Averaged Wave Function Research Articles

Disorder-Driven Transition in the ν=5/2 Fractional Quantum Hall Effect.

The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.

The 1D Schrödinger equation with a spacetime white noise: the average wave function

For the 1D Schrödinger equation with a mollified spacetime white noise, we show that the average wave function converges to the Schrödinger equation with an effective potential after an appropriate renormalization.

The Schrödinger equation with spatial white noise: The average wave function

We prove a representation for the average wave function of the Schrödinger equation with a white noise potential in d=1,2, in terms of the renormalized self-intersection local time of a Brownian motion.

The Weak Coupling Limit for the Random Schrödinger Equation: The Average Wave Function

We consider the Schr\"odinger equation with a time-independent weakly random potential of a strength $\epsilon\ll 1$, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale $t\sim\epsilon^{-2}$. This is shown under the sharp assumption that the correlation function $R(x)$ of the random potential decays slower than $1/|x|^2$, which ensures that the effective potential is finite. When $R(x)$ decays slower than $1/|x|^2$ we establish an anomalous diffusive behavior for the averaged wave function on a time scale shorter than $\epsilon^{-2}$, as long as the initial condition is "sufficiently macroscopic". We also consider the kinetic regime when the initial condition varies on the same scale as the random potential and obtain the limit of the averaged wave function for potentials with the correlation functions decaying faster than $1/|x|^2$. We use random potentials of the Schonberg class which allows us to bypass the oscillatory phase estimates.

Non-adiabatic quantum pumping by a randomly moving quantum dot

We look at the time dependent fluctuations of the electrical charge in an open 1D quantum system represented by a quantum dot experiencing random lateral motion. In essentially non-adiabatic settings we study both diffusive and ballistic (Levy) regimes of the barrier motion where the electric current as well as the net pumped electric charge experience random fluctuations over the static background. We show that in the large-time limit, , the wavefunction is naturally separated into the Berry-phase (BP) component (resulting from the singular part of the wave amplitude in the co-moving frame) and the non-adiabatic correction (arising from fast oscillating, slow decaying tails of the same amplitude). Based on this separation we report two key results: firstly, the disorder averaged wave function and current are asymptotically mainly determined by the same BP contribution that applies in the case of adiabatic motion. Secondly, after a short transition period the pumped electric charge exhibits fluctuations that grow much faster than predicted by the adiabatic theory. We also derive the exact expressions for the average propagator (in the co-moving basis representation) for the diffusive and ballistic types of motion considered.

What kind of phases does one measure in usual interference experiments?

It is pointed out that in usual optical interference experiments one does not measure phases of monochromatic light beams as is usually assumed. Rather that one measures a phase of an associated correlation function of a beam; and that, when the beam is spatially fully coherent, as is usually the case, the measured phase may be identified with the phase of an average wave function of the beam.

Spin-flavor oscillations of neutrinos in rapidly varying external fields

Spin-flavor oscillations of neutrinos in rapidly varying external fields are studied. A method for describing neutrino oscillations in arbitrary rapidly varying external fields is developed. An effective Hamiltonian that describes the evolution of the averaged neutrino wave function is obtained. Neutrino oscillations in rapidly varying magnetic fields are considered on the basis of the general formalism developed in this study. Neutrino transitions in a superposition of a constant and a rotating (in space) magnetic field that are transverse with respect to the neutrino velocity are studied. The probabilities of transitions in spin-flavor oscillations of neutrinos in the magnetic fields of the Sun are estimated. Numerical solutions to the Schrodinger equation for the Hamiltonian that describes neutrino interaction with a constant and a rotating (in space) magnetic field are given. It is shown that the approximate analytic formula obtained in the present study for the probability of neutrino transitions is consistent with the respective numerical solution to the evolution equation at high frequencies of the rotating magnetic fields.

