Classical Forbidden Region Research Articles

Hyperspherical Cluster Model for Bosons: Application to Sub-threshold Halo States in Helium Drops

To describe long-range behaviour of one particle removed from a few- or a many-body system, a hyperspherical cluster model has been developed. It has been applied to the ground and first excited states of helium drops with five, six, eight and ten atoms interacting via a two-body soft gaussian potential. Convergence of the hyperspherical cluster harmonics expansion is studied for binding energies, root-mean-squared radii and overlaps of the wave functions of two helium drops differing by one atom. It was shown that with increasing model space the functional form of such overlaps at large distances converges to the correct asymptotic behaviour. The asymptotic normalization coefficients that quantify the overlaps’ amplitudes in this region are calculated. It was also shown that in the first excited state one helium atom stays far apart from the rest forming a two-body molecule, or a halo. The probability of finding the halo atom in the classically-forbidden region of space depends on the definition of the latter and on the valence atom binding energy. The total norm of the overlap integrals, the spectroscopic factor, represents the number of partitions of a many-body state into a chosen state of the system with one particle removed. The spectroscopic factors have been calculated and their sum rules are discussed giving a further insight into the structure of helium drops.

Neutrons in the classically forbidden region in neutron-rich nucleus 68Ca * *Supported by China Scholarship Council (201906255002)

Neutrons tunneling to the classically forbidden (CF) region in the neutron-rich nucleus are investigated in the Skyrme Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) models. The definition of the CF region is examined in the HF model by using different single-particle potentials for the bound states. In the HFB model, the weakly bound and continuum states could also contribute to the neutrons in the CF region due to the pairing correlation. Their asymptotic wave functions are carefully calculated by the Green’s function method.

The quantum scattering time in a linear potential

In this paper, we calculate the quantum time delays for a neutral particle scattering off the Earth gravitational linear potential. The quantum time delays are obtained by subtracting the classical returning time (CRT) from the Wigner time, the dwell time and the Larmor time respectively. Different from the conventional definition, our Larmor time is defined by aligning the magnetic field along particle’s direction of motion, and this definition does give reasonable results for motions through a free region and a square barrier. It is worth noting that in the zero magnetic field limit, the Larmor time coincides well with the CRT, which is due to the special shape of linear barrier, and may have some relevance to the weak equivalence principle. It is also found that the classical forbidden region plays an essential role for the dwell time to match with the CRT, and the difference between the dwell and the phase times, i.e. the self-interference time delay, is barrier shape sensitive and clearly shows the peculiarity of the linear barrier. All the time delays are on the order of sub-millisecond and exhibit oscillating behaviors, signaling the self-interference of the scattering particle, and the oscillations become evident only when the de Broglie wavelength λ k = 2π/k is comparable to the characteristic length . If the time delay measurement is experimentally realizable, it can probe the quantum nature for particle scattering off the gravitational potential in the temporal domain.

Quantum Effects Dominating the Interatomic Coulombic Decay of an Extreme System.

LiHe is an extreme open-shell system. It is among the weakest bound systems known, and its mean interatomic distance extends dramatically into the classical forbidden region. Upon 1s → 2p excitation of He, interatomic Coulombic decay (ICD) takes place in which the electronically excited helium atom relaxes and transfers its excess energy to ionize the neighboring lithium atom. A substantial part of the decay is found to be to the dissociation continuum producing Li+ and He atoms. The distribution of the kinetic energy released by the ICD products is found to be highly oscillatory. Its analysis reveals that quantum phase shifts between the decaying states and the dissociating final states are controlling this ICD reaction. The semiclassical reflection principle, which commonly explains ICD reactions, fails. The process is expected to be amenable to experiment.

Convergent perturbation expansion of energy eigenfunctions on unperturbed basis states in classically-forbidden regions

We study properties of eigenfunctions of perturbed systems, given on the eigenbases of unperturbed, integrable systems. For a given pair of perturbed and unperturbed systems, with respect to the energy of each perturbed state, the unperturbed basis states can be divided into two groups: one in the classically-allowed region and the other in the classically-forbidden region; correspondingly, the eigenfunction of the perturbed state is also divided into two parts. In the semiclassical limit, it is shown that, making use of components of the eigenfunction in its classically-allowed region, its components in the classically-forbidden region can be written in the form of a convergent perturbation expansion, which is valid for all perturbation strengths.

Effective light polarization insensitive and omnidirectional properties of Si nanowire arrays developed on different crystallographic planes

In this paper, fabrication of vertical Si nanowire arrays (SiNWAs) by a facile metal assisted chemical etching approach on different crystallographic planes of Si has been reported. A very low specular reflectance (Rspec) of 0.04% and 0.03% has been achieved in the whole visible range for SiNWAs grown on Si(100) and Si(111) oriented substrates, respectively. High broadband enhancement has been detected for vertical SiNWAs due to multiple scattering paths inside the nanowire arrays. On the other hand, inclined nanowires showed a fascinating behavior at the longer wavelength regime, where light gets the longer path to reflect back-forth and ease to reflect back outward at normal incidence. Moreover, for [100] SiNWAs, transverse electric field component demonstrates the strong polarization insensitive properties at the expense of transverse magnetic field component with a minimum reflectance of <2% up to 1200 nm. The [100] SiNWAs demonstrates extraordinary omnidirectional properties at θB ≥ 58°. Theoretical validation of COMSOL with an effective medium approach reveal the effective dipole coupling and the presence of strong absorption modes for vertical SiNWs at a typical wavelength regime. The highly bound states of the particle tunneling through classical forbidden region shows a strong dependence on the gradient in the refractive index (mi) from 1 to 3.4. The high order scattering effect is observed at ∼520 cm−1 in a disordered optical medium. This novel finding of light localization properties for SiNWAs with different orientation gives a new route to support various photonic applications.

