Y. Alhassid

The statistical theory of quantum dots

A quantum dot is a sub-micron-scale conducting device containing up to several thousand electrons. Transport through a quantum dot at low temperatures is a quantum-coherent process. This review focuses on dots in which the electrons dynamics are chaotic or diffusive, giving rise to statistical properties that reflect the interplay between one-body chaos, quantum interference, and electron-electron interactions. The conductance through such dots displays mesoscopic fluctuations as a function of gate voltage, magnetic field, and shape deformation. The techniques used to describe these fluctuations include semiclassical methods, random-matrix theory, and the supersymmetric nonlinear \ensuremath{\sigma} model. In open dots, the approximation of noninteracting quasiparticles is justified, and electron-electron interactions contribute indirectly through their effect on the dephasing time at finite temperature. In almost-closed dots, where conductance occurs by tunneling, the charge on the dot is quantized, and electron-electron interactions play an important role. Transport is dominated by Coulomb blockade, leading to peaks in the conductance that at low temperatures provide information on the dots ground-state properties. Several statistical signatures of electron-electron interactions have been identified, most notably in the dots addition spectrum. The dots spin, determined partly by exchange interactions, can also influence the fluctuation properties of the conductance. Other mesoscopic phenomena in quantum dots that are affected by the charging energy include the fluctuations of the cotunneling conductance and mesoscopic Coulomb blockade.

Group theory approach to scattering

Abstract We show that both bound and scattering states of a certain class of potentials are related to the unitary representations of certain groups. In this class, several potentials of practical interest, such as the Morse and Poschl-Teller potentials, are included. The fact that not only bound states but also scattering states are connected with group representations suggests that an algebraic treatment of scattering problems similar to that of bound state problems may be possible.

An Algorithm for Finding the Distribution of Maximal Entropy

Abstract An algorithm for determining the distribution of maximal entropy subject to constraints is presented. The method provides an alternative to the conventional procedure which requires the numerical solution of a set of implicit nonlinear equations for the Lagrange multipliers. Here they are determined by seeking a minimum of a concave function, a procedure which readily lends itself to computational work. The program also incorporates two preliminary stages. The first verifies that the constraints are linearly independent and the second checks that a feasible solution exists.

Entropy and chemical change. III. The maximal entropy (subject to constraints) procedure as a dynamical theory

An equivalence between the dynamical (equations of motion) and the information theoretic (maximal entropy) approaches to collision phenomena is established. The connection is demonstrated in both directions. The variational procedure of maximal entropy is shown to converge to an exact solution of the equations of motion (be they classical or quantal) throughout the collision. In particular, a stationary precollision state is proved to be a state of maximal entropy (subject to constants of the unperturbed motion) and to remain a state of maximal entropy throughout the collision. Conversely, the exact solution of the equations of motion is shown to be of maximal entropy. In this fashion one obtains an algebraic procedure for the specification of the constraints which determine (via the procedure of maximal entropy) an exact solution of the equations of motion. Surprisal analysis does not require the solution of differential equations. These must be solved to determine the magnitude of the Lagrange parameter...

Shell-model Monte Carlo studies of fp-shell nuclei

We study the gross properties of even-even and {ital N}={ital Z} nuclei with {ital A}=48--64 using shell-model Monte Carlo methods. Our calculations account for all 0{h_bar}{omega} configurations in the {ital fp} shell and employ the modified Kuo-Brown interaction {ital KB}3. We find good agreement with data for masses and total {ital B}({ital E}2) strengths, the latter employing effective charges {ital e}{sub {ital p}}=1.35{ital e} and {ital e}{sub {ital n}}=0.35{ital e}. The calculated total Gamow-Teller strengths agree consistently with the {ital B}({ital GT}{sub +}) values deduced from ({ital n},{ital p}) data if the shell-model results are renormalized by 0.64, as has already been established for {ital sd}-shell nuclei. The present calculations therefore suggest that this renormalization (i.e., {ital g}{sub {ital A}}=1 in the nuclear medium) is universal.

An upper bound for the entropy and its applications to the maximal entropy problem

Abstract A variational principle where the Lagrange multipliers of a trial distribution are used as variational parameters is discussed as an efficient, practical route to the determination of the distribution of maximal entropy.

Phenomenology of shape transitions in hot nuclei

Abstract A phenomenological description of the temperature-driven shape transitions in heavy nuclei is presented. The general framework of the Landau theory is used to establish the free energy and entropy dependence on the deformation and the temperature-energy variables. This information is used to discuss the equilibrium quantities as well as the fluctuation effects around equilibrium shapes. Calculations are presented for the entropy, energy and the level density in the context of a typical example of a heavy nucleus undergoing shape transition. The results show considerable deviations from the standard dependences which are obtained using the assumption of a fixed nuclear shape.

SCALING PROPERTIES OF THE GIANT DIPOLE RESONANCE WIDTH IN HOT ROTATING NUCLEI

We study the systematics of the giant dipole resonance width {Gamma} in hot rotating nuclei as a function of temperature T , spin J , and mass A . We compare available experimental results with theoretical calculations that include thermal shape fluctuations in nuclei ranging from A=45 to A=208 . Using the appropriate scaled variables, we find a simple phenomenological function {Gamma}(A,T,J) which approximates the global behavior of the giant dipole resonance width in the liquid drop model. We reanalyze recent experimental and theoretical results for the resonance width in Sn isotopes and {sup 208}Pb . {copyright} {ital 1998} {ital The American Physical Society}

The potential group approach and hypergeometric differential equations

This paper proposes a generalized realization of the potential groups SO(2,1) and SO(2,2) to describe the confluent hypergeometric and the hypergeometric equations, respectively. It implies that the classes of Schrodinger equations with solvable potentials whose analytical solutions are related to the confluent hypergeometric and the hypergeometric functions can be realized in terms of the above group structure.

Effects of thermal fluctuations on giant dipole resonances in hot rotating nuclei

Abstract We present a macroscopic approach to giant dipole resonance (GDR) in highly excited nuclei, using a unified description of quadrupole shape thermal fluctuations. With only two free parameters, which are fixed by the zero-temperature nuclear properties, the-model reproduces well experimental GDR cross sections in the 100 ≤ A ≤ 170 mass range for both spherical and deformed nuclei. We also investigate the cross-section systematics as a function of both temperature and angular velocity and the sensitivity of the GDR peak to the nuclear shape. We conclude that at low temperatures (T ≈ 1 MeVfs) the GDR cross section is sensitive to changes in the nuclear energy surface. Higher-temperature (T ≳ 2 MeV) cross sections are dominated by large fluctuations (triaxial in particular) and are much less sensitive to the equilibrium shape.