scholarly journals The interaction of atoms and molecules with solid surfaces II—The evaporation of adsorbed atoms

1935 ◽  
Vol 150 (870) ◽  
pp. 456-464 ◽  
Keyword(s):  
Transition Probabilities ◽  
Average Length ◽  
Preceding Paper ◽  
Wave Functions ◽  
Solid Surfaces ◽  
Positive Energy ◽  
Matrix Elements ◽  
Discrete Energy ◽  
The Matrix ◽  
Adsorbed Atoms

This paper is an extension of the preceding one to the case of transitions from a state of discrete energy to one in the continuous region. By summing the transition probabilities from the ground state to the states in the continuous region an expression is obtained for the probability of evaporation from a solid surface. We are thus able to evaluate the average length of time spent by an adsorbed atom on a surface. 2- Transition Probabilities In order to calculate the matrix elements corresponding to transitions to states of positive energy, we must first consider the wave functions of the continuous spectrum of the equation (3a) of the preceding paper.

scholarly journals Hyperfine structure of muonic mesomolecules tdµ, tpµ, dpµ in variational method

2019 ◽  
Vol 222 ◽  
pp. 03011
Author(s):  
A.V. Eskin ◽  
V.I. Korobov ◽  
A.P. Martynenko ◽  
V.V. Sorokin
Keyword(s):  
Variational Method ◽  
Hyperfine Structure ◽  
Vacuum Polarization ◽  
Energy Levels ◽  
Computer Code ◽  
Wave Functions ◽  
Matrix Elements ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The hyperfine structure of energy levels of muonic molecules tdµ, tpµ and dpµ is calculated on the basis of stochastic variational method. The basis wave functions are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. Vacuum polarization, relativistic and nuclear structure corrections are taken into account to increase the accuracy. For numerical calculation, a computer code is written in the MATLAB system. Numerical values of energy levels of hyperfine structure in muonic molecules tdµ, tpµ and dpµ are obtained.

scholarly journals Some Generic Properties of Level Spacing Distributions of 2D Real Random Matrices

2007 ◽  
Vol 62 (9) ◽  
pp. 471-482 ◽  
Author(s):  
Siegfried Grossmann ◽  
Marko Robnik
Keyword(s):  
Distribution Function ◽  
Random Matrices ◽  
Power Law ◽  
Preceding Paper ◽  
Matrix Elements ◽  
Level Spacing ◽  
Generic Properties ◽  
Spacing Distribution ◽  
The Matrix ◽  
Level Spacing Distribution

We study the level spacing distribution P(S) of 2D real random matrices both symmetric as well as general, non-symmetric. In the general case we restrict ourselves to Gaussian distributed matrix elements, but different widths of the various matrix elements are admitted. The following results are obtained: An explicit exact formula for P(S) is derived and its behaviour close to S = 0 is studied analytically, showing that there is linear level repulsion, unless there are additional constraints for the probability distribution of the matrix elements. The constraint of having only positive or only negative but otherwise arbitrary non-diagonal elements leads to quadratic level repulsion with logarithmic corrections. These findings detail and extend our previous results already published in a preceding paper. For symmetric real 2D matrices also other, non-Gaussian statistical distributions are considered. In this case we show for arbitrary statistical distribution of the diagonal and non-diagonal elements that the level repulsion exponent ρ is always ρ = 1, provided the distribution function of the matrix elements is regular at zero value. If the distribution function of the matrix elements is a singular (but still integrable) power law near zero value of S, the level spacing distribution P(S) is a fractional exponent power law at small S. The tail of P(S) depends on further details of the matrix element statistics. We explicitly work out four cases: the uniform (box) distribution, the Cauchy-Lorentz distribution, the exponential distribution and, as an example for a singular distribution, the power law distribution for P(S) near zero value times an exponential tail.

A REALIZATION OF DYNAMIC GROUP FOR AN ELECTRON IN A UNIFORM MAGNETIC FIELD

2002 ◽  
Vol 11 (04) ◽  
pp. 265-271 ◽  
Author(s):  
SHISHAN DONG ◽  
SHI-HAI DONG
Keyword(s):  
Magnetic Field ◽  
Relativistic Electron ◽  
Uniform Magnetic Field ◽  
Wave Functions ◽  
Matrix Elements ◽  
Eigenvalues And Eigenfunctions ◽  
Radial Wave ◽  
Creation And Annihilation Operators ◽  
The Matrix ◽  
Analytical Expressions

The eigenvalues and eigenfunctions of the Schrödinger equation with a non-relativistic electron in a uniform magnetic field are presented. A realization of the creation and annihilation operators for the radial wave-functions is carried out. It is shown that these operators satisfy the commutation relations of an SU(1,1) group. Closed analytical expressions are evaluated for the matrix elements of different functions ρ2 and [Formula: see text].

