Fractional Heisenberg equation

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/?)[H,⋅], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this Letter, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/?)[H,⋅]. As a result, we obtain a fractional generalization of the Heisenberg equation. The fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. The suggested Heisenberg equation generalize a notion of quantum Hamiltonian systems to describe quantum dissipative processes.     

Axially symmetrical molecules in electric and magnetic fields: energy spectrum and selection rules

In this paper we investigate the effects of external electric and magnetic fields on a three-dimensional harmonic oscillator with axial symmetry. The energy spectrum of such a system is non-degenerate due to the presence of the magnetic field. The degeneracy of the energy spectrum in the absence of a magnetic field is discussed. The influence of electric and magnetic fields, as well as the frequencies of the oscillator on the probability distribution function is analyzed. Optical transition probabilities are examined by deriving the selection rules in dipole approximation for the quantum numbers n _p , m _l and n _z . Employing stationary perturbation theory, the effects of deformations of the potential energy function on the oscillatory states are analyzed. Such models have been used in literature in analysis of spectra of axially symmetrical molecules and cylindrical quantum dots.     

Transition energy and dipole oscillator strength of 1s23d-1s2 nf transitions for Sc18+ ion

The transition energies of the 1s²3d-1s² nf (4?n?9) transitions and fine structure splittings of 1s² nf (n?9) states for Sc¹⁸⁺ ion are calculated with the full-core plus correlation method. The quantum defect of 1s² nf series is determined by the single-channel quantum defect theory. The energies of any highly excited states with n?10 for this series can be reliably predicted using the quantum defect as function of energy. Three alternative forms of the dipole oscillator strengths for the 1s²3d-1s² nf (n?9) transitions of Sc¹⁸⁺ ion are calculated with the transition energies and wave functions obtained above. Combining the quantum defect theory with the discrete oscillator strengths, the discrete oscillator strengths for 1s²3d-1s² nf (n > 9) transitions and the oscillator strengths densities corresponding to the bound-free transitions are obtained.     

On the quantum density of states and partitioning an integer

This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one-dimensional harmonic oscillator potential, it is well known that the asymptotic (N→∞) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p^s(n), the latter being the number of partitions of n into a sum of sth powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain d^s(n) for distinct partitions. We find that the distinct square partitions d²(n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdos-Lehner formula for restricted partitions for the s=1 case, we use the modified technique to obtain a new formula for distinct restricted partitions.     

Excitation of rubidium atoms by electron impact

An analytic central potential containing two adjustable parameters is used to generate wave functions for the ground and excited states of the rubidium atom. The two parameters are chosen so that the potential reproduces accurately known values of optical oscillator strengths for the 5s?np (n≧5) transitions. Using the Born approximation, we calculate generalized oscillator strengths and integrated cross sections up to 5 keV for the 5s?np excitations. Reasonable agreement with experimental data is obtained.     

Critical assessment of the Schrödinger picture of quantum mechanics

We provide an example in which the Heisenberg and the Schrödinger pictures of quantum mechanics give different results, thus confirming the statement of P.A.M. Dirac that the two pictures may lead to inequivalent results. We consider a one-dimensional nonrelativistic charged harmonic oscillator (frequency ω₀ and mass m), and take into account the action of the radiation reaction and the vacuum electromagnetic forces on the charged oscillator. We show that the Heisenberg picture gives the correct value, ℏω₀/2, for the ground state energy of the harmonic oscillator in both cases of classical and quantized vacuum fields. In the case of the Schrödinger picture, considering classical vacuum fields, and using a simple calculation for the classical radiation reaction force that is valid in the limit of large mass (mc²⪢ℏω₀), we obtain the value ℏω₀ for the ground state energy of the harmonic oscillator. We show that the vacuum electromagnetic forces play a very important role in the understanding of this discrepancy.     

Classical Hamiltonian perturbation theory without secular terms or small denominators

We consider perturbation theory in ? for the classical Hamiltonian H = H₀ + ?H₁, where H₀ gives rise to a known motion and ? is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ? and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator (s ? L)^?1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ?, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ?ⁿ⁺¹ when the accuracy of the theory is order ?ⁿ.     

One-dimensional model of a quantum nonlinear harmonic oscillator

In this paper we study the quantization of the nonlinear oscillator introduced by Mathews and Lakshmanan. This system with position-dependent mass allows a natural quantization procedure and is shown to display shape invariance. Its energy spectrum is found by factorization. The linear harmonic oscillator appears as the λ → 0 limit of this nonlinear oscillator, whose energy spectrum and eigenfunctions are compared to the linear ones.     

Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

Gapped ground states of quantum spin systems have been referred to in the physics literature as being ‘in the same phase’ if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on \({s\in [0,1]}\), such that for each s, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that ’belong to the same phase’ are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an s-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings’ ‘quasi-adiabatic evolution’ technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the spectral flow, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models H(s), 0 ≤ s ≤ 1.     

