The Mind of a Theorist - The Blog

Work of the last two weeks

2012-02-14T10:18:00.000-08:00

Hello everyone. I have been working hard on the book with Leonard Susskind over the last two weeks. We are about two weeks away from presenting the final manuscript to the publishers at basic Books. It is scheduled for release in January 2013.

Last week I published, on the MAST web site, a paper on the Oppenheimer-Volkoff equations. These can be found here.

During the Saturday meeting we had a lot of fun writing a program to calculate the arc length of a curve and then tested lots of different curves. This work can be found here.

New Material from MAST

2012-02-02T13:59:00.001-08:00

New Post on the MAST Web Site
We have several Mathematica-based documents on our web site.
http://www.madscitech.org/mathematica/consulting/consulting.html

Book Review: Advanced University Physics

2011-10-23T15:12:00.000-07:00

Time for another book review, this time Mircea Rogalski and Stuart Palmer, Advanced University Physics. This book is for sale in the book store below.

This is the second edition of a book that I long found useful as a reference, but not as a textbook as there were no practice problems. In the second edition, this has been fixed and there are now ample problems, so I can recommend it as a textbook.

This book is almost a thousand pages, not counting the index. It has 47 chapters, five appendices, and sections on problem hints, and answers.

The first chapter is pretty introductory, but very important for that; it begins by discussing the SI system of units then dimensional analysis; one of the more important tools for a theoretical physicist. The chapter ends with a presentation of fundamental physical constants.

The next subject covered is calssical mecahnics. There are three chapters on classical mechanics: 2, 3, and 24. I recommend saving chapter 24 for after you complete the chapters on relativity. Some of the practive problems should not be done until you complete chapter 14 on Maxwell's equations, but the reading only requires chapter 3. Chapter 2 begins with an overview of kinematics and dynamics, an example on planar motion that establishes the idea of rotational components of vectors, an example on constant forces, and a lengthy example that discusses the one-dimensional harmonic oscillatory in a fairly complete way. This is followed by sections on the conservation of momentum, and the conservation of energy. The chapter ends with 14 practice problems, for which 7 have complete and detailed solutions; the remainder having hints and solutions (true for all problems in the book). Chapter 3 picks up with the conservation of angular momentum. The main theme of chapter 3 is celestial mechanics beginning with motion under a central force, then Kepler's problem, and ending with Kepler's laws. Chapter 3 has 12 practice problems, seven having complete and detailed solutions. Chapter 24 begins with D'Alembert's principle and a discussion of constraints and virtual displacements. This is followed by Hamilton's principle, also called the principle of least action. Then Lagrange's equations are presented. Then Hamilton's equations are discussed along with a connection to the previous discussion of linear oscillators. The authors then present Poisson brackets. The chapter ends with the Hamilton-Jacobi equation and its extension to systems of particles. There are 11 practice problems of which 5 have complete and detailed solutions.

There are two chapters on relativity: 4 and 5. You might want to wait until you have finished chapter 15 on wave equations before starting these. Chapter 4 begins with the postulates of special relativity. My only objection to this chapter is that if you are not familiar with electromagnetic waves you will not be able to follow the justification given for the principles of relativity and the constancy of the speed of light. These postulates are then employed in the section of Lorentz transformations, where the transformations are derived from the point of view of general linear transformations; several side issues of great important are then discussed: Simultaneity, length contraction, time dilation, and the transformation of velocities. Then there is a nice discussion of spacetime diagrams and how to represent kinematics on a spacetime diagram. Chapter 4 ends with a very important presentation of the invariance of the spacetime interval, including an example calculation of the covariant electromagnetic wave equation. There are 7 practive problems of which 5 have complete and detailed solutions. Chapter 5 begins with relativistic momentum and energy including the notion of rest mass, there is an example of the photon as a particle of zero rest mass. Relativistic dynamics is the subject of the next section, introducing 4-vectors in the process, and using the mass-energy equivalence (E=m c^2) as an example. The chapter ends with a change towards gravitation with the equivalence principle where gravitational red shift is an example. There are 10 practice problems, 5 of which have detailed and complete solutions.

