Heisenberg Operator Approach: Nano-Mechanical Oscillator (Part 2) and the Quantum LC Circuit

With the knowledge of the new design rules in Chapter 7, we use this new insight to find the eigenvectors for the nano-vibrator problem, and then we use the same approach to examine the quantum LC circuit. While the usual approach is to use Kirchhoff’s laws to analyze a simple circuit classically, we first see that Hamilton’s equations can in fact be used, giving the same classical result. But then, using the new design rules and the knowledge of the total energy in the circuit, we identify a canonical coordinate and a conjugate momentum that have nothing to do with real space and motion of a particle of mass m. At the same time, consistent with the Schrödinger picture, we continue to see that the time evolution of an observable such as position, x(t), or current, i(t), is not part of the solution. Given that Hamilton’s equations give the same result as Kirchhoff’s law but the quantum solution does not, reinforces the idea that the quantum description is showing features that cannot be imagined with a viewpoint based on classical (i.e. non-quantum) analysis.

Observable and Unobservable Mechanical Motion

A thermodynamic approach to mechanical motion is presented, and it is shown that dissipation of energy is the key process through which mechanical motion becomes observable. By studying charged particles moving in conservative central force fields, it is shown that the process of radiation emission can be treated as a frictional process that withdraws mechanical energy from the moving particles and that dissipates the radiation energy in the environment. When the dissipation occurs inside natural (eye) or technical photon detectors, detection events are produced which form observational images of the underlying mechanical motion. As the individual events, in which radiation is emitted and detected, represent pieces of physical action that add onto the physical action associated with the mechanical motion itself, observation appears as a physical overhead that is burdened onto the mechanical motion. We show that such overheads are minimized by particles following Hamilton’s equations of motion. In this way, trajectories with minimum curvature are selected and dissipative processes connected with their observation are minimized. The minimum action principles which lie at the heart of Hamilton’s equations of motion thereby appear as principles of minimum energy dissipation and/or minimum information gain. Whereas these principles dominate the motion of single macroscopic particles, these principles become challenged in microscopic and intensely interacting multi-particle systems such as molecules moving inside macroscopic volumes of gas.

Tunings for leapfrog integration of Hamiltonian Monte Carlo for estimating genetic parameters

A Hamiltonian Monte Carlo algorithm is a Markov Chain Monte Carlo method that is considered more effective than the conventional Gibbs sampling method. Hamiltonian Monte Carlo is based on Hamiltonian dynamics, and it follows Hamilton’s equations, which are expressed as two differential equations. In the sampling process of Hamiltonian Monte Carlo, a numerical integration method called leapfrog integration is used to approximately solve Hamilton’s equations, and the integration is required to set the number of discrete time steps and the integration stepsize. These two parameters require some amount of tuning and calibration for effective sampling. In this study, we applied the Hamiltonian Monte Carlo method to animal breeding data and identified the optimal tunings of leapfrog integration for normal and inverse chi-square distributions. Then, using real pig data, we revealed the properties of the Hamiltonian Monte Carlo method with the optimal tuning by applying models including variance explained by pedigree information or genomic information. Compared with the Gibbs sampling method, the Hamiltonian Monte Carlo method had superior performance in both models. We have provided the source codes of this method written in the R language.

Hamilton’s Equations & Routhian Reduction

In this chapter, the Poisson bracket and angular momentum are investigated and first integrals are used to develop conservation laws as a canonical Noether’s theorem. The Poisson bracket was developed by the French mathematician Poisson in the late nineteenth century and it is a reformulation, or at least a tidying up, of Hamilton’s equations into one neat package. The Poisson bracket of a quantity with the Hamiltonian describes the time evolution of that quantity as one moves along a curve in phase space. The Lie algebra structure of symmetries in mechanics is highlighted using this formulation. The classical propagator is derived using the Poisson bracket.

A generalized action-angle representation of wave interaction in stratified shear flows

In this paper we express the linearized dynamics of interacting interfacial waves in stratified shear flows in the compact form of action-angle Hamilton’s equations. The pseudo-energy serves as the Hamiltonian of the system, the action coordinates are the contribution of the interfacial waves to the wave action and the angles are the phases of the interfacial waves. The term ‘generalized action angle’ aims to emphasize that the action of each wave is generally time dependent and this allows for instability. An attempt is made to relate this formalism to the action at a distance resonance instability mechanism between counter-propagating vorticity waves via the global conservations of pseudo-energy and pseudo-momentum.