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Research on engineered nanomaterials (ENM) has progressed rapidly from the very early stages of studying their unique, size-dependent physicochemical properties and commercial exploration to the development of products that influence our everyday lives. We have previously reviewed various methods for synthesis, surface functionalization, and analytical characterization of ENM in a publication titled ‘Engineered Nanomaterials: Preparation, Functionalization and Characterization’. In this second, inter-linked document, we first provide an overview of important applications of ENM in products relevant to human healthcare and consumer goods, such as food, textiles, and cosmetics. We then highlight the challenges for the design and development of new ENM for bio-applications, particularly in the rapidly developing nanomedicine sector. The second part of this document is dedicated to nanotoxicology studies of ENM in consumer products. We describe the various biological targets where toxicity may occur, summarize the four nanotoxicology principles, and discuss the need for careful consideration of the biodistribution, degradation, and elimination routes of nanosized materials before they can be safely used. Finally, we review expert opinions on the risk, regulation, and ethical aspects of using engineered nanomaterials in applications that may have direct or indirect impact on human health or our environment.

Research on engineered nanomaterials (ENM) has progressed rapidly from the very early stages of studying their unique, size-dependent physicochemical properties and commercial exploration to the development of products that influence our everyday lives. We have previously reviewed various methods for synthesis, surface functionalization, and analytical characterization of ENM in a publication titled 'Engineered Nanomaterials: Preparation, Functionalization and Characterization'. In this second, inter-linked document, we first provide an overview of important applications of ENM in products relevant to human healthcare and consumer goods, such as food, textiles, and cosmetics. We then highlight the challenges for the design and development of new ENM for bio-applications, particularly in the rapidly developing nanomedicine sector. The second part of this document is dedicated to nanotoxicology studies of ENM in consumer products. We describe the various biological targets where toxicity may occur, summarize the four nanotoxicology principles, and discuss the need for careful consideration of the biodistribution, degradation, and elimination routes of nanosized materials before they can be safely used. Finally, we review expert opinions on the risk, regulation, and ethical aspects of using engineered nanomaterials in applications that may have direct or indirect impact on human health or our environment.

Research on engineered nanomaterials (ENM) has progressed rapidly from the very early stages of studying their unique, size-dependent physicochemical properties and commercial exploration to the development of products that influence our everyday lives. We have previously reviewed various methods for synthesis, surface functionalization, and analytical characterization of ENM in a publication titled 'Engineered Nanomaterials: Preparation, Functionalization and Characterization'. In this second, inter-linked document, we first provide an overview of important applications of ENM in products relevant to human healthcare and consumer goods, such as food, textiles, and cosmetics. We then highlight the challenges for the design and development of new ENM for bio-applications, particularly in the rapidly developing nanomedicine sector. The second part of this document is dedicated to nanotoxicology studies of ENM in consumer products. We describe the various biological targets where toxicity may occur, summarize the four nanotoxicology principles, and discuss the need for careful consideration of the biodistribution, degradation, and elimination routes of nanosized materials before they can be safely used. Finally, we review expert opinions on the risk, regulation, and ethical aspects of using engineered nanomaterials in applications that may have direct or indirect impact on human health or our environment.

Atomic phenomena are described mainly on the basis of non-relativistic quantum mechanics. Relativistic effects can generally be accounted for in a satisfactory way with perturbative methods. In recent years it has become increasingly apparent that a better understanding of the classical dynamics of an atomic system can lead to a deeper appreciation of various features in its observable quantum mechanical properties, see e.g. [FE97] and Sect. 5.3. This does not, however, invalidate the generally accepted point of view that quantum mechanics is the relevant theory for atomic physics.

We study the scattering of ultracold atoms by a conducting cylindrical wire with total absorption near the surface of the wire. A hierarchy of approximations for the exact nonretarded atom-wire potential is explored. We show the existence of a semiclassical region near to the surface of the wire, which allows us to unambiguously define incoming boundary conditions for the description of total absorption at the wire surface. The semiclassical region is only reproduced by an accurate treatment of the potential. We calculate the complex s-wave scattering length for the effective two-dimensional atom-wire scattering problem. The scattering length tends to the wire radius R in the thick-wire limit, and it tends to 0 nearly as (ln R)-1 in the thin-wire limit.