Approximate ansatz for the expansion of the spherically averaged wave function in terms of interelectronic separation r12 for the Hookean atom, atomic ions, and the H2 molecule

AbstractFor the two‐electron Hookean atom, it is first emphasized that, for a specific force constant k = 1/4, the ground‐state wave function has a simple dependence on the interelectronic separation r12, namely, (1 + ½r12)exp(−⅛r). For this two‐electron model, therefore, the study of Rassolov and Chipman on the electron–electron cusp conditions on the spherically averaged wave function for the N electron atomic ions can be generalized to all orders in the interelectronic separation r12. This Hookean model has therefore been used to give some justification for an ansatz for the spherically averaged wave function in atomic ions with N electrons for N ≥ 2. Several approximate two‐electron wave functions satisfying the Rassolov and Chipman conditions were tested and found to give excellent results. Another ansatz has been tested numerically on the ground state of two‐electron atomic ions and the H2 molecule. Finally, for the Hookean atom a partial differential equation that is essentially for the pair correlation density is given in the Appendix. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 95: 21–29, 2003

Effective theory of systems coupled strongly to rapidly varying external sources

We consider quantum systems that interact strongly with a rapidly varying environment and derive a Schr\"odinger-like equation that describes the time evolution of the average wave function. We show that the corresponding Hamiltonian can be taken to be Hermitian provided all states are rotated using an appropriate-unitary transformation. The formalism is applied to a variety of systems and is compared and contrasted with related results describing stochastic resonances.

Analytical gradients of a state average MCSCF state and a state average diagnostic

An efficient method for calculating the Lagrange multipliers and the analytical gradients of one state included in a state average MCSCF wave function is presented. It is demonstrated that the state average energy of an ‘equal-weight’ scheme is invariant to rotations within the state average subspace and that the corresponding rotations should be eliminated from the Lagrangian equations. Finally, a diagnostic is presented, which gauges the energy difference between a state defined by a state average calculation and the corresponding fully variational multi-configurational SCF state.

Behavior of electronic wave functions near cusps

The behavior of a nonrelativistic many-electron wave function when one electron approaches either a nucleus or another electron is well known to be governed by a cusp condition. This is a consequence of the Coulomb singularity in the potential and provides a specific relation between the wave function and its first derivative at the point of coalescence. It is shown here that the coalescence behavior also uniquely determines the third derivative of the spherically averaged wave function in terms of the lower derivatives. The new relation is valid for any atom, molecule, or electron gas in any smooth external field. Further extensions are also discussed for electron–electron coalescence in electron gas systems and for electron–nucleus coalescence in multiconfigurational self-consistent-field wave functions.

THEORY OF OPTICAL ABSORPTION IN HIGH-TEMPERATURE SUPERCONDUCTORS

We review the optical absorption spectrum of a high-temperature superconductor and present a simple theory which explains it in some detail. An experiment is proposed to determine the averaged spectral density and wave function renormalization functions of the charge carriers in the normal and in the superconducting states. This might help measure the superconducting gap.

Electron transport in quasi-one-dimensional random structure

A model of the Markovian chain of N atoms is constructed with the help of extension theory. Behavior of averaged wave functions of random Dirac operator is studied for finite and large N with respect to parameters of the model and Markovian process. Study of a random Dirac operator is connected with the study of nonrandom double-dimensional non-self-adjoint Dirac operator. Dynamics of bands of effective periodic operator with respect to the Markovian parameter is presented.

On the self-consistent-field linear combination of atomic orbitals for bounded crystal orbitals (SCF-LCAO-BCO) method

The self-consistent-field linear combination of atomic orbitais for bounded crystal orbitais (SCF-LCAO-BCO) method can be used to study the electronic structure of bounded polymers and crystals if they have a significant amount of internal translational symmetry, at the ab initio Hartree-Fock level. The direct recursion (transfer matrix) method (DRM) is applied to its derivation, here only for certain simpler models. It is shown that the application of the free boundary condition instead of the periodic (Born-von Kármán) one halves the conventional one-dimensional Brillouin zone and resolves the double degeneracy of the Bloch states. A longitudinal bulk-state distortion (LBSD) is demonstrated for a linear bounded polymer model possessing perfect translational symmetry and symmetrical intercell interaction matrices. The block diagonalizability is discussed when free boundary conditions are applied. The non-existence of limit wave functions in the overlapping regions of bands is shown for the infinite size limit. The use of certain averaged wave functions is proposed for the calculation of the charge-bond order matrices in order to resolve the above difficulty.

