QUANTUM STATES WITH CONTINUOUS SPECTRUM FOR THE TIME-DEPENDENT HARMONIC OSCILLATOR WITH A SINGULAR PERTURBATION

2007 ◽  
Vol 21 (10) ◽  
pp. 585-593 ◽  
Author(s):  
JEONG RYEOL CHOI ◽  
JUN-YOUNG OH
Keyword(s):  
Singular Perturbation ◽  
Harmonic Oscillator ◽  
Continuous Spectrum ◽  
Unitary Operator ◽  
Energy Eigenvalue ◽  
Invariant Operator ◽  
Quantum States ◽  
Wave Functions ◽  
Time Dependent ◽  
Discrete Energy

The quantum states with continuous spectrum for the time-dependent harmonic oscillator perturbed by a singularity are investigated. This system does not oscillate while the system that has discrete energy eigenvalue does. Exact wave functions satisfying the Schrödinger equation for the system are derived using invariant operator and unitary operator together.

WAVE FUNCTIONS WITH DISCRETE AND WITH CONTINUOUS SPECTRUM FOR QUANTUM DAMPED HARMONIC OSCILLATOR PERTURBED BY A SINGULARITY

2004 ◽  
Vol 18 (07) ◽  
pp. 1007-1020 ◽  
Author(s):  
JEONG-RYEOL CHOI
Keyword(s):  
Harmonic Oscillator ◽  
Continuous Spectrum ◽  
Unitary Operator ◽  
Invariant Operator ◽  
Confluent Hypergeometric Function ◽  
Quantum States ◽  
Wave Functions ◽  
The Other ◽  
Damped Harmonic Oscillator ◽  
Discrete And Continuous Spectrum

The quantum states with discrete and continuous spectrum for the damped harmonic oscillator perturbed by a singularity have been investigated using invariant operator and unitary operator together. The eigenvalue of the invariant operator for ω0≤β/2 is continuous while for ω0>β/2 is discrete. The wave functions for ω0=β/2 expressed in terms of the Bessel function and for ω0<β/2 in terms of the Kummer confluent hypergeometric function. The convergence of the probability density is more rapid for over-damped harmonic oscillator than that of the other two cases due to the large damping constant.

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.

OPERATOR METHOD FOR A NONCONSERVATIVE HARMONIC OSCILLATOR WITH AND WITHOUT SINGULAR PERTURBATION

2002 ◽  
Vol 16 (31) ◽  
pp. 4733-4742 ◽  
Author(s):  
JEONG RYEOL CHOI ◽  
BO HA KWEON
Keyword(s):  
Singular Perturbation ◽  
Harmonic Oscillator ◽  
Operator Method ◽  
Invariant Operator ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Expectation Values ◽  
Ladder Operators ◽  
Raising Operators ◽  
Quantum Mechanical Solution

We used dynamical invariant operator method to find the quantum mechanical solution of a harmonic plus inverse harmonic oscillator with time-dependent coefficients. The eigenvalue of invariant operator is obtained and is constant with time. We constructed lowering and raising operators from the invariant operator. The solution of Schrödinger equation is obtained using operator method. We have also used ladder operators to obtain various expectation values of the time-dependent system. The results in this manuscript are not only more general than the existing results in the literatures but also well match with others.

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

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.

scholarly journals Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Salim Medjber ◽  
Hacene Bekkar ◽  
Salah Menouar ◽  
Jeong Ryeol Choi
Keyword(s):  
Separation Of Variables ◽  
Type Equation ◽  
Invariant Operator ◽  
Wave Functions ◽  
Time Dependent ◽  
Harmonic Potential ◽  
Uncertainty Relations ◽  
Mathematical Methods ◽  
Potential System ◽  
Uvarov Method

The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

Exact wave functions and nonadiabatic Berry phases of a time-dependent harmonic oscillator

1995 ◽  
Vol 52 (4) ◽  
pp. 3352-3355 ◽  
Author(s):  
Jeong-Young Ji ◽  
Jae Kwan Kim ◽  
Sang Pyo Kim ◽  
Kwang-Sup Soh
Keyword(s):  
Harmonic Oscillator ◽  
Wave Functions ◽  
Time Dependent ◽  
Berry Phases

scholarly journals A CLASS OF QUANTUM STATES WITH CLASSICAL-LIKE EVOLUTION

1994 ◽  
Vol 08 (29) ◽  
pp. 1823-1831 ◽  
Author(s):  
SALVATORE DE MARTINO ◽  
SILVIO DE SIENA ◽  
FABRIZIO ILLUMINATI
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Quantum States ◽  
Time Dependent ◽  
Stationary States ◽  
Oscillator Potential ◽  
Classical Motion ◽  
Stochastic Formulation ◽  
New States

In the framework of the stochastic formulation of quantum mechanics we derive non-stationary states for a class of time-dependent potentials. The wave packets follow a classical motion with constant dispersion. The new states define a possible extension of the harmonic oscillator coherent states. As an explicit application, we study a sestic oscillator potential.

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