A systematic study of torsional potential curves in electronic ground state based on second-order Møller–Plesset energy (MP2), density functional theory (DFT), and Austin mode 1 (AM1) methods is presented for para-phenylenevinylene oligomers constructed from two to four aromatic rings. The semiempirical AM1 approach gives the correct location of potential energy minima in comparison with the reference MP2 calculations and literature data. However, the semiempirical AM1 energy barriers at perpendicular orientation are ca. 30% smaller than the MP2 ones. The DFT calculations indicate optimal planar structures and the barriers are three times higher than MP2 values. Excited-state potential energy curves evaluated from vertical excitations at the time-dependent density functional theory (TD-DFT) and ab initio CI levels exhibit much steeper increase of values in the vicinity of perpendicular orientation than in the semiempirical Zerner's intermediate neglect of differential overlap (ZINDO) and ab initio RI-CC2 cases. The effects of vibrational motion of phenylene rings on the torsional broadening of absorption spectra were estimated from semi-classical molecular dynamics simulations and harmonic oscillator sampling. The simulated spectra agree well with the experiment and allow estimating the conformer distribution of the molecules.

A systematic characterization of excited-state properties of para-phenylene oligomers constructed from two to eight aromatic rings is presented using density functional theory (DFT) and the coupled-cluster singles and doubles (CC2) method. Geometry optimizations have been performed for the ground state and for the electronically excited state. Vertical excitations and the fluorescence transitions have been calculated. Time-dependent DFT (TDDFT) method underestimates excitation and fluorescence energies systematically in comparison with experimental results. The computed TDDFT lifetime for the polymer limit (0.43 ns) is in agreement with the experimental value of 0.55 ns. The TDDFT torsional potential curves were investigated for biphenyl, terphenyl, and quarterphenyl oligomers in their electronic ground and excited states. Our calculations show an increase in the separation of the lowest excited state (S1) to the next higher one with increasing molecular size. No indication is found for state crossings of the S1 state with higher ones from planar structures up to torsional angles of 60 degrees to 70 degrees. Thus, an adiabatic description of the dynamics of the S1 state might significantly simplify any dynamics simulations of torsional broadenings.

In this chapter, arbitrary internal excitations of projectile and/or target are considered, which enables the description of inelastic scattering via the appropriate coupled-channel equations. The chapter contains the theory Feshbach resonances, as opposed to single-channel shape resonances, and an account of multichannel quantum-defect theory, which is a powerful tool for describing Coulombic systems with attractive interactions falling off as \(1/r\) for large values of the projectile-target separation \(r\).

We examine the possibility of trapping ultra-cold atoms solely by quantum reflection in front of the walls of a cavity. Using a model potential with realistic threshold properties, we calculate the enhancement of the survival probability in the trap compared to the decay of a freely spreading wave-packet. Under conditions close to a recent experiment [T.A. Pasquini, Y. Shin, C. Sanner, M. Saba, A. Schirotzek, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 93 (2004) 223201], an enhancement factor near 100 and surviving fractions of 10–20% are obtained after a time of 0.4 s.

Ultracold molecules can be associated from ultracold atoms by ramping the magnetic field through a Feshbach resonance. A reverse ramp dissociates the molecules. Under suitable conditions, more than one outgoing partial wave can be populated. A theoretical model for this process is discussed here in detail. The model reveals the connection between the dissociation and the theory of multichannel scattering resonances. In particular, the decay rate, the branching ratio, and the relative phase between the partial waves can be predicted from theory or extracted from experiment. The results are applicable to our recent experiment in 87Rb, which has a d-wave shape resonance., Added Refs.[32-38]

Due to the degeneracy of the n = 2 states, the potential between a metastable 2S hydrogen atom and a conducting surface falls off as − 1 / z 3 asymptotically, in contrast to the usual retarded van der Waals potentials, which fall off as − 1 / z 4 . This affects the quantum reflection amplitudes and causes the apparent space shifts, which express the time involved in the quantum reflection process, to diverge to large negative values at threshold.

