scholarly journals Fisher Information of Free-Electron Landau States

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 268
Author(s):  
Takuya Yamano
Keyword(s):  
Quantum Number ◽  
Fisher Information ◽  
Free Electron ◽  
Principal Quantum Number ◽  
Energy Levels ◽  
Laguerre Polynomials ◽  
Polar Coordinates ◽  
Quantum Numbers ◽  
Lowest Landau Level ◽  
Cylindrical Polar Coordinates

An electron in a constant magnetic field has energy levels, known as the Landau levels. One can obtain the corresponding radial wavefunction of free-electron Landau states in cylindrical polar coordinates. However, this system has not been explored so far in terms of an information-theoretical viewpoint. Here, we focus on Fisher information associated with these Landau states specified by the two quantum numbers. Fisher information provides a useful measure of the electronic structure in quantum systems, such as hydrogen-like atoms and under some potentials. By numerically evaluating the generalized Laguerre polynomials in the radial densities, we report that Fisher information increases linearly with the principal quantum number that specifies energy levels, but decreases monotonically with the azimuthal quantum number m. We also present relative Fisher information of the Landau states against the reference density with m=0, which is proportional to the principal quantum number. We compare it with the case when the lowest Landau level state is set as the reference.

Vacuum polarization in muonic and exotic atoms: the Lamb shift at medium Z and high n

2007 ◽  
Vol 85 (5) ◽  
pp. 551-561 ◽  
Author(s):  
E Yu. Korzinin ◽  
V G Ivanov ◽  
S G Karshenboim
Keyword(s):  
Quantum Number ◽  
Lamb Shift ◽  
Vacuum Polarization ◽  
Principal Quantum Number ◽  
Energy Levels ◽  
Exotic Atoms ◽  
Atomic States

We present new results on various asymptotics for the Uehling contribution to the energy levels in atomic states in hydrogen-like atoms that have a principal quantum number n with a high value. The results may be applied to conventional atoms (with an orbiting electron) as well as to muonic, pionic, antiprotonic, and other exotic atoms.PACS Nos.: 36.10.Gv, 31.30.Jv

The Δk = ±2 "forbidden band" and inversion–rotation energy levels of ammonia

1984 ◽  
Vol 62 (12) ◽  
pp. 1775-1791 ◽  
Author(s):  
Š. Urban ◽  
Romola D'cunha ◽  
K. Narahari Rao ◽  
D. Papoušek
Keyword(s):  
Quantum Number ◽  
State Energy ◽  
Energy Levels ◽  
Data Set ◽  
Quantum Numbers ◽  
Forbidden Transitions ◽  
Fourier Transform Spectra ◽  
The Fourier Transform ◽  
And Inversion ◽  
Rotation Energy

An entire band of perturbation allowed transitions to the ν4 state of 14NH3 has been assigned in the Fourier transform spectra recorded using a 192-m path length. More than 900 of the forbidden transitions provide necessary information on the spacing between the energy levels with different quantum numbers k, inaccessible from allowed transitions. These data were combined with all other relevant data (MW, submillimetrewave, FIR, IR) published in the literature to derive precise values of inversion–rotation energy levels. This extensive data set completely describes the ground state energy levels of 14NH3 up to quantum number J = 16 for all possible values of the quantum number k.

scholarly journals Direct Interband Light Absorption in Conical Quantum Dot

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
D. B. Hayrapetyan ◽  
A. V. Chalyan ◽  
E. M. Kazaryan ◽  
H. A. Sarkisyan
Keyword(s):  
Quantum Dot ◽  
Quantum Number ◽  
Light Absorption ◽  
Principal Quantum Number ◽  
Particle Energy ◽  
Geometrical Parameters ◽  
Quantum Numbers ◽  
Direct Light ◽  
Interband Light Absorption ◽  
Analytical Expressions

In the framework of the adiabatic approximation, the energy states of electron as well as the direct light absorption are investigated in conical quantum dot. Analytical expressions for particle energy spectrum are obtained. The dependence of the absorption edge on geometrical parameters of conical quantum dot is obtained. Selection rules are revealed for transitions between levels with different quantum numbers. In particular, it is shown that for the radial quantum number transitions are allowed between the levels with the same quantum numbers, and any transitions between different levels are allowed for the principal quantum number.

