Some Chapters

Prof. Robert Gilmore
Physics Department
The following set of isolated chapters were written by the author as an antidote for lack of similar presentations that may be readily available elsewhere. Copyright 2004 by Robert Gilmore

[PDF] Dimensional Analysis

Dimensional Analysis is presented in a matrix formulation. The exponents of products of quantities with dimensions are treated as vectors in a linear vector space. Once a choice of basis vectors is made, all exponent vectors can be expressed as linear combinations of the basis vectors. Null vectors describe products of dimensional quantities that are dimension free. Examples are given and include the hydrogen atom, the Planck scales, fluids, and some of the standard dimensionless ratios.

[PDF] Lagrange Multipliers

Lagrange multipliers provide a convenient way to treat constrained optimization problems. This discussion introduces the ideas behind their use and illustrates their use with a large variety of examples.

[PDF] Time-Independent Perturbation Theory

Time independent perturbation theory is formulated in terms of 3 x 3 matrices. The expansion of the eigenvalues and eigenvectors to third order in a smallness parameter is directly generalized to the correct form of perturbation theory to third order for any number of states.

[PDF] Time-Dependent Perturbation Theory

Time-Dependent perturbation theory is formulated in a systematic way in terms of an iterated expansion of the perturbing time-dependent hamiltonian. The expansion is carried out in the spirit of Dyson.

[PDF] The Structure of Thermodynamics

Classical equilibrium thermodynamics is presented in a geometric framework. This provides a useful bookkeeping mechanism for treating the extensive and dual intensive variables. The First Law appears as a differential relation. The Second Law appears as a condition on the curvature of the equilibrium surface. The Third Law is a boundary condition. A simple linear vector space algorithm is presented for computing any thermodynamic partial derivative. The LeChatelier reciprocal relations are derived and stated in simple, elegant form. Fluctuation moments are expressed in terms of the macroscopic static linear susceptibility coefficients. Relaxation of perturbations to the equilibrium surface occur with time constants determined by the kinetic coefficients and the equilibrium susceptibilities. Fluctuation moments and kinetic coefficients are related by the fluctuation-dissipation theorem.

[PDF] Lie Groups: general theory

This summary of the most important aspects of Lie group theory consists of the following sections 1. Introduction
2. Lie Groups
3. Matrix Lie groups
4. Linearization of a Lie Group
5. Matrix Lie algebras
6. Lie algebra tools
7. Structure of Lie algebras
8. Structure of semisimple Lie algebras
9. Root spaces
10. Canonical commutation relations
11. Dynkin diagrams
12. Real forms
13. Riemannian symmetric spaces
14. Summary

[PDF] Quadruply discrete classification for Low Dimensional Chaos

It is finally possible to classify low-dimensional strange attractors --- strange attractors with Lyapunov dimension $d_L < 3$. There are four levels of structure in this classification: (1) basis sets of orbits; (2) branched manifolds; (3) bounding tori; and (4) embeddings into $R^3$. All four levels involve links opf knots in very powerful ways. We describe these four levels of structure. There is an incomplete understanding in several levels of this organizational hierarchy. We describe how singularities form the backbone of stretching and squeezing processes that generate chaotic behavior. We ask: What is invariant about topological analysis? We conclude with a brief description of all the covers of a universal image dynamical system --- in this case the horseshoe.

[PDF] Quadruply discrete classification for Low Dimensional Chaos

Chaos is a type of behavior that can be exhibited by a large class of physical systems and mathematical models of them. These systems are deterministic. They are modeled by sets of coupled nonlinear ordinary differential equations. This review article describes the four levels of structure that have been found necessary and useful for describing low dimensional dissipative dynamical systems: these are systems that can be embedded in three dimensional phase spaces.

[PDF] Finite Element Methods for Quantum Mechanics

A simple walk-through of the finite element method using triangular tesselations of a planar region is provided.

[PDF] Ehrenfest Theorems

These theorems indicate the close relationship between Classical Mechanics and expectation values of corresponding operators that are computed in Quantum Mechanics.

[PDF] Modern Matrix Mechanics

This study shows how a physicist thinks about how the Finite Element Method can be used to transform Schrodinger's Wave Mechanics from its action representation into a matrix eigenvalue equation.

[PDF] Quantum Mechanics on Curves \& Surfaces

In order to describe the quantum mechanics of a particle confined to a curve or a surface in $R^3$ Schr\"odinger's equation must be modified in two ways. The kinetic energy operator must be written in terms of the metric $g^{ij}$ induced on the surface from the flat space metric $G_{ij}=\delta_{ij}$. The potential energy term must be modified by the inclusion of a geometric potential, necessary to constrain the particle to the curve or surface. These two modifications involve the two Fundamental Forms introduced by Gauss a century before Quantum Mechanics was developed.

Last update: 10 December, 2008