WKB wave functions are expected to be accurate approximations of exact quantum mechanical wave functions mainly near the semiclassical limit of the quantum mechanical Schrodinger equation. The accuracy of WKB wave functions is, however, a local property of the Schrodinger equation, and the failure of the WKB approximation may be restricted to a small “quantal region” of coordinate space, even under conditions which are far from the semiclassical limit and close to the anticlassical or extreme quantum limit of the Schrodinger equation. In many physically important situations, exact or highly accurate approximate wave functions are available for the quantal region where the WKB approximation breaks down, and together with WKB wave functions in residual space provide highly accurate solutions of the full problem. WKB wave functions can thus be used to derive exact or highly accurate quantum mechanical results, even far from the semiclassical limit. We present a wide range of applications, including the derivation of properties of bound and continuum states near the threshold of a potential, which are important for understanding many results observed in experiments with cold atoms.

When potential energy curves for the relative motion of two atoms in different electronic states cross, the probability for diabatic or adiabatic passage can be described by the Landau–Zener formula. For near-threshold energies, close to the asymptotic value of one of the curves, the corresponding velocity is very small at large separations. Then quantum reflection, which can prevent the atoms from coming close to the avoided crossing, becomes a dominating effect. The probabilities for transmission to smaller separations—in either electronic state—thus vanish towards threshold, a feature not contained in conventional Landau–Zener theory. We propose a correction which takes the effect of quantum reflection into account so that the correspondingly modified Landau–Zener formula remains accurate at low energies, all the way down to threshold.

Near-threshold properties of bound and continuum states in a deep potential with an attractive tail depend essentially on a few `tail parameters', which are determined by the properties of the potential tail beyond the region of r-values where WKB wavefunctions are accurate solutions of the Schrodinger equation. One of these tail parameters is a length parameter which defines the singular contribution to the level density just below threshold and the reflectivity of the tail of the potential just above threshold; another is a phase difference which, together with the length parameter, determines the mean scattering length. The near-threshold quantization rule and the actual scattering length are determined by the tail parameters together with a dimensionless constant depending on the zero-energy value of the WKB action integral. We study potentials with tails consisting of two inverse-power terms, V(r)~-Cα/rα-Cα1/rα1,α1>α>2 and we derive exact analytical expressions for the tail parameters in the special case α1 = 2(α-1). This enables us to demonstrate the effect of a significant non-homogeneity of the potential tail on the results derived previously for homogeneous tails.

The classical deflection function for scattering by a radially symmetric potential corresponds to twice the angular momentum derivative of the scattering phase shifts calculated in the conventional WKB approximation. A ``quantum-mechanical deflection function'' defined as twice the angular momentum derivative of the quantum-mechanical phase shifts (without WKB approximation) is able to incorporate into the simple trajectory picture certain quantum aspects of the scattering process, e.g., the influence of the Goos-H\"anchen shift. We demonstrate the performance of this quantum-mechanical deflection function for sharp-edged and smooth-edged two-dimensional scattering potentials.

For a weakly attractive inverse-square potential, $V(x)=\ensuremath{-}g{\ensuremath{\Elzxh}}^{2}{/(2\mathrm{mx}}^{2})$ with $0lgl~1/4,$ the standard WKB wave function shows unphysical divergence near the origin. Introducing an appropriate nonvanishing reference point and a related phase yields WKB wave functions whose deviation from the regular solution of the Schr\"odinger equation decreases asymptotically as ${1/(\mathrm{kx})}^{3}.$ This is two orders better than the alternative technique involving the Langer modification of the potential. The performance of the correspondingly modified quantization conditions is demonstrated for the bound states of vanishing angular momentum in the two-dimensional circle billiard and in a two-dimensional Woods-Saxon well.

Generalized WKB connection formulas are used to derive the transmission amplitude describing tunneling through a potential barrier via two isolated classical turning points. The resulting formulas correctly reproduce the behavior in the vicinity of the base of the barrier where the tunneling probability vanishes exactly, and where formulas available to date fail.

