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Some Chapters

## Prof.
Robert Gilmore

Physics Department

Email: robert.gilmore@drexel.edu

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**

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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**

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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**

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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**

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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**

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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
**

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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
**

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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
**

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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