The complete symmetry group of the one‐dimensional time‐dependent harmonic oscillator

1980 ◽  
Vol 21 (2) ◽  
pp. 300-304 ◽  
Author(s):  
P. G. L. Leach
Keyword(s):  
Harmonic Oscillator ◽  
Symmetry Group ◽  
Time Dependent ◽  
One Dimensional ◽  
Complete Symmetry ◽  
The One ◽  
Complete Symmetry Group

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.

scholarly journals Dynamical Invariants and Quantization of the One-Dimensional Time-Dependent, Damped, and Driven Harmonic Oscillator

2020 ◽  
Vol 50 (5) ◽  
pp. 534-540
Author(s):  
M. C. Bertin ◽  
J. R. B. Peleteiro ◽  
B. M. Pimentel ◽  
J. A. Ramirez
Keyword(s):  
Harmonic Oscillator ◽  
Time Dependent ◽  
One Dimensional ◽  
Dynamical Invariants ◽  
The One

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.

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

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