Guessing Solutions to the H-atom Schrodinger Equation

In teaching elementary quantum chemistry, the concept of eigenfunctionality is explored using the H-atom's Hamiltonian and various guessed functions. This is done in Cartesian coordinates, in Spherical Polar coordinates, and in Confocal Elliptical coordinates.

Fate of accidental symmetries of the relativistic hydrogen atom in a spherical cavity

The non-relativistic hydrogen atom enjoys an accidental SO(4) symmetry, that enlarges the rotational SO(3) symmetry, by extending the angular momentum algebra with the Runge–Lenz vector. In the relativistic hydrogen atom the accidental symmetry is partially lifted. Due to the Johnson–Lippmann operator, which commutes with the Dirac Hamiltonian, some degeneracy remains. When the non-relativistic hydrogen atom is put in a spherical cavity of radius R with perfectly reflecting Robin boundary conditions, characterized by a self-adjoint extension parameter γ, in general the accidental SO(4) symmetry is lifted. However, for R=(l+1)(l+2)a (where a is the Bohr radius and l is the orbital angular momentum) some degeneracy remains when γ=∞ or γ = 2/R. In the relativistic case, we consider the most general spherically and parity invariant boundary condition, which is characterized by a self-adjoint extension parameter. In this case, the remnant accidental symmetry is always lifted in a finite volume. We also investigate the accidental symmetry in the context of the Pauli equation, which sheds light on the proper non-relativistic treatment including spin. In that case, again some degeneracy remains for specific values of R and γ.

Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone

In a recent study of the self-adjoint extensions of the Hamiltonian of a particle confined to a finite region of space, in which we generalized the Heisenberg uncertainty relation to a finite volume, we encountered bound states localized at the wall of the cavity. In this paper, we study this situation in detail both for a free particle and for a hydrogen atom centered in a spherical cavity. For appropriate values of the self-adjoint extension parameter, the bound states localized at the wall resonate with the standard hydrogen bound states. We also examine the accidental symmetry generated by the Runge–Lenz vector, which is explicitly broken in a spherical cavity with general Robin boundary conditions. However, for specific radii of the confining sphere, a remnant of the accidental symmetry persists. The same is true for an electron moving on the surface of a finite circular cone, bound to its tip by a 1/r1/r potential.

Nature of hydrogen atom trapped inside palladium lattice

Colloid of palladium nanoparticles has been prepared by the Solvated Metal Atom Dispersion (SMAD) method. Reaction of Pd(0) nanopowder obtained upon precipitation from the colloid, with ammonia borane (H3N center dot BH3, AB) in aqueous solutions at room temperature results in the generation of active hydrogen atoms. The active hydrogen atoms either combine with one another resulting in H-2 evolution or diffuse into the Pd lattice to afford PdHx. Diffusion of hydrogen atoms leads to an expansion of the Pd lattice. The diffused hydrogen atoms are distributed uniformly over the entire particle. These features were established using powder XRD and electron microscopy studies. The H-1 NMR spectral studies of PdHx before and after desorption of H-2 revealed that the hydrogen atoms trapped inside Pd lattice are hydridic in nature. Desorption of hydrogen from PdHx did not result in complete reversibility suggesting that some hydrogen atoms are strongly trapped inside the Pd lattice. (C) 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

Hydrogen-Atom Transfer Photochemistry of Tetrakis(µ-pyrophosphito)diplatinate(II)

In many senses, the hydrogen-atom transfer reactions observed with the triplet excited state of pyrophosphito-bridged platinum(II) dimers resemble the reactions of organic ketone nπ* states. The first two chapters describe our attempts to understand the reactivity differences between these two chromophores. Reactivity of the metal dimers is strongly regulated by the detailed nature of the ligands that ring the axial site, the hydrogen-abstraction center. A hydrogen-bonded network linking the ligands facilitates H-atom transfer quenching with alcohols through the formation of a hydrogen-bonded complex between the alcohol and a dimer. For substrates of equal C-H bond strength that lack a hydroxyl group (e.g., benzyl hydrocarbons), the quenching rate is several orders of magnitude slower.

The shape and size of the axial site, as determined by the ligands, also discriminate among quenchers by their steric characteristics. Very small quenchers quench slowly because of high entropies of activation, while very large ones have large enthalpic barriers. The two effects find a balance with quenchers of "just the right size."

The third chapter discusses the design of a mass spectrometer that uses positron annihilation to ionize neutral molecules. The mass spectrometer creates positron-molecule adducts whose annihilation produces fragmentation products that may yield information on the bonding of positrons in such complexes.

Energy-level shifts of a stationary hydrogen atom in a static external gravitational field with Schwarzschild geometry

he first order perturbations of the energy levels of a stationary hydrogen atom in a static external gravitational field, with Schwarzschild metric, are investigated. The energy shifts are calculated for the relativistic 1S, 2S, 2P, 3S, 3P, 3D, 4S, 4P, 4D, and 4F levels. The results show that the energy-level shifts of the states with total angular momentum quantum number 1/2 are all zero, and the ratio of absolute energy shifts with total angular momentum quantum number 5/2 is 145. This feature can be used to help us to distinguish the gravitational effect from other effects.

Gravitational corrections to energy-levels of a hydrogen atom

The first-order perturbations of the energy levels of a hydrogen atom in central internal gravitational field are investigated. The internal gravitational field is produced by the mass of the atomic nucleus. The energy shifts are calculated for the relativistic 1S, 2S, 2P, 3S, 3P, 3D, 4S, and 4P levels with Schwarzschild metric. The calculated results show that the gravitational corrections are sensitive to the total angular momentum quantum number.

Quantum manifestations of closed orbits in the photoexcitation scaled spectrum of the hydrogen atom in crossed fields

The scaled photoexcitation spectrum of the hydrogen atom in crossed electric and magnetic fields has been obtained by means of accurate quantum mechanical calculation using a new algorithm. Closed orbits in the corresponding classical system have also been obtained, using a new, efficient and practical searching procedure. Two new classes of closed orbit have been identified. Fourier transforming each photoexcitation quantum spectrum to yield a plot against scaled action has allowed direct comparison between peaks in such plots and the scaled action values of closed orbits, Excellent agreement has been found with all peaks assigned.

The closed orbits and the photo-excitation scaled spectrum of the hydrogen atom in crossed fields

In a recent Letter to the Editor (J Rao, D Delande and K T Taylor 2001 J. Phys. B: At. Mol. Opt. Phys. 34 L391-9) we made a brief first report of our quantal and classical calculations for the hydrogen atom in crossed electric and magnetic fields at constant scaled energy and constant scaled electric field strength. A principal point of that communication was our statement that each and every peak in the Fourier transform of the scaled quantum photo-excitation spectrum for scaled energy value epsilon = -0.586 538 871028 43 and scaled electric value (f) over tilde = 0.068 537 846 207 618 71 could be identified with a scaled action value of a found and mapped-out closed orbit up to a scaled action of 20. In this follow-up paper, besides presenting full details of our quantum and classical methods, we set out the scaled action values of all 317 closed orbits involved, together with the geometries of many.

Time-dependent electron interference prior to ionization in the hydrogen atom and hydrogen molecular ion

We investigate electron dynamics in the hydrogen atom and the hydrogen molecular ion when exposed to long wavelength laser pulses yet having intensity insufficient to ionize the system. We find that the field is still able to drive the electron, leading to time-dependent interference effects.