Resonant ${J}^{\ensuremath{\pi}}{=1}^{\ensuremath{-}}$ states of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ below the $N=2$ threshold are studied in a semiempirical coupled-channel calculation, in which fine structure and QED corrections are included exactly. This leads to the termination of the dipole series of Feshbach resonances and to increasing mixing between different ${}^{2S+1}L$ states toward the end of the series. We predict a total of three resonances in the series dominantly of ${}^{1}{P}^{o}$ nature, and four resonances in the series dominantly of ${}^{3}{P}^{o}$ nature. The highest state of the latter series contains a 17% admixture of ${}^{1}{P}^{o}$ configurations, so one-photon excitation of this state from the ${}^{1}{S}^{e}$ ground state is far from forbidden.

We present a modified Bohr-Sommerfeld quantization rule for calculating energy levels in a molecular potential; it is based on the appropriate consideration of the reflection phase at the outer turning point. When the attractive tail of the potential is given by an inverse power law with a power greater than two, the reflection phase is accurately approximated by a simple algebraic formula. In an example defined by a 12-6 Lennard-Jones potential, this simple modified quantization rule gives highly accurate energy levels in the whole spectrum; the error relative to the level spacing is uniformly near . Very close to threshold, where the conventional WKB expansion breaks down completely, our simple formula can be supplemented by an accurate `effective-range' expansion. This yields energies for the most weakly bound level with residual errors which are smaller than the level spacing to the second most weakly bound level by seven powers of ten.

We compare WKB and exact wave functions for inverse power-law potentials and derive a universal algebraic formula reproducing reflection phases and scattering phase shifts with uniform high accuracy for any wave length. Appropriately modified quantization gives competitive accuracy for the spiked harmonic oscillator. The asymptotic accuracy of WKB wave functions is discussed.

We give a detailed analysis of the applicability of Hund's first rule for harmonic planar two-electron quantum dots by means of entanglement witnesses. We find that for purely harmonic confinement there is only one pair of singlet and triplet states for which it can be applied. We also discuss the origin and validity for this case and extend the discussion to a quartic confining potential, a hard-wall potential, and a combination of harmonic confinement with quartic perturbation. A generalized rule, the alternating rule, is found to be applicable and valid for vanishing angular momentum states in all these cases. Furthermore, we are able to clarify the role of entanglement in general harmonic two-electron models for vanishing interaction strength. This behavior can be attributed to the special separability properties of these models.

The phase loss of a wave reflected by a smooth potential generally varies continuously from \ensuremath{\pi} in the long-wave limit to \ensuremath{\pi}/2 in the limit of short waves. Incorporating the corresponding nonintegral multiples of \ensuremath{\pi}/2 as nonintegral Maslov indices in the formulation of the WKB approximation leads to a substantial improvement of accuracy when the conditions for applicability of the WKB method are violated only near the classical turning points. We demonstrate the efficacy of using nonintegral Maslov indices for a Woods-Saxon potential and a repulsive 1/${\mathit{x}}^{2}$ potential. The nonintegral Maslov index for a given 1/${\mathit{x}}^{2}$ potential yields far more accurate wave functions than the conventional Langer modification of the potential in conjunction with phase loss \ensuremath{\pi}/2. The energy spectrum of the radial harmonic oscillator (including the centrifugal potential), which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification when the nonintegral Maslov index is used. We suggest a method for approximately calculating the nonintegral Maslov index near the long-wave limit from the decaying WKB wave function in the classically forbidden region. \textcopyright{} 1996 The American Physical Society.

This chapter reviews the classical theory describing the scattering of a projectile particle from a target particle. Important concepts such as cross sections can already be explained in this context. Some results will be used later in the discussion of the relation between classical mechanics and quantum mechanics and of the behaviour of quantum systems in the “semiclassical” or “anticlassical” limits. These are subjects of lasting interest.

This chapter contains a detailed formulation of the quantum mechanical description of elastic scattering by a conservative potential. Particular attention is given to the influence of the large-distance behaviour of the interaction potential, which is often well known in realistic situations. The concept of channels, which is related to internal degrees of freedom of the projectile and/or the target, is introduced for the example internal orbital angular momenta and spins.

We study low-energy scattering of a neutral atom by a perfectly conducting cylindrical nanowire. Based on the exact atom-wire potential given recently by Eberlein and Zietal [ Phys. Rev. A 80 012504 (2009)] we derive tractable expressions for both the nonretarded van der Waals limit and the highly retarded limit as well as a realistic description of the transition between these two limits. For real atoms and wire radii greater than a few atomic units, the nonretarded van der Waals limit is insufficient, whereas the highly retarded limit of the atom-wire potential gives an accurate description of low-energy elastic scattering and absorption cross sections.

