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.

Quantum Radiation Field: Spontaneous Emission and Entangled Photons

2021 ◽  
pp. 247-273
Author(s):  
Duncan G. Steel
Keyword(s):  
Electromagnetic Field ◽  
Excited State ◽  
Radiation Field ◽  
Quantum Electrodynamics ◽  
Equations Of Motion ◽  
Single Mode ◽  
Vacuum Field ◽  
Level System ◽  
Quantum Radiation ◽  
System P

One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.

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.

ORIGIN OF HYSTERESIS AND PLATEAU-LIKE BEHAVIOR OF THE I-V CHARACTERISTICS OF RESONANT TUNNELING DIODES

2000 ◽  
Vol 14 (04) ◽  
pp. 411-426 ◽  
Author(s):  
PEIJI ZHAO ◽  
H. L. CUI ◽  
D. L. WOOLARD ◽  
K. L. JENSEN ◽  
F. A. BUOT
Keyword(s):  
Energy Level ◽  
Quantum Well ◽  
Resonant Tunneling ◽  
Electron Tunneling ◽  
Energy Levels ◽  
Three Dimensional ◽  
Physical Factors ◽  
Tunneling Diodes ◽  
Tunneling Devices ◽  
Resonant Tunneling Devices

In terms of numerical calculation of the coupled Wigner function–Poisson equations, the explanation to the origin of hysteresis and plateau-like behavior of the I-V characteristics of double barrier resonant tunneling devices is put forth. Several basic physical factors play key roles in the process of electron tunneling. Among these the most important factors are the interference of the injected and the reflected electron waves which leads to the formation of an emitter quantum well, the coupling between the energy level in the main quantum well and that in the emitter quantum well, and the coupling between the energy level in the main quantum well and the conduction b and edge or the three-dimensional states in the emitter. The interplay of these factors determines the form of the I-V curve of the resonant tunneling structure. The coupling between the energy levels in the emitter quantum-well and the main quantum-well leads to the plateau behavior of the I-V curves. The strength of the coupling determines the average slope of the plateau-like region in the I-V curve. The bias domain that the coupling exists determines the length of the plateau-like structure in the I-V curve. The domain can be controlled by adjusting the width of the barriers. The hysteresis is shown to be a manifestation of the above-mentioned energy level coupling, the coupling between the energy level in the main quantum well and the conduction b and edge or the three-dimensional states in the emitter, and the quantitative accumulation and distribution of electrons in the emitter region. This work provides new insight for underst and ing the nonlinear I-V behavior and establishes a foundation for the future analysis of bistability and oscillation behavior in resonant tunneling structures.

Control of Doping and Electronic Transport in Nanowires

2004 ◽  
Vol 820 ◽  
Author(s):  
Jianxin Zhong ◽  
G. Malcolm Stocks
Keyword(s):  
Energy Level ◽  
Fermi Energy ◽  
Disordered Systems ◽  
Carrier Mobility ◽  
Transition Energy ◽  
Energy Levels ◽  
Three Dimensional ◽  
Delta Doping ◽  
Surface Disorder ◽  
Novel Concept

AbstractWe propose a novel concept, namely, delta-doping of nanowires, to control the carrier mobility in nanowires. Different from the traditional doping, our approach features doping of a nanowire only on its surface. Our calculations based on Anderson models for nanowires with surface disorder showed remarkably different results from the traditional doping where impurities are distributed inside the nanowire. We found that there exist transition energy levels similar to the mobility edges in three-dimensional disordered systems. If the Fermi energy is below the transition energy level, the delta-doped nanowire is simply an insulator. But once the Fermi energy exceeds this energy level, the carrier mobility increases significantly. The transition levels are almost independent of the degree of disorder in the regime of strong disorder.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

Casimir forces and vacuum energy

2017 ◽  
Author(s):  
Serge Reynaud ◽  
Astrid Lambrecht
Keyword(s):  
Casimir Force ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Thermal Fluctuations ◽  
Model Systems ◽  
Vacuum Field ◽  
Force Measurements ◽  
Casimir Forces ◽  
Current State ◽  
Field Fluctuations

The Casimir force is an effect of quantum vacuum field fluctuations, with applications in many domains of physics. The ideal expression obtained by Casimir, valid for perfect plane mirrors at zero temperature, has to be modified to take into account the effects of the optical properties of mirrors, thermal fluctuations, and geometry. After a general introduction to the Casimir force and a description of the current state of the art for Casimir force measurements and their comparison with theory, this chapter presents pedagogical treatments of the main features of the theory of Casimir forces for one-dimensional model systems and for mirrors in three-dimensional space.

scholarly journals Simulation of three-dimensional quantum systems with projected entangled-pair states

2021 ◽  
Vol 103 (20) ◽  
Author(s):  
Patrick C. G. Vlaar ◽  
Philippe Corboz
Keyword(s):  
Three Dimensional ◽  
Quantum Systems

scholarly journals Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

2021 ◽  
Vol 9 (1) ◽  
pp. 76
Author(s):  
Duoc Nguyen ◽  
Niels Jacobsen ◽  
Dano Roelvink
Keyword(s):  
Surface Waves ◽  
Deep Water ◽  
Surf Zone ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Breaking Waves ◽  
Successful Implementation ◽  
Random Waves ◽  
Mean Motion ◽  
Generalized Lagrangian

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

scholarly journals SPECTRAL CORRELATIONS IN DISORDERED MESOSCOPIC METALS AND THEIR RELEVANCE FOR PERSISTENT CURRENTS

1996 ◽  
Vol 10 (28) ◽  
pp. 1397-1406 ◽  
Author(s):  
AXEL VÖLKER ◽  
PETER KOPIETZ
Keyword(s):  
Energy Levels ◽  
Three Dimensional ◽  
Average Density ◽  
Electron Interaction ◽  
Energy Window ◽  
Persistent Currents ◽  
Dependent Part ◽  
The Difference ◽  
Aharonov Bohm ◽  
Grand Canonical

We use the Lanczos method to calculate the variance σ2(E, ϕ) of the number of energy levels in an energy window of width E below the Fermi energy for noninteracting disordered electrons on a thin three-dimensional ring threaded by an Aharonov–Bohm flux ϕ. We confirm numerically that for small E the flux-dependent part of σ2(E, ϕ) is well described by the Altshuler–Shklovskii-diagram involving two Cooperons. However, in the absence of electron–electron interactions this result cannot be extrapolated to energies E where the energy-dependence of the average density of states becomes significant. We discuss consequences for persistent currents and argue that for the calculation of the difference between the canonical- and grand canonical current it is crucial to take the electron–electron interaction into account.

Radical induced intermolecular linkage and energy level modifications of a porphyrin monolayer

2017 ◽  
Vol 53 (6) ◽  
pp. 1104-1107 ◽  
Author(s):  
Abdolreza Jahanbekam ◽  
Colin Harthcock ◽  
David Y. Lee
Keyword(s):  
Energy Level ◽  
Scanning Tunneling Microscopy ◽  
Surface Structure ◽  
Energy Levels ◽  
New Method ◽  
Scanning Tunneling ◽  
Tunneling Microscopy ◽  
Molecular Scale

A new method to directly modify the surface structure and energy levels of a porphyrin monolayer was examined with molecular-scale resolution using scanning tunneling microscopy and spectroscopy (STM and STS) and presented in this communication.

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