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.

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.

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.

The exact evolution of the time-dependent driven damped harmonic oscillator with time-dependent mass and frequency

2002 ◽  
Vol 80 (12) ◽  
pp. 1559-1569 ◽  
Author(s):  
M Liang ◽  
B Yuan ◽  
K Zhong
Keyword(s):  
Wave Function ◽  
Harmonic Oscillator ◽  
Classical Trajectory ◽  
Wave Functions ◽  
Time Dependent ◽  
Quantization Scheme ◽  
Dynamical Phase ◽  
Damped Harmonic Oscillator ◽  
Total Phase ◽  
Dependent Mass

Under a new quantization scheme, the exact wave functions of the time-dependent driven damped harmonic oscillator with time-dependent mass and frequency are obtained. The wave functions are shape-unchanging wave packet with the center moving along the classical trajectory. The total phase of the wave function is explicitly expressed as the sum of the dynamical phase and the geometrical phase. PACS Nos.: 03.65-w, 05.40-a

QUANTUM AND THERMAL STATE FOR EXPONENTIALLY DAMPED HARMONIC OSCILLATOR WITH AND WITHOUT INVERSE QUADRATIC POTENTIAL

2002 ◽  
Vol 16 (09) ◽  
pp. 1341-1351 ◽  
Author(s):  
J. R. CHOI
Keyword(s):  
Harmonic Oscillator ◽  
Density Operator ◽  
Thermal State ◽  
Mechanical Energy ◽  
Invariant Operator ◽  
Diagonal Element ◽  
Quadratic Potential ◽  
Damped Harmonic Oscillator ◽  
Energy Expectation ◽  
Quantum Mechanical Energy

By taking advantage of dynamical invariant operator, we derived Schrödinger solution for exponentially damped harmonic oscillator with and without inverse quadratic potential. We investigated quantum mechanical energy expectation value, uncertainty relation, partition function and density operator of the system. The various expectation values in thermal state are calculated using the diagonal element of density operator.

C. The Continuous Spectrum of Pulsating Variable Stars

1967 ◽  
Vol 28 ◽  
pp. 177-206
Author(s):  
J. B. Oke ◽  
C. A. Whitney
Keyword(s):  
Continuous Spectrum ◽  
Variable Stars ◽  
Velocity Fields ◽  
The Other ◽  
Line Spectrum ◽  
Direct Information ◽  
Thermodynamic Structure ◽  
Diagnostic Interpretation ◽  
Pulsating Variable Stars ◽  
The Continuum

Pecker:The topic to be considered today is the continuous spectrum of certain stars, whose variability we attribute to a pulsation of some part of their structure. Obviously, this continuous spectrum provides a test of the pulsation theory to the extent that the continuum is completely and accurately observed and that we can analyse it to infer the structure of the star producing it. The continuum is one of the two possible spectral observations; the other is the line spectrum. It is obvious that from studies of the continuum alone, we obtain no direct information on the velocity fields in the star. We obtain information only on the thermodynamic structure of the photospheric layers of these stars–the photospheric layers being defined as those from which the observed continuum directly arises. So the problems arising in a study of the continuum are of two general kinds: completeness of observation, and adequacy of diagnostic interpretation. I will make a few comments on these, then turn the meeting over to Oke and Whitney.

Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

2014 ◽  
Vol 4 (1) ◽  
pp. 404-426
Author(s):  
Vincze Gy. Szasz A.
Keyword(s):  
Harmonic Oscillator ◽  
Classical Mechanics ◽  
Universal Time ◽  
Variation Theory ◽  
Quantum Oscillator ◽  
Variation Principle ◽  
Poisson Brackets ◽  
Mathematical Meaning ◽  
Damped Harmonic Oscillator ◽  
Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

Quantum Theory

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Principle ◽  
Quantum Spin ◽  
Quantum States ◽  
Wave Functions ◽  
Symbolic Representations ◽  
Quantum Numbers ◽  
The Subject ◽  
Heisenberg's Uncertainty Principle

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.

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