Group algebra, convolution algebra, and applications to quantum mechanics.
Hamiltonian symmetry group role in quantum theory in stationary states classification, transition probabilities and selection rules
Loewdin, P.-O.↗
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Results for “Hamiltonian mechanics”
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Unitary representation of the Poincaré group for classical relativistic dynamics
We give a unitary irreducible representation of the proper Poincaré group that leads to an operational version of the classical relativistic dynamics of a massive spinless particle. Unlike quantum mechanics, in this operational theory there is no uncertainty principle between position and momentum. It will be shown that the theory contains the Koopman–von Neumann formalism as a particular case, and an explicit connection with relativistic Hamiltonian mechanics will be given.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Hamiltonian cosmology.
The study of cosmological models by means of equations of motion in Hamiltonian form is considered. Hamiltonian methods applied to gravity seem to go back to Rosenfeld (1930), who constructed a quantum-mechanical Hamiltonian for linearized general relativity theory. The first to notice that cosmologies provided a simple model in which to demonstrate features of Hamiltonian formulation was DeWitt (1967). Applications of the ADM formalism to homogeneous cosmologies are discussed together with applications of the Hamiltonian formulation, giving attention also to Bianchi-type universes. Problems involving the concept of superspace and techniques of quantization are investigated.
Ryan, M.↗
New treatment of the critical argument in resonance problems
Traditional methods for isolating the critical argument in Hamiltonian mechanics involve the introduction of a special set of action variables which are both complicated in form and arbitrary in nature. The new variables are in general obscure in their interpretation. This paper introduces a new canonical system corresponding to the original one, but with one more degree of freedom, in which all the variables of the original Hamiltonian can be treated on the same footing. It appears to have significant advantages for machine calculations.
Jefferys, W. H.↗
Electron Drift Kinetics in Space and in the Lab [Slides]
In a magnetized plasma, dynamics are fundamentally split between the directions parallel and perpendicular to the magnetic field. Generally, gyration about the magnetic field will occur on smaller spatial and time scales than the bulk dynamics of the system. From Hamiltonian mechanics, a periodic motion on a time scale much shorter than the other system dynamics leads to an adiabatic invariant. The adiabatic invariant for this motion is the magnetic moment $\mu=\frac{mv⟂^2}{2B}$, which will be approximately conserved.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Canonical orbital elements in terms of an arbitrary independent variable
Within the framework of the Hamiltonian mechanics in the extended phase space, a set of canonical elements of the Delaunay type is developed in terms of an arbitary independent angular variable. Application to the four classical anomalies - eccentric, true, elliptic, and mean - is presented. Particular attention is given to the generalized time equation and its conjugate energy equation.
Bond, V. R.↗
Proper-time relativistic dynamics
Proper-time relativistic single-particle classical Hamiltonian mechanics is formulated using a transformation from observer time to system proper time which is a canonical contact transformation on extended phase space. It is shown that interaction induces a change in the symmetry structure of the system which can be analyzed in terms of a Lie-isotopic deformation of the algebra of observables.
Gill, Tepper L.↗
Microwave, Millimeter, Submillimeter, and Far Infrared Spectral Databases
The spectrum of most known astrophysical molecules is derived from transitions between a few hundred to a few hundred thousand energy levels populated at room temperature. In the microwave and millimeter wave regions. spectroscopy is almost always performed with traditional microwave techniques. In the submillimeter and far infrared microwave technique becomes progressively more technologically challenging and infrared techniques become more widely employed as the wavelength gets shorter. Infrared techniques are typically one to two orders of magnitude less precise but they do generate all the strong features in the spectrum. With microwave technique, it is generally impossible and rarely necessary to measure every single transition of a molecular species, so careful fitting of quantum mechanical Hamiltonians to the transitions measured are required to produce the complete spectral picture of the molecule required by astronomers. The fitting process produces the most precise data possible and is required in the interpret heterodyne observations. The drawback of traditional microwave technique is that precise knowledge of the band origins of low lying excited states is rarely gained. The fitting of data interpolates well for the range of quantum numbers where there is laboratory data, but extrapolation is almost never precise. The majority of high resolution spectroscopic data is millimeter or longer in wavelength and a very limited number of molecules have ever been studied with microwave techniques at wavelengths shorter than 0.3 millimeters. The situation with infrared technique is similarly dire in the submillimeter and far infrared because the black body sources used are competing with a very significant thermal background making the signal to noise poor. Regardless of the technique used the data must be archived in a way useful for the interpretation of observations.
Pearson, J. C.↗
Hamilton's Equations with Euler Parameters for Rigid Body Dynamics Modeling
A combination of Euler parameter kinematics and Hamiltonian mechanics provides a rigid body dynamics model well suited for use in strongly nonlinear problems involving arbitrarily large rotations. The model is unconstrained, free of singularities, includes a general potential energy function and a minimum set of momentum variables, and takes an explicit state space form convenient for numerical implementation. The general formulation may be specialized to address particular applications, as illustrated in several three dimensional example problems.