On the Dynamical Spin Susceptibility of Paramagnetic La2CuO4

1990 ◽  
Vol 193 ◽  
Author(s):  
H. Winter ◽  
Z. Szotek ◽  
W. M. Temmerman
Keyword(s):  
Local Density ◽  
Spin Susceptibility ◽  
The Body ◽  
Wave Functions ◽  
Matrix Elements ◽  
Electron Wave ◽  
Energy Bands ◽  
Fermi Surface Nesting ◽  
The Matrix ◽  
Dynamical Spin

ABSTRACTThe self-consistent one-electron wave functions and energy bands obtained by the LMTO-ASA method within the local density approximation (LDA) are used to calculate the wave vector and frequency dependent non-interacting spin susceptibility of paramagnetic La2CuO4 in the body-centred tetragonal (bct) structure. We show that the tendency towards the antiferromagnetic instability is strongly dependent on the effects of the matrix elements which lead to a substantial depression of the susceptibility, especially near the X-point. The Fermi surface nesting properties, although important for the susceptibility, are by far not sufficient for the instability and the interband transitions turn out to be of great significance. Our results indicate that the susceptibility is at least 3 times too small to drive this system through a transition to the antiferromagnetic state, and we discuss possible reasons for this failure.

Mikroskopische Theorie der Kernreaktionen für einen Mehrteilchen-Aufbruch / Microscopic Theory of Nuclear Reactions for a Multi-particle-break-up

1974 ◽  
Vol 29 (6) ◽  
pp. 859-866 ◽  
Author(s):  
A. Grauel
Keyword(s):  
Nuclear Reactions ◽  
Microscopic Theory ◽  
Bound State ◽  
Wave Functions ◽  
Matrix Elements ◽  
S Matrix ◽  
The Matrix ◽  
Nucleon Emission ◽  
The Continuum ◽  
Continuum Wave

Introducing correlated continuum wave functions for the two- and re-particle-continuum a microscopic theory of nuclear reactions based on a method of Fano is developed. The S-matrix-elements are given by the matrix-elements between correlated continuum wave functions and bound state wave functions. The antisymmetrization of the continuum wave functions with more than one particle in the continuum is included. The theory can be straightforwardly applied on the n-nucleon-emission process following photo- and particle excitations.

THE HIDDEN SYMMETRY FOR A QUANTUM SYSTEM WITH A PÖSCHL–TELLER-LIKE POTENTIAL

2003 ◽  
Vol 12 (06) ◽  
pp. 809-815 ◽  
Author(s):  
SHI-HAI DONG ◽  
GUO-HUA SUN ◽  
YU TANG
Keyword(s):  
Quantum System ◽  
Wave Functions ◽  
Matrix Elements ◽  
Eigenvalues And Eigenfunctions ◽  
Creation And Annihilation Operators ◽  
Commutation Relations ◽  
Hidden Symmetry ◽  
The Matrix ◽  
The Creation ◽  
Analytical Expressions

The eigenvalues and eigenfunctions of the Schrödinger equation with a Pöschl–Teller (PT)-like potential are presented. A realization of the creation and annihilation operators for the wave functions is carried out. It is shown that these operators satisfy the commutation relations of an SU(1,1) group. Closed analytical expressions are evaluated for the matrix elements of different functions, sin (ρ) and [Formula: see text] with ρ=πx/L.

scholarly journals The interaction of atoms and molecules with solid surfaces VII—The diffraction of atoms by a surface

1937 ◽  
Vol 158 (894) ◽  
pp. 253-268 ◽  
Keyword(s):  
Lithium Fluoride ◽  
Preceding Paper ◽  
Solid Surfaces ◽  
Precise Knowledge ◽  
Yield Information ◽  
Adsorbed Atoms ◽  
Atomic Beams ◽  
Explicit Formulae ◽  
And Migration ◽  
Surface Fields

In the hands of Stern and his collaborators, an accurate technique has been evolved for the measurement of the intensities of reflected and diffracted beams of atoms from the surfaces of crystals, but surprisingly little use has been made of these results to deduce information of theoretical interest except to verify the validity of the de Broglie relation for atomic beams. In this paper we attempt a theory of the diffraction of atoms at surfaces with a view to finding explicit formulae for the intensities of diffracted beams in terms of the constants of the surface field in order that a comparison of the theory and experiment might yield information as to the magnitudes of these constants. A precise knowledge of surface fields would be of value in that it would make possible a calculation of other properties of atoms on surfaces, such as the adsorption and migration of adsorbed atoms, as has already been done in the preceding paper (Part VI) for helium on lithium fluoride.

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