The next five chapters are on bulk matter: 6, 7, 8, 9, and 10. You can start these as soon as you finish chapter 3. You should be completely comfortable with partial derivatives and matrices before starting chapter 6. The chapter begins by defining strain as a tensor, and establishing volume expansions and the equation of continuity. The second section defines the stress tensor. The third section describe elasticity and introduces waves on a string. The final section of the chapter is a nice introduction to fluid mechanics. The chapter has thirteen practice problems of which 5 have complete and detailed solutions. Chapter 7 begins with the zeroth law of thermodynamics and the temperature scale. Then the first law of thermodynamics is presented along with a calculation of the heat capacity of a hystrostatic system. The chapter ends with the second law of thermodynamics and the Carnot cycle. Chapter 7 has ten practice problems, 5 of which have detailed and complete solutions. Chapter 8 begins with Clausius' theorem then moves on to entropy. This is followed by a discussion of thermodynamic potentials, including internal energy, the Helmholtz function, and the Gibbs function. These ideas are then applied to hydrostatic systems through the Gibbs-Helmholtz equation, the Maxwell relations, the reciprocity theorem, and the relation between the entropy and temperature. The chapter ends with a discussion of heat capacity. This chapter has 10 practice problems, 5 of which are solved in detail. Chapter 9 begins with the ideal gas laws, including the equation of state and the polytropic equation (of great use in astrophysics). The results of chapter 8 are then extended to ideal gases including the temperature-entropy diagram, energy transfer in polytropic processes (including adiabatic, isobaric, and isochoric processes), the internal energy, the enthalpy, and the Helholtz and Gibbs functions. The chapter ends with some "real" gases, the van der Waals and Dieterici gases, the equation of corresponding states, and the entropy of a van der Waals gas. This chapter has 10 practice problems, 5 with complete and detailed solutions.

The next four chapters form a nice introduction to electrodynamics. Chapter 11 begins with the electric field and it begins, reasonably enough, with Coulomb's law then moves on to Gauss' law. The next section introduces several important concepts beginning with the electrostatic potential, then the electric dipole, and then Laplace's equation. The authors then present a section on polarization leading to the permittivity. The chapter ends with a section on electrostatic energy. This chapter has 10 problems, 6 of which have complete and detailed solutions. Chapter 12 begins with the flow of current, Ohm's law, dielectrics and conductors, and Kirchhoff's rules for circuits. The next section covers magnetic induction and uniform magnetic fields, the magnetic induction is defined as what most authors call the magnetic field. The chapter ends with a discussion of the magnetic vector potential. The chapter has 10 practice problems, 5 of which have complete and detailed solutions. Chapter 13 begins its exploration of magnetic fields with magnetization and includes the Biot-Savart law, the magnetic dipole moment, and the magnetic field. The emphasis shifts to Faraday's law and inductance, introducing the Lorentz force (important for relativity). The chapter ends with energy in magnetic fields. The chapter has 10 practice problems, 5 solved in detail. Chapter 14 begins with Maxwell's equations both in vacuum and inside matter, along with a significant example of their application to circuit theory, and the introduction to the electromagnetic wave equation (thus forming the basis for the chapters on special relativity). The next section is on electromagnetic energy and culminates in the Poynting vector and a discussion of the momentum and energy of a charge in an electromagnetic field. The chapter ends with a section entitled Potential Equations that introduces the Coulomb and Lorentz Gauges. The chapter has 10 practice problems, 5 of which are solved in complete detail.

The next nine chapters form a unit on waves and optics. Chapter 15 begins with the wave equation, then applies that to elastic waves on a string, acoustical waves in fluids, and electromagnetic waves in isotropoc dielectrics. This chapter has 10 practice problems, 5 solved in complete detail. Chapter 16 begins with a discussion of harmonic waves deriving d'Alembert's formula and introducing such concepts as the local phase constant and the complex amplitude, then it tackles wave propagation in three dimensions, a derivation of Snell's law, stationary waves as an exercise in separation of variable solutions to differential equations, Fourier analysis is introduced and applied to the problem of continuous waves, and the chapter ends with dispersion. The chapter has 12 practice problems, 6 have detailed and complete solutions. Chapter 17 begins by studying the energy density of waves, the flow rate and characteristic impedences, the intensity of electromagnetic waves, the momentum of waves, electromagnetic radiation pressure, wave attenuation, and it ends with transmission and reflection at boundaries. This chapter has 10 problems, 5 of which have detailed and complete solutions. Chapter 18 begins with the interference of two monochromatic waves, Young's experiment, Newton's rings, and ends with interference with multiple beams. This chapter has 10 practice problems, 5 of which have detailed and complete solutions. Chapter 19 begins with Fresnel and Faunhofer diffraction of scalar waves, the superposition of spherical waves, the linear approximation of diffraction, single-slit diffraction, and the diffraction grating. This chapter has 10 practice problems, 5 of which have detailed and complete solutions. Chapter 20 begins with a treatment of the transverse nature of electromagnetic waves, then it goes into the intensity of electromagnetic waves, then polarization, the Jones vectors, and it ends with the Poincaré sphere and Pauli spin matrices. This chapter has 11 practice problems, 5 of which have detailed and complete solutions. Chapter 21 begins with electromagnetic waves at an interface culminating in Snell's law, then Fresnel's equation, polarization by reflection, and ending in total internal reflection. This chapter has 12 problems, 5 of which have detailed and complete solutions. Chapter 22 begins with a presentation of plane waves in anisotropic media, then the dielectric tensor for optical anisotropic media is presented, and the chapter ends with the ray direction, phase and ray velocities, and then the ray-velocity surface. This chapter has 10 practice problems, 5 of which have detailed and complete solutions. The final chapter of this group begins with the optical properties of conducting media, then the skin effect, the origin of the complex constutive parameters, ending with the optical properties of a plasma. This chapter has 10 practice problems, 5 of which have detailed and complete solutions.

