total angle, which is the sum of the dynamical and Hannay angles, is equal to the one calculated from the correct energy.

scholarly journals Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures

2021 ◽  
Vol 3 (3) ◽  
pp. 458-472
Author(s):  
Kevin Zelaya ◽  
Oscar Rosas-Ortiz
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Exactly Solvable ◽  
Point Transformations

We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectra, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola–Kanai model as a particular case. Explicit solutions are given in both the classical and quantum pictures.

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