Imaging the He2 quantum halo state using a free electron laser

Quantum tunneling is a ubiquitous phenomenon in nature and crucial for many technological applications. It allows quantum particles to reach regions in space which are energetically not accessible according to classical mechanics. In this "tunneling region," the particle density is known to decay exponentially. This behavior is universal across all energy scales from nuclear physics to chemistry and solid state systems. Although typically only a small fraction of a particle wavefunction extends into the tunneling region, we present here an extreme quantum system: a gigantic molecule consisting of two helium atoms, with an 80% probability that its two nuclei will be found in this classical forbidden region. This circumstance allows us to directly image the exponentially decaying density of a tunneling particle, which we achieved for over two orders of magnitude. Imaging a tunneling particle shows one of the few features of our world that is truly universal: the probability to find one of the constituents of bound matter far away is never zero but decreases exponentially. The results were obtained by Coulomb explosion imaging using a free electron laser and furthermore yielded He2's binding energy of [Formula: see text] neV, which is in agreement with most recent calculations.

Relativistic features and time delay of laser-induced tunnel ionization

The electron dynamics in the classically forbidden region during relativistic tunnel-ionization is investigated. The classical forbidden region in the relativistic regime is identified by defining a gauge invariant total energy operator. Introducing position dependent energy levels inside the tunneling barrier, we demonstrate that the relativistic tunnel-ionization can be well described by a one-dimensional intuitive picture. This picture predicts that, in contrast to the well-known nonrelativistic regime, the ionized electron wave packet arises with a momentum shift along the laser's propagation direction. This is compatible with results from a strong field approximation calculation where the binding potential is assumed to be zero-ranged. Further, the tunneling time delay, stemming from Wigner's definition, is investigated for model configurations of tunneling and compared with results obtained from the exact propagator. By adapting Wigner's time delay definition to the ionization process, the tunneling time is investigated in the deep-tunneling and in the near-threshold-tunneling regimes. It is shown that while in the deep-tunneling regime signatures of the tunneling time delay are not measurable at remote distance, it is detectable, however, in the latter regime.

A Priori Estimation of the Resolvent on Approximation of Born-Oppenheimer

In this study, we estimate the resolvent of the two bodies Shrodinger operator perturbed by a potential of Coulombian type on Hilbert space when h tends to zero. Using the Feschbach method, we first distorted it and then reduced it to a diagonal matrix. We considered a case where two energy levels cross in the classical forbidden region. Under the assumption that the second energy level admits a non degenerate point well and virial conditions on the others levels, a good estimate of the resolvent were observed.

Classically-forbidden processes in photoabsorption spectra

We review two classically-forbidden processes that give rise to prominent structures in atomic photoabsorption spectra. In one, the electron undergoes classically-forbidden reflection above a potential-energy barrier. In the other, absorption occurs into a classically-forbidden region of the electron's motion.

Exact relativistic time evolution for a step potential barrier

We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth $2x_p$ inside the potential, where $x_p$ is the penetration length of the stationary solution. Beyond the characteristic distance $2x_p$, a Sommerfeld-type precursor travels along the potential at the speed of light, $c$. However, no spatial propagation of a main wavefront along the structure is observed. We also find a non-causal time evolution of the wavefront peak. The effect is only an apparent violation of Einstein causality.

Relativistic Thomas-Fermi calculations of finite nuclei including quantum corrections.

Relativistic Thomas-Fermi calculations for finite nuclei including quantum corrections up to second order in \ensuremath{\Elzxh}, i.e., Wigner-Kirkwood and exchange corrections, have been performed. A linear \ensuremath{\sigma}-\ensuremath{\omega} model is used, in case of exchange-corrected calculations extended by \ensuremath{\pi}-nucleon and tensor \ensuremath{\rho}-nucleon contributions. A detailed discussion of the outcome shows that the inclusion of quantum corrections improves the description of the nuclear surface and the classical forbidden region in comparison to the standard relativistic Thomas-Fermi model. Furthermore, special attention is devoted to the investigation of the spin-orbit interaction and the influence of the \ensuremath{\sigma}-meson mass on nuclear properties.

The momentum of scalar field and the wave function of the universe

Using the minisuperspace model we analyse the effect of the momentum of scalar field on the wave function of the universe. He find that the classical forbidden region is sensitive to the value of the momentum and that, under certain conditions, there exists a classical allowed region in the a ~ 0 vicinity. Also, the wave function depends on the value of the operator ordering factor p. Under the Vilenkin boundary condition, the distribution probablility of scalar field φ is found to be independent of φ and different from the result of neglecting the momentum. Different implications of the initial conditions of the inflationary universe follow and are compared with the classical version.

Quantum effects in semiclassical treatment of molecular reactions

By the use of the known Coulomb wavefunctions, “quantal” Coulomb streamlines are defined for the eikonal of the system. Comparisons of such streamlines with the classical Coulomb trajectories are made. Use of such “quantum” Coulomb streamlines is proposed for reaction systems containing the Coulomb interaction U. The derived equations of motion are of particular use for cases where the residual potential V 1 ( V 1 = V − U) has either no classical forbidden region or has relatively small effects on trajectories.