Uma introdução à espectroscopia atômica: II - o espectro do sódio

Much of the physics determined for hydrogen and hydrogen-like ions is relevant to atoms and ions with many electrons. In a multi-electron atom or ion, the quantum state energies and wavefunctions are dominated by the central potential arising from the nuclear charge as is the case for hydrogen and hydrogen-like ions. Consequently, quantum wavefunctions similar in form to those found for hydrogen or hydrogen-like ions are produced. There are some corrections needed to the exact electron energies associated with the nuclear electric field experienced by each electron due to shielding effects by other electrons – and some perturbing energy effects due to electron–electron Coulomb repulsion. The electron wavefunctions have similar orbital or angular shapes as for hydrogen and hydrogen-like ions (see Figures 7.2 and 7.3). The same electron-configuration system (i.e. 1s, 2s, 2p, …) can be used and the same degeneracies are associated with different values of principal quantum number n (degeneracy 2 n 2 ) and orbital angular momentum quantum number l (degeneracy 2×2 l(l +1 ) ) as found for hydrogen and hydrogen-like ions. As multi-electron atoms and ions have quantum states following the ‘rules’ established for hydrogen and hydrogen-like ions, the quantum state energy values are primarily determined by a principal quantum number n with values n =1, 2, 3, … . Angular quantum numbers l with l =0, 1, 2, … n − 1 are found to have a small effect on energy values due to shielding of the nuclear charge. As for hydrogen, the number of individual quantum states at a particular energy involves the different magnetic quantum numbers m with values m = − l ,− l + 1, … , 0, 1 … + l and the two possible spin orientations (relative to the orbital angular momentum). The states which the electrons occupy are called the electronic configuration for the atom or ion. An important feature of multi-electron atoms is established by the Pauli exclusion principle which requires that only one electron can occupy an individual quantum state. In the ground state of an atom or ion, the required number of electrons effectively fills the lowest energy quantum states designated by the n , l , m and s values according to the hydrogen degeneracies.

The transformation coefficients connecting the Stark and the angular momentum states belonging to the positive spectrum of the hydrogen atom are calculated by a group- theoretical method which makes use of the O(3,1) symmetry of the states and also by a purely analytical method. The results of the two calculations agree except for an undetermined factor not containing the orbital quantum number l. Complete agreement between the results is achieved by taking the normalization factors for the continuum states to be analytic continuations of those for the bound states. The transformation coefficients turn out to be SU(2) Clebsch-Gordan coefficients (j1j2l; m1m2m) with complex j1j2m1m2, and physical l,m. From the general theory the well known expansion of the Coulomb scattering function is obtained by giving the magnetic quantum number m and one of the electric quantum numbers n2, the special values, 0 and -1, respectively.

The CN(B 2Σ+–X 2Σ+) tail band system was observed in the reaction of argon metastable atoms with BrCN. The observed transitions in the 17-14 and 17-16 bands were analyzed in terms of the B 2Σ+,v=17∼A 2Πinv,v=34 and 4Π,v=p+11 perturbations, and the molecular constants for these levels and the interaction constants were determined. Intensity anomalies were also observed in the 9-8, 9-9, and 12-12 bands. The quantum numbers of the total angular momentum, J, of the perturbed rotational levels of B 2Σ+ were assigned as (vB,N;J)=(9,56;55.5) and (12,10;10.5). The vibrational quantum number was determined as p=0, and the crossing spin sublevels at the anomalies of vB=9 and 12 were determined as 4Π5/2 and 4Π1/2 by examination of the electronic part of the B 2Σ+∼4Π perturbation constant, HSOel, for the (vB,vΠ)=(14,6) and (17,11) perturbations estimated by use of the Rydberg–Klein–Rees (RKR) potentials for B 2Σ+ and 4Π.

Mapping Coriolis-coupled quantum dynamics onto parallel computer architectures

As we have already indicated, experimental knowledge on the proton-neutron system is acquired by studying the scattering of neutrons by protons and by investigations of the properties of the deuteron. An analysis of these data, the goal of which is the determination of phase shifts, requires, first of all, a resolution of the scattered waves in terms of spherical functions. Because of the spins of the proton and neutron this procedure turns out to be somewhat more complicated than in the case of spinless particles which have been described. First of all we must introduce a system for classifying the possible states of the system. In the case of spinless particles the state was determined by the azimuthal quantum number L. In the presence of spin the state of the system is also determined by the spin quantum number S. If the spin and coordinates are independent, the orbital angular momentum and the spin angular momentum are conserved individually. Any coupling between the spin and orbital motion means that the vectors L and S are not conserved individually; rather the total angular momentum J is conserved;* this is the sum of the orbital and spin angular momenta.

We perform a rigorous analysis of the πK scattering at an unphysical pion mass 391 MeV using the Roy-Steiner equations, which satisfy unitarity, analyticity, and crossing symmetry, for the first time. Stable solutions of the Roy-Steiner equations with different quantum numbers of isospin and angular momentum are obtained in the elastic energy region, by taking inputs from the πK lattice data in the inelastic region, the lattice data from the crossed ππ→KK¯ channels, the masses of f0(500) and K* at the same pion mass from previous study, and the Regge model. Predictions on the elastic πK scattering phase shifts and the K0*(700) pole content are made. Contrary to the virtual pole scenario obtained using the K-matrix method in the literature, we find that lightest strange scalar meson K0*(700) remains a broad resonance at mπ=391 MeV. The cross-channel dynamics is found to play a crucial role in deriving the proper pole position.

