Approximate energy levels of central field bound quantum systems

1983 ◽  
Vol 51 (12) ◽  
pp. 1150-1151 ◽  
Author(s):  
Francisco M. Fernández ◽  
Eduardo A. Castro
Keyword(s):  
Energy Levels ◽  
Quantum Systems ◽  
Central Field

Approximate energy levels and sizes of bound quantum systems

1982 ◽  
Vol 50 (1) ◽  
pp. 53-59 ◽  
Author(s):  
H. A. Gersch ◽  
C. H. Braden
Keyword(s):  
Energy Levels ◽  
Quantum Systems

Finite element distributions in statistical theory of energy levels in quantum systems

1999 ◽  
Vol 125 (3-4) ◽  
pp. 260-274 ◽  
Author(s):  
Maciej M. Duras ◽  
Krzysztof Sokalski
Keyword(s):  
Finite Element ◽  
Statistical Theory ◽  
Energy Levels ◽  
Quantum Systems

Simplified P-representation operator correspondence applied to quantum systems with generalized Kerr nonlinearity

2019 ◽  
Vol 33 (28) ◽  
pp. 1950340 ◽  
Author(s):  
S. Chiangga ◽  
S. Pitakwongsaporn ◽  
Till D. Frank
Keyword(s):  
Energy Levels ◽  
Kerr Nonlinearity ◽  
Quantum Systems ◽  
Matter Wave ◽  
Quantum Oscillator ◽  
Text Representation ◽  
Nonlinear Quantum ◽  
Quantum Optical ◽  
Field Oscillation ◽  
Quadratic And Cubic Nonlinearities

A simplified operator correspondence scheme is derived to address nonlinear quantum systems within the framework of the [Formula: see text]-representation. The simplified method is applied to a general nonlinear quantum oscillator model that has been used in the literature to describe nonlinear quantum optical and matter wave systems. The [Formula: see text]-representation evolution equation for the model is derived for arbitrary nonlinearity exponents. It is shown that in the high temperature case, the [Formula: see text]-representation is sufficient to describe the model and its associated systems such that there is no need to use alternative, mathematically more involved representations. Systems with quadratic and cubic nonlinearities are considered in more detail. Distributions for the energy levels and photon and particle numbers are obtained within the framework of the [Formula: see text]-representation. Moreover, the electrical field oscillation frequency dependency is studied numerically when interpreting the model as model for quantum optical nonlinear oscillators.

scholarly journals ON THE STABILITY OF PERIODICALLY TIME-DEPENDENT QUANTUM SYSTEMS

2008 ◽  
Vol 20 (06) ◽  
pp. 725-764 ◽  
Author(s):  
P. DUCLOS ◽  
E. SOCCORSI ◽  
P. ŠŤOVÍČEK ◽  
M. VITTOT
Keyword(s):  
Transition Probabilities ◽  
Initial Conditions ◽  
Energy Levels ◽  
Sufficient Conditions ◽  
Energy Operator ◽  
Mean Value ◽  
Quantum Systems ◽  
Time Dependent ◽  
Dense Subspace ◽  
Periodically Driven

The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e. conditions under which it holds true sup t ∈ ℝ|〈ψt, H(t)ψt〉| < ∞ where ψt denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next, we show, under certain assumptions, that if the spectrum of the monodromy (Floquet) operator U(T, 0) is pure point then there exists a dense subspace of initial conditions for which the mean value of the energy is uniformly bounded in the course of time. Further, we show that if the propagator admits a differentiable Floquet decomposition then ‖H(t)ψt‖ is bounded in time for any initial condition ψ0, and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.

scholarly journals A New Kind of Shift Operators for Infinite Circular and Spherical Wells

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Guo-Hua Sun ◽  
K. D. Launey ◽  
T. Dytrych ◽  
Shi-Hai Dong ◽  
J. P. Draayer
Keyword(s):  
Energy Levels ◽  
Quantum Systems ◽  
Angular Momentum Operator ◽  
Spatial Variables ◽  
Shift Operators ◽  
The Third ◽  
Commuting Operators ◽  
Complete Set ◽  
Orbital Angular Momentum Operator

A new kind of shift operators for infinite circular and spherical wells is identified. These shift operators depend on all spatial variables of quantum systems and connect some eigenstates of confined systems of different radiiRsharing energy levels with a common eigenvalue. In circular well, the momentum operatorsP±=Px±iPyplay the role of shift operators. ThePxandPyoperators, the third projection of the orbital angular momentum operatorLz, and the HamiltonianHform a complete set of commuting operators with the SO(2) symmetry. In spherical well, the shift operators establish a novel relation betweenψlm(r)andψ(l±1)(m±1)(r).

An analysis of the states in the phase space: the energy levels of quantum systems

1996 ◽  
Vol 111 (2) ◽  
pp. 193-215 ◽  
Author(s):  
S. Tosto
Keyword(s):  
Phase Space ◽  
Energy Levels ◽  
Quantum Systems

scholarly journals Generalized Dirac oscillator with κ-Poincaré algebra

2020 ◽  
Vol 29 (05) ◽  
pp. 2050033
Author(s):  
Jing Wu ◽  
Chao-Yun Long ◽  
Zheng-Xue Wu ◽  
Zheng-Wen Long
Keyword(s):  
Deformation Parameter ◽  
Singular Potential ◽  
Dimensional Space ◽  
Energy Levels ◽  
Quantum Systems ◽  
Energy Spectra ◽  
Dirac Oscillator ◽  
Radial Equation ◽  
Poincaré Algebra ◽  
Type Potential