The next set of three chapters form a good introduction to classical statistical mechanics and a good preview to quantum mechanics. Chapter 25 begins with Liouville's theorem, then motion in phase space, some statistical notions of dynamics in phase space, Boltzmann's principle, the microcanonical ensemble, the canonical ensemble, and then the grand canonical ensemble. This chapter has 11 practice problems, 5 of which have detailed and complete solutions. Chapter 26 begins with a presentation on the equipartition theorem and heat capacities from a statistical point of view, then it goes into the statistical ensembles for ideal gases, then the Maxwell-Boltzmann law, the barometric equation, and molecular speeds, and the chapter ends with Gibb's paradox. This chapter has 10 practice problems, 5 of which have detailed and complete solutions. Chapter 27 begins with the thermodynamics of blackbody radiation, the spectral energy density, and Wein's law, this is extended into the statistics of radiation, and then the Planck radiation formula. This chapter has 10 practice problems, 5 of which have detailed and complete solutions.

The next ten chapters form a logical group on quantum mechanics and atomic physics. Chapter 28 begins with a presentation of the Einstein theory of electromagnetic radiation and establishes the photon, then it presents the Bohr theory of the atom, the deBroglie wave, and then the uncertainty principle. This chapter has 12 practice problems, 6 with detailed solutions. Chapter 29 begins with the wave function and Dirac notation, then it presents operator notation, then eigenvalues, and it ends with commutation relations. This chapter has 10 practice problems, 5 of which are solved in detail. Chapter 30 begins with the time-dependent Schroedinger equation, then the time-independent case, then unbound states and the probability current density, quantum tunneling, and it ends with bound states and the quantum harmonic oscillator. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 31 begins by extending the discussion of operators to include orbital angular momentum, then it presents the eigenvalue equations for a central field and introduces spherical harmonics, this is then extended to the quantization of angular momentum through the example of spatial quantization, a very important concept for doing quantum mechanics. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 32 begins to apply quantum mechanics to hydrogen-like atoms with a radial equation, thorughout the discussion fo quantum mechanics new special functions like the associeted Laguerre polynomial are presented, then the ground state and excited states of hydrogen are prented, then the focus switches to atoms in a magnetic field and Zeeman splitting. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 33 introduces Heisenberg's matrix mechanics, then it presents the angular momentum representation with the example of orbital angular momentum matrices, the focus then changes to spin and explicitly presents the expectation values of spin components. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 34 begins with adding angular momenta, including the Clebsch-Gordon coefficients, then spin-orbit coupling, the chapter ends with the anomalous Zeeman effects and the Paschen-Back effect. Once again, this chapter has 10 practice problem and 5 solved in detail. Chapter 35 begins by extending the quantum mechanics so far presented to systems of particles, then to identical particles in the form of fermions and bosons through the application of the Pauli principle, it then presents the actual Fermi-Dirac and Bose-Einstein statistics following a discussion of distribution functions; this is another point of the book I am not wild about, the presentation of the Pauli principle before discussing the distribution functions. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 36 introduces perturbation theory with the time-independent version, then there is a discussion of the Helium atom, and finally the Hartree equations and LS couping. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 37 deals with emission and absorption of radiation by atoms and begins with time-dependent perturbation theory, then how the notion of a constant perturbation leads to Fermi's golden rule, then the emission and absorption of electromagnetic radiation, there is a detailed discussion of the selection rules for electric dipole transitions, and the chapter ends with a discussion of spontaneous emissions. This chapter also has 10 practice problems, 5 of which are solved in detail.

