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Applications of the Formalism-II
Published in Shabnam Siddiqui, Quantum Mechanics, 2018
Thus, L^2f=cfwhere c = the eigenvalue of the operator L^2, and f = s an eigenfunction. It is customary to write the eigenvalue of this operator as ℏ2l(l + 1), where l is a positive integer and is called the total angular momentum quantum number. It is also called the azimuthal quantum number. The word azimuth refers to the spherical angle. Thus, L^2f=ℏ2l(l+1)f
Basic Atomic and Nuclear Physics
Published in Douglas S. McGregor, J. Kenneth Shultis, Radiation Detection, 2020
Douglas S. McGregor, J. Kenneth Shultis
Equation (3.68) in θ has normalizable solutions only if the separation constant has the form β = ℓ(ℓ + 1) where ℓ is a positive integer or zero, and is called the angular momentum quantum number. Moreover, the azimuthal quantum number m must be restricted to 2ℓ + 1 integer values, namely, m = 0, ± 1, ± 2, …, ± ℓ. The corresponding solutions of Eq. (3.68) are denoted by Θ ℓm(θ) and are known to mathematicians as the associate Legendre functions of the first kind; however, these details are not of concern here.
The Orbital Exponent Efficacy to Study the Periodic Parameters
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Reed (1999) modified Slater’s method (1930) by simply refining the Slater’s (1930) grouping of the orbital. He (1999) also proposed two sets of rules for the computation of the screening constants. Reed (1999) also considered the pairing energy for p and d orbitals. Now relying upon the periodic law and following Reed, we have evaluated the effective nuclear charge and the orbital exponents for the atoms of the 118 elements of the Periodic Table with some modifications. We propose a simple approach for the calculation of the ionization potential, electronegativity, atomic radius of atoms in terms of their orbital exponents. The prescription is simple and utilizes only the simple Bohr equation with some modifications. We have pointed out that the atomic first ionization energy does not only depend on the principal quantum number (n) but also on the azimuthal quantum number (l) of the orbital (n, l) on which the electron of interest is present. The formalism is tested through the calculation of the atomic ionization potentials of 118 elements of the Periodic Table. The orbital exponent values for 118 elements of the Periodic Table are computed following the suggestions of Reed. The calculated numerical results for a number of atoms are shown to agree quite well with their experimental counterparts. To perform the validity tests of the present scale of ionization potential, various physicochemical properties of the atoms are also correlated on the basis of the computed ionization potential data. It is found that the stability of the half-filled configuration depends on the orbital (n, l) on which the electron is present. The express periodic behavior and correlation of the most important physicochemical properties of elements suggest that present method of evaluation of the periodic parameters in terms of their orbital exponents’ ionization potential of the atoms is a quite a successful venture.
Nonlinear optical properties of coupled quantum dots in peanut configuration
Published in Philosophical Magazine, 2023
E. S. Hakobyan, D. A. Baghdasaryan, E. M. Kazaryan, P. A. Mantashyan, D. B. Hayrapetyan
For the fixed value z of the slow subsystem the motion of fast subsystem will be in the disk with effective radius . The radial Schrodinger equation for the fast subsystem will be: where is azimuthal quantum number. The general solution is: where , and are Bessel functions of the first and second kinds, respectively. Second solution do not satisfy as it diverges, thus for the radial wave function of the fast subsystem is expressed as: where is the radial quantum number. The effective energy for the slow subsystem will have the following form: where is -the zero of the Bessel function .
Effect of Sr-doping on electronic and thermal properties of Pr2-xSrxFeCrO6 (0≤x≤1) oxide materials synthesized by using sol-gel technique
Published in Journal of Asian Ceramic Societies, 2023
Lav Kush, Sanjay Srivastava, Sanjay Kumar Vajpai, Serguei V. Savilov
The aforementioned XPS discovery also showed that the propensity of spin-orbit splitting in the 2p edges of Cr and Fe may be caused by either the final 2p5 core hole state or by their various oxidation states. While Fe and Cr share a similar valence of + 3 in BO6 and B’O6 octahedra, respectively, these elements can alter their oxidation states by a super interchange of valence between the two octahedra. Because of this, Fe shows up as 2+ and Cr as 4+, and these two can begin their spin-orbit coupling as a result of unpaired cations with various oxidation states. Moreover, the quantity of unpaired electrons in each orbital can affect the values of the L-S coupling as well as the l and s quantum {∑l (l: azimuthal quantum number) and ∑s (s: spin quantum number)} [16]. Such an exciting finding could help to explain the Cr and Fe disorder in the Pr2-xSrxFeCrO6 lattice. Given that Pr and Sr, unlike Cr and Fe, have stable
Limitations in the 2D description of the electromagnetic waves propagation in thin dielectric and magnetic layers
Published in Journal of Modern Optics, 2018
Tomasz Radożycki, Piotr Bargieła
The condition for a large cylinder () leads to the curves shown in Figure 6. They are drawn for a dielectric layer with and and for two values of azimuthal quantum number: and . They turn out, however, to be insensitive to the value of m. The propagation with just corresponds to the propagation in the slab in a direction other than x and there is no qualitative difference between the waves of various m. The situation will change, when R is not large as compared to d, and the cylindrical layer due to its large curvature may not be treated locally as a plane.