Pseudo-harmonic oscillator potential in Wigner-Weyl transforms

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 commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of nonzero minimal uncertainties in the positions and momenta. The usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta.

We propose a mechanism to use scanning tunneling microscopy (STM) for direct measurements of the two-electron singlet-triplet exchange splitting $J$ in diatomic molecular systems, unsing the coupling between the molecule and the substrate electrons. The different pathways for electrons lead to interference effects and generate kinks in the differential conductance at the energies for the singlet and triplet states. These features are related to Fano resonance due to the branched electron wave functions. The ratio between the tunneling amplitudes through the two atoms can be modulated by spatial movements of the tip along the surface.

Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms

Electronic Eigenenergies of the Hydrogen Molecular Ion

The cylindrically and the spherically symmetric harmonic oscillators are considered by means of matrix techniques in a basis that is appropriate for each symmetry. For example, the spherically symmetric oscillator is described by the radial and angular momentum quantum numbers. The energy eigenvalues and the matrix elements of the promotion-demotion operators are found in each case. By expressing the operators in curvilinear coordinates, the promotion-demotion operators are used to deduce the form of the corresponding wave functions.

In this paper we generalize our previous approach to electromagnetic interactions within empirical tight-binding theory to encompass molecular solids and isolated molecules. In order to guarantee physically meaningful results, we rederive the expressions for relevant observables using commutation relations appropriate to the finite tight-binding Hilbert space. In carrying out this generalization, we examine in detail the consequences of various prescriptions for the position and momentum operators in tight binding. We show that attempting to fit parameters of the momentum matrix directly generally results in a momentum operator which is incompatible with the underlying tight-binding model, while adding extra position parameters results in numerous difficulties, including the loss of gauge invariance. We have applied our scheme, which we term the Peierls-coupling tight-binding method, to the optical dielectric function of the molecular solid PPP, showing that this approach successfully predicts its known optical properties even in the limit of isolated molecules.

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.

Theoretical formulation of finite-dimensional discrete phase spaces: I. Algebraic structures and uncertainty principles

A procedure to obtain the operational solutions of second order differential equations related with Sturm–Liouville problems is presented. The method is based on the commutation relation between the ladder operators themselves, with a certain structure, and the position and momentum operators. Even though the creation and annihilation operators, derived by the proposed approach, factorize as expected the corresponding differential equation, the method does not use, as original premise factoring, the differential relation under consideration. That is, the displayed procedure is quite different, simple, and direct when compared with other procedures such as the factorization method of Infeld and Hull. To illustrate the above, the usefulness of the proposed procedure is shown by finding the ladder operators associated to the quantum numbers n and I for various potential wave functions. © 1992 John Wiley & Sons, Inc.

We construct, analytically and numerically, the Wigner distribution functions for the exact solutions of the position-dependent effective mass Schrödinger equation for two cases belonging to the generalized Laguerre polynomials. Using a suitable quantum canonical transformation, expectation values of position and momentum operators are obtained analytically in order to verify the universality of Heisenberg’s uncertainty principle.

A representation of Weyl–Heisenberg Lie algebra in the quaternionic setting

This work addresses a method for constructing supersymmetric vector coherent states (VCS) for a two-dimensional electron gas in a perpendicular magnetic field in the presence of both Rashba and Dresselhaus spin-orbit (SO) interactions, with an effective Zeeman coupling. The model Hamiltonian, decomposed into two conveniently defined operators, acts on a tensor product of two Hilbert spaces associated with corresponding chiral sectors. Supersymmetric (SUSY) VCS, related to a SUSY pair of Hamiltonians, are built. A generalized oscillator algebra is provided using quadrature operators. Mean-values of position and momentum operators and uncertainty relation in the SUSY VCS are obtained and discussed. Besides, the SUSY VCS time evolution is also given.

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.

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.