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 Time-dependent quantum harmonic oscillator: a continuous route from adiabatic to sudden changes

2021 ◽  
Author(s):  
Daniel M. Tibaduiza ◽  
Luis Barbosa Pires ◽  
Carlos Farina
Keyword(s):  
Harmonic Oscillator ◽  
Simple Expression ◽  
Time Dependent ◽  
Frequency Change ◽  
Quantum Harmonic Oscillator ◽  
Transition Analysis ◽  
Adiabatic Theorem ◽  
Significant Difference ◽  
The One ◽  
Proper Interpretation

Abstract In this work, we give a quantitative answer to the question: how sudden or how adiabatic is a frequency change in a quantum harmonic oscillator (HO)? We do that by studying the time evolution of a HO which is initially in its fundamental state and whose time-dependent frequency is controlled by a parameter (denoted by ε) that can continuously tune from a totally slow process to a completely abrupt one. We extend a solution based on algebraic methods introduced recently in the literature that is very suited for numerical implementations, from the basis that diagonalizes the initial hamiltonian to the one that diagonalizes the instantaneous hamiltonian. Our results are in agreement with the adiabatic theorem and the comparison of the descriptions using the different bases together with the proper interpretation of this theorem allows us to clarify a common inaccuracy present in the literature. More importantly, we obtain a simple expression that relates squeezing to the transition rate and the initial and final frequencies, from which we calculate the adiabatic limit of the transition. Analysis of these results reveals a significant difference in squeezing production between enhancing or diminishing the frequency of a HO in a non-sudden way.

scholarly journals The complete symmetry group of a forced harmonic oscillator

1980 ◽  
Vol 22 (1) ◽  
pp. 12-21 ◽  
Author(s):  
P. G. L. Leach
Keyword(s):  
Harmonic Oscillator ◽  
Symmetry Group ◽  
Lie Groups ◽  
Time Dependent ◽  
One Dimensional ◽  
Complete Symmetry ◽  
Point Transformations ◽  
The One ◽  
Dependent Point ◽  
Complete Symmetry Group

AbstractThe complete symmetry group of a forced harmonic oscillator is shown to be Sl(3, R) in the one-dimensional case. Approaching the problem through the Hamiltonian invariants and the method of extended Lie groups, the method used is that of time-dependent point transformations. The result applies equally well to the forced repulsive oscillator and a particle moving under the influence of a coordinate-free force. The generalization to na-dimensional systems is discussed.

Classical Mechanics According to Newton and Hamilton

2020 ◽  
pp. 14-26
Author(s):  
Jochen Autschbach
Keyword(s):  
Angular Momentum ◽  
Harmonic Oscillator ◽  
Rotational Motion ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Equation Of Motion ◽  
Minimal Action ◽  
Newton Equation ◽  
Acceleration Force ◽  
The One

This chapter introduces classical mechanics, starting with the familiar definitions of position, momentum, velocity, acceleration force, kinetic, potential, and total energy. It is shown how the Newton equation of motion is solved for the one-dimensional harmonic oscillator, which is a point mass oscillating around the position x = 0 driven by a force that is proportional to x (Hooke’s law). Next, the minimal action principle, the Lagrange equation of motion, and the classical Hamilton function (Hamiltonian) and conjugated variables are introduced. The chapter also discusses angular momentum and rotational motion.

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