Parity doubling in open-shell van der Waals complexes

Past calculations of the anisotropic superexchange interaction between Yb3+ and Fe3+ in the garnets have made use of drastic assumptions about the form of the exchange potential. A more fundamental calculation has been impossible until now because of a lack of detailed knowledge of the relative magnitudes of the superexchange integrals. We have calculated the nineteen exchange integrals between the different combinations of O2− p orbitals and Yb3+ f orbitals. The O2− wave functions were taken from Watson's analytic Hartree-Fock calculation and the Yb3+ wave functions were calculated using a Hartree-Fock computer program of F. Herman. The values of the integrals span a range of three orders of magnitude. They exhibit a strong dependence on the azimuthal quantum numbers m and m′ of the oxygen p orbitals along the bond, and a much weaker dependence on the f-electron quantum numbers. The integrals involving m = 0 and m′ = 0 clearly predominate, and those involving m = 0 and m′ = ±1 are smaller by almost an order of magnitude. The transfer integrals and the implications of these results for the form of the exchange potential are discussed.

Electronic Eigenenergies of the Hydrogen Molecular Ion

Coefficients of fractional parentage (CFP's) in an L-S (or J-T) coupling scheme have recently been computed by using a new algorithm extending two procedures introduced many years ago: a factorization procedure introduced by Jahn and a combination of numerical and analytic methods introduced by Bayman and Lande. As a consequence of the factorization, coefficients of fractional parentage between states of arbitrary permutational symmetry must be computed within each angular momentum subspace separately. The new algorithm is extended here to calculate Hamiltonian matrix elements in the second quantized formalism. The matrix elements of the creation and annihilation operators, within each individual subspace, between states of arbitrary symmetry, are defined to be proportional to the corresponding coefficients of fractional parentage. The matrix elements for the one- and two-body operators in the Hamiltonian are constructed by combining the matrix elements in the individual subspaces using a new sum-over-path-overlaps technique. This procedure is extremely efficient for quark shell model calculations, where the quarks carry multiple quantum numbers of angular momentum, or more general, type (L, S, color, and flavor).

The main purpose of this chapter is to present shift operators for the quantum numbers of angular momentum. These operators have useful applications, some of which are discussed in this and subsequent chapters. We also give a brief discussion of the non-relativistic treatment of spin angular momentum.

Light-front holographic QCD and emerging confinement

The Pfaffian determinant is sometimes used to multiply the Laughlin’s wave function at the half filled Landau level. The square of the Pfaffian gives the ordinary determinant. We find that the Pfaffian wave function leads to four times larger energies and two times faster time. By the same logic, the Pfaffian breaks the supersymmetry of the Dirac equation. By using the spin properties and the Landau levels, we correctly interpret the state with 5/2 filling. The quantum numbers which represent the state vectors are now products of n (Landau level quantum number), l(orbital angular momentum quantum number and the spin, s |n, l, s>. In a circuit, the noise measures the resistivity and hence the charge. The Pfaffian velocity is different from that of the single‐particle states and hence it has important consequences in the measurement of the charge of the quasiparticles.

More matter, with less art. Hamlet II ii The quantum mechanics of a single electron in a central potential leads to classification of the one-electron states in terms of four quantum numbers. The individual electrons' spin and orbital angular momenta are coupled in an isolated manyelectron ion to give total spin and orbital quantum numbers S and L . Spin-orbit coupling then operates to split the energy levels into a series of J -multiplets, the lowest of which is specified by Hund's rules. Curie-law susceptibility, χ = C /T , is calculated for a general value of J . When placed in a solid, the ion experiences a crystal field due to the charge environment which disturbs the spin-orbit coupling, leaving either S or J as the appropriate quantum number. The crystal field modifies the structure of the lowest M S or M J magnetic sublevels which are split by the Zeeman interaction and it introduces single-ion anisotropy. Atomic physics is concerned with the energy levels of an atom or ion and the possible transitions between them, which are usually in the optical or ultraviolet energy range (1–10 eV). Magnetism is concerned with the energy levels that are occupied at ambient temperature, which usually means only the ground state , and its sublevels resulting from interactions with magnetic or electric fields, which produce a splitting of less than 0.1 eV. At ambient temperature k B T is about 25 meV.

Rydberg states of atomic helium with principal quantum numbers ranging from n=20 to n=100 have been prepared by non-resonance-enhanced single-color two-photon excitation from the metastable 2 {^3}S{_1} state. Photoexcitation was carried out using linearly and circularly polarized pulsed laser radiation. In the case of excitation with circularly polarized radiation, Rydberg states with azimuthal quantum number |m_{\ell}|=2 were prepared in zero electric field, and in homogeneous electric fields oriented parallel to the propagation axis of the laser radiation. In sufficiently strong electric fields, individual Rydberg-Stark states were resolved spectroscopically, highlighting the suitability of non-resonance-enhanced multiphoton excitation schemes for the preparation of long-lived high-|m_{\ell}| hydrogenic Rydberg states for deceleration and trapping experiments. Applications of similar schemes for Doppler-free excitation of positronium atoms to Rydberg states are also discussed.