The next eight chapters form a logical unit on molecular and condensed matter physics. Chapter 38 begins by considering large systems of atoms that collectively exhibit the properties of bulk matter—this leads to the adiabatic approximation, this is then extended to treating systems as linear lattices undergoing vibrations, this leads to the continuum approximation (also known as the Debye approximation), the quanta of energy within each normal mode of oscillation is a phonon—these are next introduced, the chapter ends with a calculation of the heat capacity of a lattice. This chapter has 10 practice problems, 5 of which are solved in detail. Chapter 39 begins with a nice treatment of the crystal lattice, then applies this to simple crystal structures, it then discusses the process of defining a lattice with a reciprocal basis like you would get from experimental data—called the reciprocal lattice, this leads into a discussion of structure determination, ending with X-ray diffraction. This chapter has 11 practice problems, 5 of which are solved in detail. Chapter 40 begins with the one- and free-electron approximations, the electron gas model and the density of states, the Fermi energy and the Fermi temperature, the thermodynamics of the free-electron model, then it turns to paramgnetism and the chapter ends with electron conduction. This chapter also has 11 practice problems, 5 of which are solved in detail. Chapter 41 begins with the theory of Bloch waves which are then applied to a one-dimensional periodic potential, then the authors present the weak-binding approximation (also known as the nearly-free-electron model), this is then applied to the effective mass of the electron, the chapter ends with the tight-binding approximation which is then applied the determine the energy bands in a cubic lattice. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 42 is an introduction to semiconductor physics and begins with free charge carriers and applies that to the effective mass of such carriers, the authors then present both intrinsic and impurity semiconductors. This chapter has 9 practice problems, 5 of which are solved in detail. Chapter 43 extends semiconductor physics to sold state electronics beggining with carrier transposrt by diffusion, energy-band diagrams, steady-state diffusion, then the authors turn to the pn junction, and they end with the junction transistor. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 44 switches topics a bit beginning with diamagnetism and paramagnetism, then ferromagnetism with the exchange interaction in a two-electron system, then the authors present antiferromagnetism, the chapter ends with ferrimagnetism. This chapter also has 10 practice problems, 5 of which are solved in detail. Chapter 45 deals with superconductivity beginning with the Meissner effect and the London theory (or two-fluid model), then it presents Cooper pairs, ending with high-temperature superconductivity. This chapter also has 10 practice problems, 5 of which are solved in detail.

The final two chapters are about nuclear physics. Chapter 46 deals with nuclear structure beginning with the binding fraction, then the liquid drop model, the gas model, then nuclear fission are presented as semi-classical theories, the chapter ends with the shell model. This chapter has 10 practice problems, 5 of which are solved in detail. Chapter 47 deals with nuclear dynamics, beginning with radiative decay, this is then applied to nuclear resonance, then the chapter covers alpha decay, then the chapter ends with beta decay. This chapter also has 10 practice problems, 5 of which are solved in detail.

The book ends with an appendix containing several matheamtical articles and a list of symbols. Appendix I covers vector calculus, Appendix II covers matrices, Appendix III gives some results for partial derviatives, and Appendix IV covers Gaussian integrals, evaluating integrals using the Riemann zeta function, and Stilring's approximation.

I personally think that this huge book is very good, though it is lacking in any true introduction to either general relativity or to particle physics.

Goodbye to a Dear Friend

2011-10-15T00:51:00.000-07:00

I learned today that a dear friend passed away. Hakki Ogelman was not just a colleague, he was a dear friend. A world-renowned astrophysicist, he was responsible for my first poster presentation at a professional conference; an analysis of the time series data from a gamma-ray burst. That was way back in 1995. Hakki was in charge of the research group Compact Objects at Wisconsin (COW). Hakki suffered a debilitating stroke in the mid 1990's, but perservered. He lost a months-long battle to cancer on 4 September 2011. He passed quietly, for which I am grateful. Goodbye, old friend, I will miss you...

Been Away Again

2011-10-01T18:09:00.000-07:00

I have been away for a couple of months. In that time my co-author on the book, Leonard Susskind, and I have been working on the final draft of our book THEORETICAL MINIMUM: Claswsical Mechanics and hope to have it done in a few weeks.

