arXiv subjects
Publications and source records attributed to S. M. Blinder.
Simplified Hartree-Fock computations are carried out on the atoms He through Ne, using orthonormalized basis functions for the 1s, 2s and 2p orbitals dependent on three parameters. Using Mathematica with the new Apple M1 chip, computations require about 0.005 seconds of CPU time. Approximate energies within 1% of the best H-F values are thereby obtained, with an order of magnitude less computational effort.
The eigenvalues of a pure quartic oscillator are computed, applying a canonical operator formulation, generalized from the harmonic oscillator. Solving a 10x10 secular equation produces eigenvalues in agreement, to at least 4 significant figures, with accurate computations given in the literature.
This year marks the 100th anniversary of Einstein's General Theory of Relativity (1915-2015). The first nontrivial solution of the Einstein field equations was derived by Karl Schwarzschild in 1916. This Note will focus mainly on the Schwarzschild solution and the remarkable developments which it inspired, the most dramatic being the prediction of black holes. Later extensions of Schwarzschild's spacetime structure has led to even wilder conjectures, such as white holes and passages to other universes. Penrose diagrams are introduced as compact representations of extended spacetime structures. Stephen Hawking's derivations of quantum effects in black holes might provide clues to an eventual "Theory of Everything" encompassing both general relativity and quantum mechanics.
In the first part of this paper, we review the assumption of the L-S coupling scheme, with which we derive the electronic states arising from a given atomic configuration. Then, with the aid of the spectral data of Group 15 elements, it becomes clear that the assumption of the L-S coupling scheme is no longer valid as we go farther and farther down the Periodic Table. In the second part, we introduce the j-j coupling scheme, which is seldom covered in standard inorganic chemistry texts, and contrast the assumptions of the two schemes. Next, we use two worked examples to demonstrate the derivation of electronic states with the j-j coupling scheme. Finally, the correlation between the states derived by L-S and j-j schemes is pictorially shown. It is believed a student, by also studying j-j coupling schemes (by no means a difficult task) along with the L-S scheme, will gain a better understanding of the concept of atomic electronic states.
The atomic orbitals, hybridization and chemical bonding of the most common hydrides of boron, carbon, nitrogen and oxygen are described. This can be very instructive for beginning students in chemistry and chemical physics.
Hylleraas in 1929 carried out a variational computation on the Schrodinger equation for the helium atom which gave, for the first time, a ground-state energy in essential agreement with experimental results. Coolidge and James in 1933, likewise did the first accurate computation for the hydrogen molecule. These are considered epoch-making contributions in the development of ab initio quantum chemistry, since they provided definitive evidence for the validity of the multiple-particle Schrodinger equation for atoms and molecules. Before then, exact solutions had been obtained only for one-electron hydrogenlike atoms. The helium and hydrogen work was done long before the advent of electronic computers and required many months of drudgery, using hand-cranked calculating machines. Nowadays, students of chemistry and physics can carry out these same computations in a matter of hours, or even minutes, using relatively straightforward Mathematica routines. Moreover, the results can be easily improved far beyond the capabilities of the original workers.
An electromagnetic analog of the Kerr-Newman solution in general relativity is derived, based on Minkowski's formulation for electromagnetic fields in moving media. The equivalent system is a distribution of charges and currents largely localized within a spinning disk of radius a. This occurs in a rotating medium of inhomogeneous index of refraction and "frame dragging" at exactly half the angular velocity of the electrical charge.
It is shown how point charges and point dipoles with finite self-energies can be accomodated into classical electrodynamics. The key idea is the introduction of constitutive relations for the electromagnetic vacuum, which actually mirrors the physical reality of vacuum polarization. Our results reduce to conventional electrodynamics for scales large compared to the classical electron radius $r_0\approx 2.8\times10^{-13}$ cm. A classical simulation for a structureless electron is proposed, with the appropriate values of mass, spin and magnetic moment.
The region very close to an electron ($r << r_0 = e^2/mc^2 \approx 2.8\times 10^{-13}$ cm) is, according to quantum electrodynamics, a seething maelstrom of virtual electron-positron pairs flashing in and out of existence. To take account of this well-established physical reality, a phenomenological representation for vacuum polarization is introduced into the framework of classical electrodynamics. Such a model enables a consistent picture of classical point charges with finite electromagnetic self-energy. It is further conjectured that the reaction of a point charge to its own electromagnetic field is tantamount to interaction with its vacuum polarization charge or "aura". This leads to a modification of the Lorentz-Dirac equation for the force on an accelerating electron, a new differential-difference equation which avoids the pathologies of preacceleration and runaway solutions.
A model is proposed for the classical electron as a point charge with finite electromagnetic self-energy. Modifications of the Reissner-Nordstrøm (spin 0) and Kerr-Newman (spin 1/2) solutions of the Einstein-Maxwell equations are derived. It is conjectured that spacetime curvature very close to the point charge deforms the electric and magnetic fields such as to reduce the self-energy to a finite value by means of Hawking polarization of the vacuum, much like that around a black hole.