On the Possible Electronic Structure of Atoms in a Space of Higher Dimension

2018 ◽  
pp. 222-238
Keyword(s):  
Schrödinger Equation ◽  
Quantum Number ◽  
Schrodinger Equation ◽  
Dimensional Space ◽  
Principal Quantum Number ◽  
Higher Dimension ◽  
Quantum Numbers ◽  
Spin Quantum ◽  
Spin Quantum Number ◽  
Quantum Cells

It is shown that the stationary Schrödinger equation describing the distribution of electrons in the vicinity of the atomic nucleus has a solution, in principle, for any dimensionality of the space around the nucleus. As an example, a solution of the Schrödinger equation in a five - dimensional space is obtained. It is shown that the solution of the Schrödinger equation in p - dimensional space has p quantum numbers: the principal quantum number, the orbital quantum number and p - 2 magnetic quantum numbers. Taking into account the spin quantum number, the total number of quantum numbers in p - dimensional space is p + 1. This leads to the possibility of increasing the number of quantum cells of orbitals and, consequently, to the possibility of increasing the valence of the elements.

scholarly journals Landau-level quantization of the yellow excitons in cuprous oxide

2018 ◽  
Vol 60 (8) ◽  
pp. 1585
Author(s):  
J. Heckotter ◽  
J. Thewes ◽  
D. Frohlich ◽  
M. Abmann ◽  
M. Bayer
Keyword(s):  
Magnetic Field ◽  
Quantum Number ◽  
Landau Level ◽  
Cuprous Oxide ◽  
Principal Quantum Number ◽  
Bohr Radius ◽  
Quantum Numbers ◽  
Magnetic Length ◽  
Level Quantization ◽  
The Magnetic Field

AbstractLately, the yellow series of P -excitons in cuprous oxide could be resolved up to the principal quantum number n = 25. Adding a magnetic field, leads to additional confinement normal to the field. Thereby, the transition associated with the exciton n is transformed into the transition between the electron and hole Landau levels with quantum number n , once the associated magnetic length becomes smaller than the related exciton Bohr radius. The magnetic field of this transition scales roughly as n ^–3. As a consequence of the extended exciton series, we are able to observe Landau level transitions with unprecedented high quantum numbers of more than 75.

Precise energies of highly excited hydrogen and deuterium

2002 ◽  
Vol 80 (11) ◽  
pp. 1373-1382 ◽  
Author(s):  
S Kotochigova ◽  
P J Mohr ◽  
B N Taylor
Keyword(s):  
Least Squares ◽  
Quantum Number ◽  
Excited States ◽  
Quantum Electrodynamics ◽  
Theoretical Calculations ◽  
Principal Quantum Number ◽  
Energy Levels ◽  
Fundamental Constants ◽  
Relativistic Corrections ◽  
Least Squares Adjustment

The energy levels of hydrogen and deuterium atoms are calculated to provide frequencies for transitions between highly excited states with principal quantum number n up to 200. All known quantum electrodynamics and relativistic corrections have been included in the calculation. In some cases, contributions originally calculated for a few states have been extrapolated to highly excited states. The fundamental constants necessary for the calculation are taken from the 1998 CODATA least-squares adjustment. Evaluated uncertainties take into account uncertainties in the theoretical calculations, uncertainties in the fundamental constants, and covariances between the various contributions and input parameters. PACS Nos.: 31.15Pf, 31.30Jv, 32.10Hq

Bound States in Three Dimensions: The Atom

2021 ◽  
pp. 82-92
Author(s):  
Duncan G. Steel
Keyword(s):  
Angular Momentum ◽  
Quantum Number ◽  
Bound States ◽  
Laguerre Polynomials ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Spherical Coordinates ◽  
Quantum Numbers ◽  
Energy Plus ◽  
Charged Nucleus

In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

scholarly journals Semiclassical calculations of the quadruple excited four-electron systems

2007 ◽  
pp. 33-44
Author(s):  
N. Simonovic ◽  
M. Predojevic ◽  
V. Pankovic ◽  
P. Grujic
Keyword(s):  
Energy Levels ◽  
Point Of View ◽  
Interstellar Space ◽  
Vibrational Modes ◽  
Electron Systems ◽  
Atomic Systems ◽  
Excited Atoms ◽  
Relative Contribution ◽  
Quantum Numbers ◽  
Semiclassical States

Highly excited atoms acquire very large dimensions and can be present only in a very rarified gas medium, such as the interstellar space. Multiply excited beryllium-like systems, when excited to large principal quantum numbers, have a radius of r ? 10 ?. We examine the semiclassical spectrum of quadruple highly excited four-electron atomic systems for the plane model of equivalent electrons. The energy of the system consists of rotational and vibrational modes within the almost circular orbit approximation, as used in a previous calculation for the triply excited three-electron systems. Here we present numerical results for the beryllium atom. The lifetimes of the semiclassical states are estimated via the corresponding Lyapunov exponents. The vibrational modes relative contribution to the energy levels rises with the degree of the Coulombic excitation. The relevance of the results is discussed both from the observational and heuristic point of view.

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