MAST is working on developing education modules for basic science.

I have had a paper on numerical modeling of the diffusion equation accepted for publication in The Mathematica Journal.

I will be working on the show The Wonders of Physics this year.

I will be doing more in the coming days.

Lots of Things: Mathematica Consulting to Quantum Gravity to Storm Chasing

2011-06-30T14:32:00.000-07:00

It has been a month and a half since my last post. Sounds almost like confession, doesn't it?

I could say that I have been snowed under with work, I have been sick for a week, I have been planning and executing the storm chase from Hell and another that was very successful. I could say that I have been trying to get MAST into the Mathematica Consulting game. I could say that I have been working on a new idea in quantum gravity. All of these things are true, and yet none of them are sufficient.

The truth is that I have been somewhat lazy. For that I apologize. I will not promise not to do it again.

Book Deal

2011-05-13T11:33:00.000-07:00

Today I received the executed book deal for my collaboration with Leonard Susskind on the book Theoretical Minimum: Classical Mechanics. The book will be published by Basic Books. We have until January of 2012 to finish it, but don't be surprised if we finish early.

Book News

2011-05-12T10:11:00.000-07:00

Leonard Susskind and I have now signed a contract with Basic Books for the Theoretical Minimum: Classical Mechanics. This book is contracted to be completed by January of 2012 (don't be surprised if it is finished before then).

Check Out the Research Log on MAST

2011-05-09T13:42:00.000-07:00

There is a research log that allows you to view our research as we conduct it. You can find it at: http://www.madscitech.org/rlog/rlog1.html

More Fun With Tensors

2011-05-08T03:44:00.000-07:00

Tensor Session 2

More Examples from David McMahon’s Book, “Relativity DeMysified”

Example 2-4

Given,

the metric in plane polar coordinates, and its inverse,

and the vectors,

find

We begin,

Expanding in coordinates we have,

We also have,

So,

Now we have,

Expanding in coordinates we have,

and,

Thus,

Then,

Example 2-5

Given,

in Cartesian coordinates, show that the dot products of the basis vectors in spherical polar coordinates form the metric.
We begin with the identity,

(1)

where we have the transformation matrix,

sometimes called the Jacobi matrix. So we have the transformations between Cartesian and spherical polar coordinates,

We now determine the basis vector transformations, using (1), expanding in components

This give us the metric component,

Then we have,

This give us the metric component,

Then we have,

This give us the metric component,

We also have the metric components,

and,

This gives us the metric,

Fun With Tensors

2011-05-01T01:53:00.000-07:00

Fun with Tensors

George E. Hrabovsky

James Firmiss

Example 2-3 from David McMahon’s book Relativity Demystified.

Given the line element

we seek to find the metric tensors for raising and lowering indices for vectors and one-forms. We begin by noting that the metric tensor is symmetrical,

We then identify the terms of the metric tensor. We begin with the first term,

We also have the last term,

Then we have the middle term, 2 dx dy. You might jump at this and say it is 2, but that would neglect that product of the differentials—each of which contributes half, so we have,

We can write the metric tensor,

We note that the product of the metric tensors is,

so,

We expand this term-by-term,

(1)

(2)

(3)

(4)

By (3) we get,

Rewriting (1)

or,

so,

Rewriting (2)

so,

or,

This gives us all of the components of the raised index metric tensor,

Given the vector

we can raise the index by this formula,

Expanding in coordinates,

This gives us

Given

we can lower the index by,

Expanding this in components,

This gives us the answer

Note that V=W.

I have been busy

2011-04-08T13:31:00.000-07:00

Between working on the book deal, writing the second draft of the book, finishing The Wonders of Physics, working a contract, and working on the Institute for Scientific Computing, I have not had a lot of time to work on the blog. We will be doing some storm chasing this weekend. I will post if we gets some good storm pictures.

Wonders of Physics

2011-02-16T00:28:00.000-08:00

I am busy, this week, with the Wonders of Physics (click here to go to the web site). This is a public demonstration show. Go to the web site and check it out!

The Book Is DONE!!!

2011-01-31T03:19:00.000-08:00

At 05:17 on Monday, 31 January 2011 I finished the first draft of the book Theoretical Minimum I: Classical Mechanics by Leonard Susskind and George Hrabovsky. This book was written in Mathematica 7 and Mathematica 8.