Optical transitions of Ag− centers in alkali halides

Ag^? centers in alkali halides give rise to a strong absorption band in the 300 nm region (formerly called “B band”). Its resolved triplet structure in CsCl suggests that it corresponds to the C band of the isoelectronic In⁺ center. Two very weak bands are found in several alkali halides in the 400 nm region. These new bands are assigned to the A and B transitions of the In⁺-type centers. This is supported by the doublet structure in the A band, and by the temperature dependence of the oscillator strength of the B band. In KCl∶Ag^? the ratios of the oscillator strengths are found to bef _c /f _A =610 andf _c /f _B =3,400 at low temperatures. The energy parameters of Ag^? centers are computed and compared with those of others ²-type centers. The electron-lattice coupling parameters are estimated from the Jahn-Teller splitting of the C band in CsCl and of the A band in KC1. The temperature dependence of the lifetime of the visible fluorescence suggests that a metastable state is involved in the emission process after a C band excitation.     

Weighted Hardy-Littlewood Theorems for Conjugate A-Harmonic Tensors

In this paper, we establish some versions of A(φ ₁(x), φ ₂(x), τ, Ω)-weighted Hardy-Littlewood inequalities for conjugate A?harmonic tensors, the theory of harmonic analysis and A?harmonic differential forms largely pertain to applications in mathematical physics, quantum field theory, elementary particle physics, etc.     

Wave function(al)s in the large-N limit

The energy spectrum and correlation functions in various quantum theories have been determined within the large-N limit. Here we study the wave functions. Explicit results are presented for a quantum mechanical rotor, invariant under adjoint transformations of U(N), as well as for O(N) invariant vector models in quantum mechanics and quantum field theory.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

Exact classical and quantum solutions for a covariant oscillator near the black hole horizon in Stueckelberg-Horwitz-Piron theory

We found exact solutions for canonical classical and quantum dynamics for general relativity in Horwitz general covariance theory. These solutions can be obtained by solving the generalized geodesic equation and Schrödinger-Stueckelberg-Horwitz-Piron (SHP) wave equation for a simple harmonic oscillator in the background of a two dimensional dilaton black hole spacetime metric. We proved the existence of an orthonormal basis of eigenfunctions for generalized wave equation. This basis functions form an orthogonal and normalized (orthonormal) basis for an appropriate Hilbert space. The energy spectrum has a mixed spectrum with one conserved momentum p according to a quantum number n. To find the ground state energy we used a variational method with appropriate boundary conditions. A set of mode decomposed wave functions and calculated for the Stueckelberg-Schrodinger equation on a general five dimensional blackhole spacetime in Hamilton gauge.     

Quantization of Poisson-Lie groups and applications

LetG be a connected Poisson-Lie group. We discuss aspects of the question of Drinfel'd:can G be quantized? and give some answers. WhenG is semisimple (a case where the answer isyes), we introduce quantizable Poisson subalgebras ofC ^∞(G), related to harmonic analysis onG; they are a generalization of F.R.T. models of quantum groups, and provide new examples of quantized Poisson algebras.     

Mass Parameter Quantization in the FRW Cosmological Model

Using the canonical quantum theory apply to spherically symmetric pure gravitational systems, we present the study of the closed Friedmann-Robertson-Walker (FRW) cosmological model filled with pressureless matter (dust) content as a toy model. The Wheeler-DeWitt equation is view as the Schrödinger equation for the linear harmonic oscillator with energy E. We show that such type of universe has a quantized masses of the order of the Planck mass and harmonic oscillator wave functions, where a dual symmetry emerge among the quantum parameters.     

Interpretation of the low even configurations of neutral protactinium (Pa I)

A new analysis of the spectrum of neutral protactinium led to revise and to extend the earlier level list. The Slater-Condon method was used to interpret the low even group of configurations 5f ²6d7s ²+5f ²6d ²7s with a root mean square deviation of 134cm^?1 for 64 energy levels.     

Fluorescence Properties of Twenty Fluorescein Derivatives: Lifetime,Quantum Yield,Absorption and Emission Spectra

The fluorescence lifetime (τ_f), emission quantum yield (Φ_f), absorption and emission spectral data of 20 fluorescein derivatives were measured under the same conditions by using time-correlated single photon counting, steady state fluorescence and absorption methods to get comparable data. Based on the results, the factors and mechanism that control the fluorescence properties of the fluorescein dyes are discussed. Both Φ_f and τ_f are remarkably dependent on the substitution on either xanthene or phenyl rings, but their ratio (Φ_f/τ_f), i.e. rate constant of radiation process, is a constant value (0.20?×?10⁹ s^?1). The rate constant of nonradiation process, on the other hand, is varied with both the structure and the solvent used.     

Computer simulation of the structure and raman spectra of GaAs polytypes

The structure, energy of formation, and Raman spectra of several polytypes (3C, 2H, 4H, and 8H) of gallium arsenide GaAs have been investigated using quantum mechanical calculations based on the local density functional theory. It has been found that the energy of the formation of hexagonal polytypes increases with an increase in the length of the periodicity and approaches the value corresponding to the ground state, i.e., to the structure of the 3C polytype. It has been shown that the calculated frequencies of normal vibrations of different polytypes are consistent, with good accuracy (±6 cm^?1), with the scheme of folding of the phonon branches. In the calculated Raman spectra of the polytypes, there are new lines (forbidden in the spectrum of the 3C polytype) which can serve as characteristic lines of other polytypes. Similar lines can be found in the Raman spectra of GaAs nanowhiskers. This result has opened up new prospects for the application of Raman spectroscopy to the characterization of the structure of these nano-objects.