We give a simple description of quantum states near the dissociation threshold of deep interatomic potentials. The influence of the potential tail is separated from effects due to the interaction at short distances. We present a general formalism which is valid for weakly bound and low-energy continuum states, both for vanishing and non-vanishing angular momentum. Its applicability is demonstrated for the example of the electronic ground state of the strontium dimer (88)Sr(2). With an appropriate choice of the potential tail that includes higher-order dispersion coefficients, all short-range effects are incorporated via the threshold quantum number ν(D) and one further parameter accounting for residual short-range effects.

In a recent work, we derived a formula for the elastic $s$-wave phase shifts for deep potentials with attractive tails vanishing faster than $1/{r}^{2}$ asymptotically [Phys. Rev. A 84, 032701 (2011)]. As an addendum to that work, we reformulate its key result, i.e., the formula for the scattering phase shift ${\ensuremath{\delta}}_{0}$, in such a way that the contributions independent of the short-range influence of the full potential are explicitly expressed in terms of the amplitudes for transmission through and quantum reflection by the nonclassical region of the potential tail.

For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the $s$-wave phase shift ${\ensuremath{\delta}}_{0}$. It exposes a universal dependence of ${\ensuremath{\delta}}_{0}(E)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length $a$, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number ${n}_{\mathrm{th}}$. The formula accurately reproduces ${\ensuremath{\delta}}_{0}(E)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\ensuremath{\delta}}_{0}(E)$ and demonstrate the validity of the formula for attractive potential tails proportional to $1/{r}^{6}$ or to $1/{r}^{4}$, and also for a mixed potential tail consisting of a $1/{r}^{4}$ term together with a non-negligible $1/{r}^{6}$ contribution.

For potential wells with long-range attractive tails proportional to $\ensuremath{-}1/{r}^{3}$, as occur in the resonant dipole-dipole interaction in homonuclear alkali-metal dimers, we present a highly accurate analytical expression for the tail contribution to the quantization function $F(E)$. This quantization function determines the near-threshold bound-state energies via the quantization rule ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$. The performance of the quantization function derived in this paper is demonstrated by applying it to a model Lennard-Jones potential and to vibrational bound-state spectra of sodium dimers (Na${}_{2}$). These results are compared with those obtained via the semiclassical LeRoy-Bernstein formula which neglects quantum effects that are important in the near-threshold regime.

We study the scattering of a polarizable atom by a conducting cylindrical wire with incoming boundary conditions, that is, total absorption, near the surface of the wire. Based on the explicit expression given recently [C. Eberlein and R. Zietal, Phys. Rev. A75, 032516 (2007)] for the nonretarded atom-wire potential, we formulate a hierarchy of approximations that enables the numerical determination of this potential to any desired accuracy as economically as possible. We calculate the complex s-wave scattering length for the effectively two-dimensional atom-wire scattering problem. The scattering length a depends on the radius R of the wire and a characteristic length beta related to the polarizability of the atom via a simple scaling relation, a = R (a) over tilde(beta/R). The "scaled scattering length" (a) over tilde tends to unity in the thick-wire limit beta/R -> 0, and it grows almost proportional to 1/R in the opposite thin-wire limit.

In the early decades of this century physics was revolutionised by the advent of quantum mechanics. Observed atomic spectra could only be explained if the seemingly unshakeable concept of a particle moving along a path (trajectory), sharply defined and uniquely determined by the initial position and momentum coordinates, was dropped in favour of a description of electrons as standing waves or wave packets, whose position and momentum carry inherent uncertainties. In the course of the quantum revolution, generations of physicists were taught that the concept of classical paths is wrong on the atomic scale.

As a contribution to a current controversial discussion we show that the smoothness properties of the quantum-mechanical and the WKB phase shifts for elastic scattering by a smooth repulsive radially symmetric potential in two spatial dimensions are different.

A Comment on the Letter by Holger R. Dullin and Holger Waalkens, Phys. Rev. Lett. 101, 070405 (2008). The authors of the Letter offer a Reply.