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Quantum states can be described equivalently by density matrices, Wigner functions, or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance, and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.

This study analyzed the scar-like localization in the time-average of a time-evolving wavepacket on a desymmetrized stadium billiard. When a wavepacket is launched along the orbits, it emerges on classical unstable periodic orbits as a scar in stationary states. This localization along the periodic orbit is clarified through the semiclassical approximation. It essentially originates from the same mechanism of a scar in stationary states: piling up of the contribution from the classical actions of multiply repeated passes on a primitive periodic orbit. To achieve this, several states are required in the energy range determined by the initial wavepacket.

For the general $D$-dimensional radial anharmonic oscillator with potential $V(r)=\frac{1}{g^2}\widehat{V}(gr)$ the perturbation theory (PT) in powers of coupling constant $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) is developed constructively in $r$-space and in $(gr)$-space, respectively. The Riccati–Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate $r$-space and in $(gr)$-space, respectively, exploring the logarithmic derivative of wave function $y$. It is shown that PT in powers of $g$ developed in RB equation leads to Taylor expansion of $y$ at small $r$ while being developed in GB equation leads to a new form of semiclassical expansion at large $(gr)$: it coincides with loop expansion in path integral formalism. In complementary way PT for large $g$ developed in RB equation leads to an expansion of $y$ at large $r$ and developed in GB equation leads to an expansion at small $(gr)$. Interpolating all four expansions for $y$ leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at $r\in [0,\infty )$ for any coupling constant $g\ge 0$ and dimension $D>0$. As a concrete application, the low-lying states of the cubic anharmonic oscillator $V=r^2+g{r}^3$ are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is $\lesssim 10^{-4}$ for $r\in [0,\infty )$ for coupling constant $g\ge 0$ and dimension $D=1,2,\dots .$ In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7–8 s.d. for $g\ge 0$ and $D=1,2,\dots .$

In our previous paper I (del Valle–Turbiner, 2019) a formalism was developed to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)=\frac{1}{g^2}\widehat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As a result, the Approximant was introduced — a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations, it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper, the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials $V(r)=r^2+g^{2(m-1)}{r}^{2m}$, $m=2,3$, respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8–12 figures for any $D=1,2,3,\dots$ and $g\ge 0$, while the relative deviation of the Approximant from the exact eigenfunction is less than $10^{-6}$ for any $r\ge 0$.

In this paper, we discuss the effects of the topology of a disclination on quantum revivals when a quantum particle is subject to an infinite cylindrical well and a logarithmic potential. By using the WKB (Wentzel, Kramers and Brillouin) approximation, we obtain the eigenvalues of energy and show that the influence of the topology of the disclination on them gives rise to an Aharonov–Bohm-type effect. Further, we discuss how the topology of a disclination influences the classical periods and the revival times.

We consider the semiclassical model of an extended tight-binding Hamiltonian comprising nearest- and next-to-nearest-neighbor interactions for a charged particle hopping in a lattice in the presence of a static arbitrary field and a rapidly oscillating uniform field. The application of Kapitza’s method yields a time-independent effective Hamiltonian with long-range hopping elements that depend on the external static and oscillating fields. Our calculations show that the semiclassical approximation is quite reliable as it yields, for a homogeneous oscillating field, the same effective hopping elements as those derived within the quantum approach. Besides, by controlling the oscillating field, we can engineer the interactions so as to suppress the otherwise dominant interactions (nearest neighbors) and leave as observable effects those due to the otherwise remanent interactions (distant neighbors).

In this paper, we theoretically study the ballistic transport through a weakly open circular mesostructure within the framework of pseudopath semiclassical approximation. By defining the interference factor and comparing it numerically with the transmission amplitude, we demonstrate that the fluctuations in transmission amplitude are mainly due to the interference effects between different classical trajectories. In addition, we calculate the Fano factor versus cutoff length of classical trajectories in the energy domain to illustrate that the fluctuations in transmission amplitude mainly arise from the contribution of short-length classical trajectories. We further show that the transmission length power spectra not only associate with a lot of classical trajectories but with a series of nonclassical trajectories due to the diffraction scattering effects. Moreover, we show that the classical trajectories distributed in the mode–mode coupling function give the largest contribution to the corresponding transmission amplitude. We hope that our results and analysis can be used to reveal new effects of mesoscopic systems and to provide theoretical basis for the design of mesoscopic devices.

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

We analyze the interaction of the induced electric dipole moment of a neutral particle with an electric field in elastic medium with a charged disclination from a semiclassical point of view. We show that the interaction of the induced electric dipole moment of a neutral particle with an electric field can yield an attractive inverse-square potential, where it is influenced by the topology of the disclination. Then, by using the Wentzel, Kramers and Brillouin approximation based on the Langer transformation, we show that the centrifugal term of the radial equation must be modified due to the influence of the topology of the disclination. Besides, we obtain the bound states solutions to the Schrödinger equation.

To obtain a sufficiently rich class of nonlinear functionals of white noise, resp. the Wiener process, we study riggings of the L² space with the white noise measure. Particular examples are local functionals such as e.g. the ‘square of white noise’ and its exponential with applications in the theory of Feynman Integral.

This is a survey of our work in Ref. 1. We give a fairly general class of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of the phase space. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the Riemann integrals, the interchange of the order with a limit and the perturbation expansion formula hold in the phase space path integrals. The use of piecewise bicharacteristic paths naturally leads us to the semiclassical approximation on the phase space.

The particle flux redistribution in oriented crystals leads to arising of various orientation effects in the yields of processes connected to small impact parameters (nuclear reactions, knocking out δ-electrons etc). Orientation effects of that type in the incoherent bremsstrahlung were discussed earlier in several experimental and theoretical articles. In this paper we present some improved and new results obtained using numerical simulation.