Nonlocal and quasilocal potentials in the spontaneous emission of molecular exciplexes coupled to the phonon bath of a solid matrix

Molecular exciplexes coupled to the phonon bath of a solid matrix are treated by a two-state model allowing for radiative relaxation from the excited to ground state and thermal relaxation from each of the states to the phonon bath. Molecular dynamics is considerably simplified by a canonical (Duke–Soules) transformation on the model Hamiltonian. This transformation (i) renormalizes the bath-free exciplex potential surfaces to phonon-dressed surfaces, and (ii) combines the radiative and phonon couplings into a single effective coupling. This leads to an effective nonlocal potential whose kernel is a product of radiative and nonradiative parts, and which can be interpreted pictorially in an intuitive manner when individual terms in the Born series solution of the corresponding Schrödinger equation are linked to special ‘‘Feynman diagrams.’’ Under the small phonon energy and the weak correlation (between phonon and nuclear degrees of freedom) approximations, a thermally averaged wave function for nuclear motion can be constructed, and the nonlocal potential reduces to a quasilocal potential in which the phonon bath ‘‘filters out’’ certain nonlocal transitions between the exciplex states.

Smoothed wave functions of chaotic quantum systems

It is shown that wave functions of quantum systems as ħ → 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic trajectories. The contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory. The results are compared with the numerical calculations of the wave functions for the stadium billiard.

Scattering from a model fractal lattice of dimension less than one

Scattering cross sections are obtained for scattering of waves from one-dimensional lattices chosen as simplistic models of Cantor-like sets. The self-similar geometry is described in terms of a fractal dimension. A parametric dependence of the cross-section on the fractal dimension is proposed. The average wavefunction method is used to calculate the differential scattering cross section. The integrated cross section shows a fractional power-law dependence on the wave-vector of the incoming wave. Results are presented for a wide range of energies and two different regimes exhibiting different fractal behaviour are observed.

Average wave function method for multiple scattering

A general approximation scheme applicable to scattering of atoms and molecules off multicenter targets, is proposed. The total potential is replaced by a sum of nonlocal, separable interactions. Each term in the sum projects the wave function onto a weighted average in the vicinity of a given scattering center. The resultant solution is an infinite order approximation to the true solution, and choosing the weighting function as the zeroth order solution guarantees agreement with Born approximation to second order. In addition, the approximation also becomes increasingly more accurate in the low energy long wavelength limit. A nonlinear, nonperturbative iterative scheme for the wave function is proposed. An extension of the scheme to multichannel scattering suitable for treating inelastic scattering is also presented.

Average wave function method for gas–surface scattering

The average wave function method (AWM) is applied to scattering of a gas off a solid surface. The formalism is developed for both periodic as well as disordered surfaces. For an ordered lattice an explicit relation is derived for the Bragg peaks along with a numerical illustration. Numerical results are presented for atomic clusters on a flat hard wall with a Gaussian-like potential at each atomic scattering site. The effect of relative lateral displacement of two clusters upon the scattering pattern is shown. The ability of AWM to accommodate disorder through statistical averaging over cluster configurations is illustrated. Enhanced uniform backscattering is observed with increasing roughness on the surface.

Temperature effects in diffractive atom-surface scattering.

A theory is presented of the diffractive scattering of a light atom by a semi-infinite crystal in which the atoms are executing thermal vibrations about their equilibrium positions. The intensities of the diffracted beams can be expressed in terms of the average of the scattered wave function over the canonical ensemble defined by the vibrational Hamiltonian of the crystal (this is not true for the inelastic component of the scattered intensity). With the aid of projection operators we obtain the integral equation satisfied by the averaged atomic wave function. The effective potential entering this equation is nonlocal in space and time, complex, and temperature dependent. The change in intensity of each diffracted beam due to scattering out of the beam by the thermal vibrations of the atoms constituting the crystal is related to this effective potential.