Dynamical Symmetries of the 2D Newtonian Free Fall Problem Revisited

Among the few exactly solvable problems in theoretical physics, the 2D (two-dimensional) Newtonian free fall problem in Euclidean space is perhaps the least known as compared to the harmonic oscillator or the Kepler–Coulomb problems. The aim of this article is to revisit this problem at the classical level as well as the quantum level, with a focus on its dynamical symmetries. We show how these dynamical symmetries arise as a special limit of the dynamical symmetries of the Kepler–Coulomb problem, and how a connection to the quartic anharmonic oscillator problem, a long-standing unsolved problem in quantum mechanics, can be established. To this end, we construct the Hilbert space of states with free boundary conditions as a space of square integrable functions that have a special functional integral representation. In this functional space, the free fall dynamical symmetry algebra is shown to be isomorphic to the so-called Klink’s algebra of the quantum quartic anharmonic oscillator problem. Furthermore, this connection entails a remarkable integral identity for the quantum quartic anharmonic oscillator eigenfunctions, which implies that these eigenfunctions are in fact zonal functions of an underlying symmetry group representation. Thus, an appropriate representation theory for the 2D Newtonian free fall quantum symmetry group may potentially open the way to exactly solving the difficult quantization problem of the quartic anharmonic oscillator. Finally, the initial value problem of the acoustic Klein–Gordon equation for wave propagation in a sound duct with a varying circular section is solved as an illustration of the techniques developed here.

Mathieu-state reordering in periodic thermodynamics

A periodically driven, moderately anharmonic oscillator constitutes an ideal model system for investigating quantum resonances, which are amenable to a quantum pendulum approximation. In the present paper, I study the quasi-stationary Floquet-state occupation probabilities which emerge when such a resonantly driven system is coupled to a heat bath. It is demonstrated that the Floquet state which is associated with the ground state of the pendulum turns into an effective ground state, carrying the highest population in the strong-driving regime. Moreover, the population of this effective Floquet ground state can even exceed that of the undriven system’s true ground state at the same bath temperature. These effects can be optimized by suitably engineering the properties of the bath.

Harmonic and Anharmonic Oscillator Models for Pure Vibrational Infrared Spectroscopy of Diatomic Molecules

In molecular vibrational infrared spectroscopy, absorption spectra arise from molecular vibration and correspond to transitions between the vibrational energy levels associated with a given electronic state of the molecule. The vibrational transitions, which fall in the near infrared region, are induced through the interaction of the molecular electric dipole with the electric vector of the electromagnetic radiation. The near infrared region extends roughly from 1?m to ?10?^2 ?m. The article explains the pure vibrational absorption spectra of diatomic molecules such as HCl, HBr, HI, CO, … etc. In order to explain the vibrational spectra, diatomic molecules are treated as harmonic oscillator and anharmonic oscillator. In the harmonic oscillator model, we get only one absorption band at the wavenumber value? ?_osc corresponding to frequency of oscillation?_osc while in the actual experimental data, there are many absorption bands corresponding to wave numbers slightly lesser than ? ?_osc, 2? ?_osc, 3? ?_osc, ……..The occurrence of these additional bandsis attributed to the selection rule ?v=±2, ±3, ±4, ……The additional bands are having lesser intensity and are called overtone bands.

Stable solvers for real-time Complex Langevin

This study explores the potential of modern implicit solvers for stochastic partial differential equations in the simulation of real-time complex Langevin dynamics. Not only do these methods offer asymptotic stability, rendering the issue of runaway solution moot, but they also allow us to simulate at comparatively large Langevin time steps, leading to lower computational cost. We compare different ways of regularizing the underlying path integral and estimate the errors introduced due to the finite Langevin time steps. Based on that insight, we implement benchmark (non-)thermal simulations of the quantum anharmonic oscillator on the canonical Schwinger-Keldysh contour of short real-time extent.