In this paper, the generalized Dirac oscillator with [Formula: see text]-Poincaré algebra is structured by replacing the momentum operator p with [Formula: see text] in [Formula: see text]-deformation Dirac equation. The deformed radial equation is derived for this model. Particularly, by solving the deformed radial equation, the wave functions and energy spectra which depend on deformation parameter [Formula: see text] have been obtained for these quantum systems with [Formula: see text] being a Yukawa-type potential, inverse-square-type singular potential and central fraction power singular potential in two-dimensional space, respectively. The results show that the deformation parameter [Formula: see text] can lead to decreasing of energy levels for the above quantum systems. At the same time, the degeneracy of energy spectra has been discussed and the corresponding conditions of degeneracy have been given for each case.

scholarly journals The effect of a nuclear spin on the optical spectra

1929 ◽  
Vol 124 (795) ◽  
pp. 568-591 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Hyperfine Structure ◽  
Nuclear Spin ◽  
Energy Levels ◽  
Principal Series ◽  
Wave Functions ◽  
Central Field ◽  
Similar Work ◽  
Electronic Spin ◽  
Working Out

Some recent work has been done by Back and Goudsmidt on the “hyperfine” structure of the optical spectrum of bismuth,* and more recently similar work has been carried out for cæsium by Jackson. In each of these investigations the line structure was examined closely with a view to revealing a still finer structure, and it was found in both cases that the lines attributed to electronic spin were themselves composed of several distinct lines. In fact, for cæsium, each of the fine (electron spin) lines of the principal series was found to split up into two ; for bismuth the hyperfine structure was more complicated. Back and Goudsmidt attributed the structure to a nuclear spin, and working out the consequences of this on the lines of the old quantum mechanics they found that a nuclear spin of 41/2 quanta is necessary to account for the facts ; a spin of a 1/2 quantum is similarly attributed by Jackson to the nucleus of cæsium. The hypothesis explains very satisfactorily in a qualitative way the results of observation. In the work described in the present paper the methods of the new quantum mechanics have been applied to the problem. More precisely, we consider the motion of a single electron in a Coulombian field due to a nucleus possessing a 1/2 quantum of spin. It will be seen that the results can easily be extended to the case of any central field, and the principle could also be extended to the case of an atom with a nuclear spin of 1/2( nh /2π), but the detailed working out would be very heavy for n > I (at any rate, using the methods explained in this paper), owing to the large number of wave functions which would be necessary to specify any state of the atom. It will be seen that the results we obtain are substantially the same as Jackson’s so far as the energy levels are concerned, but the calculated intensities are not consistent with the observed transitions, and we deduce a combination rule which is radically different from Jackson’s.

scholarly journals History and Some Aspects of the Lamb Shift

Physics ◽  
2020 ◽  
Vol 2 (2) ◽  
pp. 105-147 ◽  
Author(s):  
G. Jordan Maclay
Keyword(s):  
Energy Level ◽  
Quantum Electrodynamics ◽  
Lamb Shift ◽  
Energy State ◽  
Energy Levels ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Zero Point ◽  
Quantum Systems ◽  
Vacuum Field

Radiation is a process common to classical and quantum systems with very different effects in each regime. In a quantum system, the interaction of a bound electron with its own radiation field leads to complex shifts in the energy levels of the electron, with the real part of the shift corresponding to a shift in the energy level and the imaginary part to the width of the energy level. The most celebrated radiative shift is the Lamb shift between the 2 s 1 / 2 and the 2 p 1 / 2 levels of the hydrogen atom. The measurement of this shift in 1947 by Willis Lamb Jr. proved that the prediction by Dirac theory that the energy levels were degenerate was incorrect. Hans Bethe’s calculation of the shift showed how to deal with the divergences plaguing the existing theories and led to the understanding that interactions with the zero-point vacuum field, the lowest energy state of the quantized electromagnetic field, have measurable effects, not just resetting the zero of energy. This understanding led to the development of modern quantum electrodynamics (QED). This historical pedagogic paper explores the history of Bethe’s calculation and its significance. It explores radiative effects in classical and quantum systems from different perspectives, with the emphasis on understanding the fundamental physical phenomena. Illustrations are drawn from systems with central forces, the H atom, and the three-dimensional harmonic oscillator. A first-order QED calculation of the complex radiative shift for a spinless electron is explored using the equations of motion and the m a s s 2 operator, describing the fundamental phenomena involved, and relating the results to Feynman diagrams.

scholarly journals MODELING DISORDERED QUANTUM SYSTEMS WITH DYNAMICAL NETWORKS

1999 ◽  
Vol 10 (04) ◽  
pp. 577-606 ◽  
Author(s):  
ROCHUS KLESSE ◽  
MARCUS METZLER
Keyword(s):  
Energy Levels ◽  
Quantum Hall ◽  
Three Dimensional ◽  
Network Models ◽  
Quantum Systems ◽  
Electronic Systems ◽  
Dynamical Models ◽  
Static Properties ◽  
Three Dimensional Model ◽  
Metal Insulator

It is the purpose of the present article to show that so-called network models, originally designed to describe static properties of disordered electronic systems, can be easily generalized to quantum-dynamical models, which then allow for an investigation of dynamical and spectral aspects. This concept is exemplified by the Chalker–Coddington model for the quantum Hall effect and a three-dimensional generalization of it. We simulate phase coherent diffusion of wave packets and consider spatial and spectral correlations of network eigenstates as well as the distribution of (quasi-)energy levels. Apart from that, it is demonstrated how network models can be used to determine two-point conductances. Our numerical calculations for the three-dimensional model at the Metal-Insulator transition point delivers, among others, an anomalous diffusion exponent of η=3-D2=1.7±0.1. The methods presented here in detail have been used partially in earlier work.

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