The Book is Almost Done!

2011-01-28T21:36:00.000-08:00

I am almost done with the final two cahpters of the book I am writing with Leonard Susskind! I will be done by Sunday!

Book Project Update

2011-01-23T23:42:00.000-08:00

Well here I am again. Sorry about the lengthy delay in posting, I am in a full-scale assault on the book to finish it by the end of this week. I have completed chapter 16, am finishing up chapter 15, and will finish 17 and 18 by week's end! The first draft is nearly done!

Here is the book now:

Lecture 1: Dynamical systems.

Interlude 1: Calculus.

Interlude 2: Newtonian mechanics.

Lecture 2: Conservation laws, the principle of least action, and the Lagrangian.

Interlude 3: Vectors and vector spaces.

Lecture 3: Conservation laws imply symmetry.

Interlude 4: Symmetry.

Lecture 4: Time translation symmetry, the conservation of energy, and the Hamiltonian.

Interlude 5: Transformations and matrices.

Lecture 5: Pendulums and springs.

Interlude 6: Complex numbers and ordinary differential equations.

Lecture 6: The Hamiltonian method of mechanics.

Interlude 7: Vector analysis.

Lecture 7: Liouville's Theorem and Information Conservation

Interlude 8: Thermodynamics (almost complete)

Lecture 8: Charged Particles in Electromagnetic Fields and an Introduction to Poisson Brackets

Interlude 9: Chaos (not done yet)

Lecture 9: Poisson Brackets and Canonical Transformations

Major New result on Dinosaur Extinction

2011-01-09T20:32:00.000-08:00

MAST member Jack Horner (not the paleontologist) has written a paper establishing that the asteroid impact theory of dinosaur exctintion is not necessary to explain the exctintion.

His paper can be found here: http://www.madscitech.org/csg/dinosaur.pdf

Some new results

2011-01-08T16:25:00.000-08:00

MAST has begun to post some research results!

1) I have started to do some dripping handrail models of accretion disks. You can find them here: http://www.madscitech.org/csg/accretion.html

2) James Firmiss and I have begun serious work on neutron star equations of state, results will be put up shortly.

3) Chris Winfield is giving a talk on local solvability at the American Mathematical Society meeting in New Orleans, that talk is located here: http://www.madscitech.org/csg/AMS10mintalk.pdf

4) We are doing research and development of database capabilities in Mathematica, mostly due to the efforts of Rocky Wenz.

More results will be reported as they occur!

Book Update

2011-01-04T20:04:00.000-08:00

I have now completed 13 chapters out of 18!

Here is the chapter list so far:

Lecture 1: Dynamical systems.

Interlude 1: Calculus.

Interlude 2: Newtonian mechanics.

Lecture 2: Conservation laws, the principle of least action, and the Lagrangian.

Interlude 3: Vectors and vector spaces.

Lecture 3: Conservation laws imply symmetry.

Interlude 4: Symmetry.

Lecture 4: Time translation symmetry, the conservation of energy, and the Hamiltonian.

Interlude 5: Transformations and matrices.

Lecture 5: Pendulums and springs.

Interlude 6: Complex numbers and ordinary differential equations.

Lecture 6: The Hamiltonian method of mechanics.

Interlude 7: Vector analysis.

I am now working on Lecture 7: Liouville's Theorem and Information Conservation

Summer School in Scientific Computation

2011-01-02T01:09:00.000-08:00

I would like to announce the Madison Area Science and Technology Summer School for Computational Science.

This will be a long Summer School, June 15, 2011 through August 15, 2011.

Tentatively, this will be held at the University of Wisconsin, Madison, Department of Physics.

Details can be found at the web site: www/madscitech.org/isc/sschool1.htmlThis web site will be completely functional by January 7, 2011.
Here are the details as they stand now:
Week One: Mathematica Orientation, Symbolic and Numerical Methods, and Good Programming Practices
Monday Morning: Introduction to Mathematica Workshop (3 hr)
Monday Afternoon: Constructing a Program in Mathematica (1 hr)
Tuesday: Breakout sessions
Wednesday Morning: Workshop on Document Preparation in Mathematica (3 hr)
Wednesday Afternoon: Image Processing and Analysis in Mathematica (1 hr)
Thursday: Breakout sessions.
Friday Morning: Workshop on Dynamic Programming in Mathematica (3 hr)
Friday Afternoon: Good Programming Practices (1 hr)