Shivarama, Ravishankar↗
Dirac's generalized hamiltonian dynamics and the pontryagin principle fifth semiannual report
Dirac-Hamiltonian dynamics - celestial mechanics
CELESTIAL MECHANICS↗
Deep learning of dynamically responsive chemical Hamiltonians with semiempirical quantum mechanics
Conventional machine-learning (ML) models in computational chemistry learn to directly predict molecular properties using quantum chemistry only for reference data. While these heuristic ML methods show quantum-level accuracy with speeds several orders of magnitude faster than traditional quantum chemistry methods, they suffer from poor extensibility and transferability; i.e., their accuracy degrades on large or new chemical systems. Incorporating quantum chemistry frameworks into the ML models directly solves this problem. Here we take the structure of semiempirical quantum mechanics (SEQM) methods to construct dynamically responsive Hamiltonians. SEQM methods use empirical parameters fitted to experimental properties to construct reduced-order Hamiltonians, facilitating much faster calculations than ab initio methods but with compromised accuracy. By replacing these static parameters with machine-learned dynamic values inferred from the local environment, we greatly improve the accuracy of the SEQM methods. Trained on molecular energies and atomic forces, these dynamically generated Hamiltonian parameters show a strong correlation with atomic hybridization and bonding. Trained with only about 60,000 small organic molecular conformers, the resulting model retains interpretability, extensibility, and transferability when testing on much larger chemical systems and predicting various molecular properties. Overall, this work demonstrates the virtues of incorporating physics-based descriptions with ML to develop models that are simultaneously accurate, transferable, and interpretable.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Flux-driven algebraic damping of m = 2 diocotron mode
Experiments with pure electron plasmas in a Malmberg–Penning trap have observed linear in time, algebraic damping of m = 2 diocotron modes. Transport due to small field asymmetries produces a low-density halo of electrons moving radially outward from the plasma core, and the mode damping begins when the halo reaches the resonant radius of the mode. The damping rate is proportional to the flux of halo particles through the resonant layer. The damping is related to, but distinct from spatial Landau damping in which a linear wave–particle resonance produces exponential damping. This paper reports an analytic theory that captures the main signatures reported for this novel damping, namely, that the damping begins when the halo particles reach the resonant radius and that the damping is algebraic in time with nearly constant damping rate. The model also predicts a nonlinear frequency shift. The model provides two ways to think about the damping. It results from a transfer of canonical angular momentum from the mode to halo particles being swept by the mode field through the nonlinear cat's eye orbits of the resonant region. More mechanistically, the electric field produced by the perturbed charge density of the resonant particles acts back on the plasma core causing E × B drift that gives rise to the damping and nonlinear frequency shift.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Rotational pumping revisited
This paper considers a pure electron plasma, with a small admixture of negative ions, confined in a Penning–Malmberg trap. When a diocotron mode is excited on the plasma, the end sheaths of the plasma are azimuthally distorted. During reflection at a distorted end sheath, an ion steps off of the surface characterizing the drift motion in the plasma interior, and this step produces transport. The diocotron mode transfers canonical angular momentum to the ions, and in response damps. These transport mechanism and associated damping are called rotational pumping. It is particularly strong when the axial bounce motion and the rotational drift motion, in the rotating frame of the mode, satisfy a resonance condition. This paper calculates the transport flux of ions and the associated damping rate of the mode in the resonant regime. Previous papers have discussed the theory and the experimental observation of rotational pumping for the special case of a diocotron mode with azimuthal wave number l = 1, and this paper extends the theory to modes with l≠1, which may sound like a trivial extension, but in fact is not.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Analysis of the classical trajectory treatment of photon dynamics for polaritonic phenomena
Simulating photon dynamics in strong light–matter coupling situations via classical trajectories is proving to be powerful and practical. Here, in this work, we analyze the performance of the approach through the lens of the exact factorization approach. Since the exact factorization enables a rigorous definition of the potentials driving the photonic motion, it allows us to identify that the underestimation of photon number and intensities observed in earlier work is primarily due to an inadequate accounting of light–matter correlation in the classical Ehrenfest force rather than errors from treating the photons quasiclassically per se. The latter becomes problematic when the number of photons per mode begins to exceed a half.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Geometric adiabatic angle in anisotropic oscillators
We discuss a classical anisotropic oscillator and the Foucault pendulum as examples illustrating non-conservation of action variables in integrable classical mechanical systems with adiabatically slow evolution. We also emphasize the importance of the mass parameter of a harmonic oscillator, alongside its frequency, in explicitly time-dependent situations.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
One-center and Two-center Expansions of the Breit-pauli Hamiltonian
One and two center expansions for orbit-orbit, spin-spin, and spin-orbit Hamiltonians
QUANTUM MECHANICS↗
Modified long-period behavior due to tesseral harmonics.
Von Zeipel method and Hamiltonian perturbation mechanics used for orbits at resonance with tesseral harmonics of geopotential
GEOPOTENTIAL↗
Local analysis of fast magnetic reconnection
Fast magnetic reconnection is defined by the topology of the magnetic field lines changing on a timescale that is approximately an order of magnitude longer than the topology-conserving ideal-evolution timescale. Fast reconnection is an intrinsic property of Faraday's law when the evolving magnetic field depends non-trivially on all three spatial coordinates and is commonly observed—even when the effects that allow topology breaking are arbitrarily small. The associated current density need only be enhanced by a factor of approximately ten and flows in thin but broad ribbons along the magnetic field. These results follow from the variation in the separation of neighboring pairs of magnetic field lines, which in an ideal evolution typically increases exponentially with time, and the existence of a spatial scale below which magnetic field lines freely change their identities due to non-ideal effects, such as resistivity. Traditional reconnection theory ignores exponentially large variations and relies on the current density reaching a magnitude that is exponentially larger than is actually required. Here, an analysis of the behavior of magnetic field lines in the neighborhood of an arbitrarily chosen line is used to obtain more precise and rigorous results on intrinsic reconnection. The maximum parallel kinetic energy of collisionless charged particles is shown to have an exponential increase in time during a generic magnetic evolution.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