Positive Fractal Dimensions of Bound States Spectral Measures: Application to the Hydrogen Atom

2022 ◽  
Vol 52 (2) ◽  
Author(s):  
Moacir Aloisio ◽  
Silas L. Carvalho ◽  
César R. de Oliveira
Keyword(s):  
Hydrogen Atom ◽  
Bound States ◽  
Fractal Dimensions ◽  
Spectral Measures

scholarly journals Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1323 ◽  
Author(s):  
G. Jordan Maclay
Keyword(s):  
Hydrogen Atom ◽  
Quantum Electrodynamics ◽  
Particle Physics ◽  
Bound States ◽  
Integrated Treatment ◽  
Invariance Group ◽  
Elementary Particle Physics ◽  
Background Theory ◽  
Modern Physics ◽  
History Of

Understanding the hydrogen atom has been at the heart of modern physics. Exploring the symmetry of the most fundamental two body system has led to advances in atomic physics, quantum mechanics, quantum electrodynamics, and elementary particle physics. In this pedagogic review, we present an integrated treatment of the symmetries of the Schrodinger hydrogen atom, including the classical atom, the SO(4) degeneracy group, the non-invariance group or spectrum generating group SO(4,1), and the expanded group SO(4,2). After giving a brief history of these discoveries, most of which took place from 1935–1975, we focus on the physics of the hydrogen atom, providing a background discussion of the symmetries, providing explicit expressions for all of the manifestly Hermitian generators in terms of position and momenta operators in a Cartesian space, explaining the action of the generators on the basis states, and giving a unified treatment of the bound and continuum states in terms of eigenfunctions that have the same quantum numbers as the ordinary bound states. We present some new results from SO(4,2) group theory that are useful in a practical application, the computation of the first order Lamb shift in the hydrogen atom. By using SO(4,2) methods, we are able to obtain a generating function for the radiative shift for all levels. Students, non-experts, and the new generation of scientists may find the clearer, integrated presentation of the symmetries of the hydrogen atom helpful and illuminating. Experts will find new perspectives, even some surprises.

scholarly journals Relationship between the Behavior of Hydrogen and Hydrogen Bubble Nucleation in Vanadium

Materials ◽  
2020 ◽  
Vol 13 (2) ◽  
pp. 322
Author(s):  
Zhengxiong Su ◽  
Sheng Wang ◽  
Chenyang Lu ◽  
Qing Peng
Keyword(s):  
Hydrogen Atom ◽  
First Principles ◽  
Bound States ◽  
Hydrogen Molecule ◽  
Vacancy Cluster ◽  
Bubble Nucleation ◽  
Hydrogen Bubble ◽  
First Principles Calculations ◽  
Hydrogen Atoms ◽  
Macroscopic Deformation

Hydrogen plays a significant role in the microstructure evolution and macroscopic deformation of materials, causing swelling and surface blistering to reduce service life. In the present work, the atomistic mechanisms of hydrogen bubble nucleation in vanadium were studied by first-principles calculations. The interstitial hydrogen atoms cannot form significant bound states with other hydrogen atoms in bulk vanadium, which explains the absence of hydrogen self-clustering from the experiments. To find the possible origin of hydrogen bubble in vanadium, we explored the minimum sizes of a vacancy cluster in vanadium for the formation of hydrogen molecule. We show that a freestanding hydrogen molecule can form and remain relatively stable in the center of a 54-hydrogen atom saturated 27-vacancy cluster.

scholarly journals The de Sitter groupL 4,1 and the bound states of hydrogen atom

1966 ◽  
Vol 41 (2) ◽  
pp. 222-234 ◽  
Author(s):  
H. Bacry
Keyword(s):  
Hydrogen Atom ◽  
Bound States ◽  
De Sitter

scholarly journals REVIEW OF THE N-QUANTUM APPROACH TO BOUND STATES

2011 ◽  
Vol 26 (06) ◽  
pp. 935-945 ◽  
Author(s):  
O. W. GREENBERG
Keyword(s):  
Hydrogen Atom ◽  
Bound States ◽  
Bound State ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Asymptotic Field ◽  
Quantum Approach ◽  
Future Work ◽  
Asymptotic Fields

We describe a method of solving quantum field theories using operator techniques based on the expansion of interacting fields in terms of asymptotic fields. For bound states, we introduce an asymptotic field for each (stable) bound state. We choose the nonrelativistic hydrogen atom as an example to illustrate the method. Future work will apply this N-quantum approach to relativistic theories that include bound states in motion.

scholarly journals Nonrelativistic bound states at finite temperature: The hydrogen atom

2008 ◽  
Vol 78 (3) ◽  
Author(s):  
Miguel Ángel Escobedo ◽  
Joan Soto
Keyword(s):  
Hydrogen Atom ◽  
Finite Temperature ◽  
Bound States

scholarly journals Fractal dimensions of wave functions and local spectral measures on the Fibonacci chain

2016 ◽  
Vol 93 (20) ◽  
Author(s):  
Nicolas Macé ◽  
Anuradha Jagannathan ◽  
Frédéric Piéchon
Keyword(s):  
Fractal Dimensions ◽  
Wave Functions ◽  
Spectral Measures

Hydrogen-like Atoms

2020 ◽  
pp. 328-339
Author(s):  
Jochen Autschbach
Keyword(s):  
Organic Chemistry ◽  
Fixed Point ◽  
Angular Momentum ◽  
Hydrogen Atom ◽  
Total Energy ◽  
Bound States ◽  
Principal Quantum Number ◽  
Momentum Equation ◽  
Polar Coordinates ◽  
Real Linear

This chapter shows how the electronic Schrodinger equation (SE) is solved for a hydrogen-like atom, i.e. an electron moving in the field of a fixed point-like nucleus with charge number Z. The hydrogen atom corresponds to Z = 1. The potential in atomic units is –Z/r, with r being the distance of the electron from the nucleus. The SE is not separable in Cartesian coordinates, but in spherical polar coordinates it separates into a radial equation and an angular momentum equation. The bound states have a total energy of –Z2/(2n2), with n = nr + ℓ being the principal quantum number (q.n.), ℓ = 0,1,2,… the angular momentum q.n., and nr = 1,2,3,… being a radial q.n. Each state for a given ℓ is 2ℓ+1-fold degenerate, with the components labelled by the projection q.n. mℓ. The wavefunctions for mℓ ≠ 0 are complex, but real linear combinations can be formed. This gives the atomic orbitals known from general and organic chemistry. Different ways of visualizing the real wavefunctions are discussed, e.g. as iso-surfaces.

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