Week Two: Databases, Data Analysis, and Bioinformatics
Monday Morning: Workshop on Database Management in Mathematica (3 hr)
Monday Afternoon: Using Databases (1 hr)
Tuesday: Breakout sessions
Wednesday Morning; Workshop on Data Analysis in Mathematica (3 hr)
Wednesday Afternoon: Visualization in Mathematica (1 hr)
Thursday: Breakout sessions.
Friday Morning: Workshop on Probablity Theory in Mathematica (3 hr)
Friday Afternoon: Bioinformatics in Mathematica (1 hr)

Week Three: System Simulation and Wavelets
Monday (holiday)
Tuesday: Breakout sessions
Wednesday Morning; Workshop on Control Systems and Feedback (3 hr)
Wednesday Afternoon: System Simulation (1 hr)
Thursday: Breakout sessions.
Friday Morning: Workshop on Signals and Systems (3 hr)
Friday Afternoon: Wavelets for Data, Wavelets for Theory (1 hr)

Week Four: Ordinary Differential Equations, Cellular Automata, and Chaotic Dynamical Systems
Monday Morning: Workshop on DSolve and ODEs (3 hr)
Monday Afternoon: Global Analysis of ODEs (1 hr)
Tuesday: Breakout sessions
Wednesday Morning; Workshop on NDSolve and ODEs (3 hr)
Wednesday Afternoon: Cellular Automata (1 hr)
Thursday: Breakout sessions.
Friday Morning: Workshop on Systems of ODEs in Mathematica (3 hr)
Friday Afternoon: Chaotic Dynamical Systems (1 hr)

Week Five: PDES, Fluids, and Shocks
Monday Morning: Workshop on DSolve, NDSolve, and PDEs(3 hr)
Monday Afternoon: Exploring the Solvability of PDEs (1 hr)
Tuesday: Breakout sessions
Wednesday Morning; Workshop on PDEs(3 hr)
Wednesday Afternoon: Fluid Dynamics in Mathematica (1 hr)
Thursday: Breakout sessions.
Friday Morning: More About PDEs (3 hr)
Friday Afternoon: Shockwaves (1 hr)

Week Six: Discrete Mathematics, Operations Research, and Groups
Monday Morning: Workshop on Discrete Mathematics (3 hr)
Monday Afternoon: Graphs (1 hr)
Tuesday: Breakout sessions
Wednesday Morning; Workshop on Optimization (3 hr)
Wednesday Afternoon: Operations Simulation (1 hr)
Thursday: Breakout sessions.
Friday Morning: Workshop on Abstract Algebra in Mathematica (3 hr)
Friday Afternoon: Groups and Symmetries (1 hr)

Week Seven: Information Theory, Quantum Computing, and Scattering
Monday Morning: Workshop on Statistical Mechanics in Mathematica (3 hr)
Monday Afternoon: Information Theory (1 hr)
Tuesday: Breakout sessions
Wednesday Morning; Workshop on Quantum Mechanics in Mathematica (3 hr)
Wednesday Afternoon: Quantum Computing Simulation (1 hr)
Thursday: Breakout sessions.
Friday Morning: Workshop on Abstract Mathematics in Mathematica (3 hr)
Friday Afternoon: Scattering Theory (1 hr)

Week 8: Student Presentations
Monday: 6 One-Hour Presentations Chosen Prior to the Summer School
Tuesday: 12 Half-Hour Presentations Chosen Prior to the Summer School
Wednesday: 16 Twenty-Minute Presentations Chosen Prior to the Summer School
Thursday: 16 Twenty-Minute Presentations Chosen During the Summer School
Friday: 32 Ten-Minute Presentations Chosen During the Summer School
The proceedings of the school will be given to any registrants.

Project Updates

2010-11-30T18:00:00.000-08:00

Hello!

I have been insanely busy the last week. Had a great Thanksgiving!

Just completed Chapter 9 in the book project, this is a survey of linear algebra. I am about to start in on chapter 10, Leonard Susskind's classical mechanics Lecture 5.

I am also developing the program for the Institute for Scientific Computing Summer School program. I am also working on developing courses for teaching Mathematica 8 online.

That's all for now.

Mathematica 8 Is HERE!!!

2010-11-16T20:42:00.000-08:00

The new version of Mathematica has arrived! I have been playing with it for a day now. It is a substantial upgrade from version 7. I will write more about the new features as I play with them. One new feature is the integration with Wolfram Alpha allowing the user to use free-form commands to produce results without the use of syntax; this is limited and for people who know Mathematica, it will not be much of an addition, but it does allow people who do not know the system to begin exploring. The results also include the correct Mathematica syntax for what it thinks you want.

