Corrections to Wigner type phase space methods

Over decades, the time evolution of Wigner functions along classical Hamiltonian flows has been used for approximating key signatures of molecular quantum systems. Such approximations are for example the Wigner phase space method, the linearized semiclassical initial value representation, or the statistical quasiclassical method. The mathematical backbone of these approximations is Egorov's theorem. In this paper, we reformulate the well-known second order correction to Egorov's theorem as a system of ordinary differential equations and derive an algorithm with improved asymptotic accuracy for the computation of expectation values. For models with easily evaluated higher order derivatives of the classical Hamiltonian, the new algorithm's corrections are computationally less expensive than the leading order Wigner method. Numerical test calculations for a two-dimensional torsional system confirm the theoretical accuracy and efficiency of the new method

Discretising the Herman--Kluk Propagator

The Herman--Kluk propagator is a popular semi-classical approximation of the unitary evolution operator in quantum molecular dynamics. In this paper we formulate the Herman--Kluk propagator as a phase space integral and discretise it by Monte Carlo and quasi-Monte Carlo quadrature. Then, we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with Fourier integral operator calculus. Numerical experiments for two- and six-dimensional model systems support our theoretical results

On the higher derivates of arctan

WWe give a rational closed form expression for the higher derivatives of the inverse tangent function and discuss its relation to Chebyshev polynomials, trigonometric expansions and Appell sequences of polynomials.Comment: 7 page

Quasi-classical description of molecular dynamics based on Egorov's theorem

Egorov's theorem on the classical propagation of quantum observables is related to prominent quasi-classical descriptions of quantum molecuar dynamics as the linearized semiclassical initial value representation (LSC-IVR), the Wigner phase space method or the statistical quasiclassical method. The error estimates show that different accuracies are achievable for the computation of expectation values and position densities. Numerical experiments for a Morse model of diatomic iodine and confined Henon-Heiles systems in various dimensions illustrate the theoretical results.Comment: 14 pages, 12 figure

Symmetric Kronecker products and semiclassical wave packets

We investigate the iterated Kronecker product of a square matrix with itself and prove an invariance property for symmetric subspaces. This motivates the definition of an iterated symmetric Kronecker product and the derivation of an explicit formula for its action on vectors. We apply our result for describing a linear change in the matrix parametrization of semiclassical wave packets

A new Phase Space Density for Quantum Expectations

We introduce a new density for the representation of quantum states on phase space. It is constructed as a weighted difference of two smooth probability densities using the Husimi function and first-order Hermite spectrograms. In contrast to the Wigner function, it is accessible by sampling strategies for positive densities. In the semiclassical regime, the new density allows to approximate expectation values to second order with respect to the high frequency parameter and is thus more accurate than the uncorrected Husimi function. As an application, we combine the new phase space density with Egorov's theorem for the numerical simulation of time-evolved quantum expectations by an ensemble of classical trajectories. We present supporting numerical experiments in different settings and dimensions.Comment: 26 pages, 7 figure

Quantum expectations via spectrograms

We discuss a new phase space method for the computation of quantum expectation values in the high frequency regime. Instead of representing a wavefunction by its Wigner function, which typically attains negative values, we define a new phase space density by adding a first-order Hermite spectrogram term as a correction to the Husimi function. The new phase space density yields accurate approximations of the quantum expectation values as well as allows numerical sampling from non-negative densities. We illustrate the new method by numerical experiments in up to $128$ dimensions.Comment: 6 pages, 5 figure

Semiclassical resonances for a two-level Schr\"odinger operator with a conical intersection

We study the resonant set of a two-level Schr\"odinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr-Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.Comment: 45 pages 4 figure

Landau-Zener type surface hopping algorithms

A class of surface hopping algorithms is studied comparing two recent Landau-Zener (LZ) formulas for the probability of nonadiabatic transitions. One of the formulas requires a diabatic representation of the potential matrix while the other one depends only on the adiabatic potential energy surfaces. For each classical trajectory, the nonadiabatic transitions take place only when the surface gap attains a local minimum. Numerical experiments are performed with deterministically branching trajectories and with probabilistic surface hopping. The deterministic and the probabilistic approach confirm the good agreement of both the LZ probabilities as well the good approximation of the reference solution computed solving the Schroedinger equation via a grid based pseudo-spectral method. Visualizations of position expectations and superimposed surface hopping trajectories with reference position densities illustrate the effective dynamics of the investigated algorithms

A Gentle Introduction to Deep Learning in Medical Image Processing

This paper tries to give a gentle introduction to deep learning in medical image processing, proceeding from theoretical foundations to applications. We first discuss general reasons for the popularity of deep learning, including several major breakthroughs in computer science. Next, we start reviewing the fundamental basics of the perceptron and neural networks, along with some fundamental theory that is often omitted. Doing so allows us to understand the reasons for the rise of deep learning in many application domains. Obviously medical image processing is one of these areas which has been largely affected by this rapid progress, in particular in image detection and recognition, image segmentation, image registration, and computer-aided diagnosis. There are also recent trends in physical simulation, modelling, and reconstruction that have led to astonishing results. Yet, some of these approaches neglect prior knowledge and hence bear the risk of producing implausible results. These apparent weaknesses highlight current limitations of deep learning. However, we also briefly discuss promising approaches that might be able to resolve these problems in the future.Comment: Accepted by Journal of Medical Physics; Final Version after revie