ON THE BEHAVIOR AT INFINITY OF THE FUNDAMENTAL SOLUTION OF TIME DEPENDENT SCHRÖDINGER EQUATION

2001 ◽  
Vol 13 (07) ◽  
pp. 891-920 ◽  
Author(s):  
KENJI YAJIMA
Keyword(s):  
Asymptotic Behavior ◽  
Integral Operator ◽  
Fundamental Solution ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Fourier Integral Operator ◽  
Fourier Integral ◽  
Time Dependent ◽  
Initial Value

We show that the asymptotic behavior at infinity of the fundamental solution of the initial value problem for the free Schrödinger equation or of the harmonic oscillator at non-resonant time is stable under subquadratic perturbations. We also show that the same is true for the phase and the amplitude of the Fourier integral operator representing the propagator.

scholarly journals Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

2018 ◽  
Vol 4 (1) ◽  
pp. 47-55
Author(s):  
Timothy Brian Huber
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Mechanical System ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

scholarly journals QUANTUM STATES OF A GENERALIZED TIME-DEPENDENT INVERTED HARMONIC OSCILLATOR

2004 ◽  
Vol 18 (09) ◽  
pp. 1379-1385 ◽  
Author(s):  
I. A. PEDROSA ◽  
I. GUEDES
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Quantum States ◽  
Time Dependent ◽  
Constant Frequency ◽  
Special Case ◽  
Generalized Time

We discuss the extension of the Lewis and Riesenfeld method of solving the time-dependent Schrödinger equation to cases where the invariant has continuous eigenvalues and apply it to the case of a generalized time-dependent inverted harmonic oscillator. As a special case, we consider a generalized inverted oscillator with constant frequency and exponentially increasing mass.

The symmetry group of the quantum harmonic oscillator in an electric field

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Juan Lejarreta ◽  
Jose Cerveró
Keyword(s):  
Schrödinger Equation ◽  
Electric Field ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Group Structure ◽  
Time Dependent ◽  
Frequency Function ◽  
General Group ◽  
Dependent Potential ◽  
The Relationship

AbstractIn this paper we present two results. First, we derive the most general group of infinitesimal transformations for the Schrödinger Equation of the general time-dependent Harmonic Oscillator in an electric field. The infinitesimal generators and the commutation rules of this group are presented and the group structure is identified. From here it is easy to construct a set of unitary operators that transform the general Hamiltonian to a much simpler form. The relationship between squeezing and dynamical symmetries is also stressed. The second result concerns the application of these group transformations to obtain solutions of the Schrödinger equation in a time-dependent potential. These solutions are believed to be useful for describing particles confined in boxes with moving boundaries. The motion of the walls is indeed governed by the time-dependent frequency function. The applications of these results to non-rigid quantum dots and tunnelling through fluctuating barriers is also discussed, both in the presence and in the absence of a time-dependent electric field. The differences and similarities between both cases are pointed out.

scholarly journals Quantum harmonic oscillator with time-dependent mass

2018 ◽  
Vol 32 (20) ◽  
pp. 1850235 ◽  
Author(s):  
I. Ramos-Prieto ◽  
A. Espinosa-Zuñiga ◽  
M. Fernández-Guasti ◽  
H. M. Moya-Cessa
Keyword(s):  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Quantum Harmonic Oscillator ◽  
Frequency Dependent ◽  
Fourier Operator ◽  
Time Dependencies ◽  
Dependent Mass

We use the Fourier operator to transform a time-dependent mass quantum harmonic oscillator into a frequency-dependent one. Then we use Lewis–Ermakov invariants to solve the Schrödinger equation by using squeeze operators. Finally, we give two examples of time dependencies: quadratically and hyperbolically growing masses.

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