More later.

Recent Papers of Interest 16 November 2010

2010-11-16T20:36:00.000-08:00

Here are some ArXiv papers I found interesting of late.

1) Takami Kuroda and Hideyuki Umeda, Three Dimensional Magneto Hydrodynamical Simulations of Gravitational Collapse of a 15Msun Star

This is an important paper for those of us who study compact objects. It begins with a nice overview of the theory of core-collapse supernovae (CCSN) and the efforts to model these. The next two sections details the code used, both MHD and GR. This is followed by the adaptive mesh scheme used. Finally they discuss the tests of the code. This accounts for half of the paper and makes a very nice overview of modeling MHD in a GR environment. The results are very nice, great paper!

2) Manjit Bhatia, P. Narayana Swamy, Elementary Quantum Mechanics in a Space-time Lattice

This paper explores the particle-in-a-box problem under the assumption of a fundamental length scale forming a spacetime lattice. This is an interesting approach.

3) Oleg Korobkin, Ernazar B. Abdikamalov, et al, Stability of general-relativistic accretion disks,

This paper is a thorough treatment of a model of unstable accretion disks near a black hole.

4) T. Fischer, I. Sagert, et al, The revival of an explosion mechanism of massive stars - the quark hadron phase transition during the early post bounce phase of core collapse supernovae,

This is a long paper (30 pages) explores the bounce during a supernova core collapse caused by the so-called quark hadron phase transition. The introduction is a nicce overview of the topic and its history. The second section, no pun intended, forms the theoretical core of the paper, describing the hybrid equation of state and other issues. The third section describes the simulation scheme. The fourth section explores the results of the model runs. The conclusion is that the quark matter description leads to early onset of deconfinement, a quark-gluon plasma. A very interesting paper!

Project Updates

2010-11-05T13:09:00.000-07:00

I am about to finish chapter six of the book project.

We (MAST) are planning a Summer School on Computational Physics; this will tentaively be held at the University of Wisconsin, Madison, Department of Physics and will be held from 15 June 2011 to 15 August 2011. Check out the MAST website for details.

We are exploring the possibility of Mathematica training over the web. This will include beginner training all the way up to advanced applications.

The Week of 24-30 October in Astrophysics

2010-10-29T21:36:00.000-07:00

Here are four interesting papers from arXiv astrophysics:

1) Francesca Valsecchi, Evert Glebbeek, et al, "Formation of the black-hole binary M33 X-7 via mass-exchange in a tight massive system",

M-33 X-7 is a mystery that this paper proports to solve, maybe. This binary object seems to be a ~16 solar mass black hole orbiting a ~70 solar mass hydrogen-rich O-type companion star. This binary system has a period of ~3.5 days. Up until now, no model explained all of the weird things about this system. This paper contrives a set of circumstances to explain it all.

2) Shigehiro Nagataki, "Rotating BHs as Central Engines of Long GRBs: Faster is Better,"

This paper relates the rotational rate of a black hole to the strength of any relativistic jet produced by it. This is based on numerical relativity calculations using a magnetohydrodynamic code. Four different models are run.

3) Joan M. Centrella, John G. Baker, "Black-hole binaries, gravitational waves, and numerical relativity,"

This is a very nice review of black hole modeling and numerical relativity. Of course, one of the primary applications is the support of gravitational wave detectors.

4) Rachid Ouyed, Mathew Kostka, et al, "Quark nova imprint in the extreme supernova explosion SN 2006gy: the advent of the Quark Star,"

A fascinating possibility is that at the center of neutron stars a special form of matter is produced, one where there is a region of free quarks surrounded by a screen of gluons. This is called a quark-gluon plasma. Such a state of matter was speculative for many decades, but recent results from RHIC (Relativistic Heavy Ion Collider) have given indications that this has been achieved. The authors of this paper suggest that the supernova SN2006gy was in fact a supernova that included a boost to the explosion through the release of energy by the conversion of neutons at the center of the developing neutron star, converting the center into more stable strange quark matter. If true, this is a significant event!

Update

2010-10-27T06:41:00.000-07:00

Well, we just turned in our NSF proposal to develop the MAST Institute for Scientific Computing. We have letters of interest from Jack Horner, a software engineer from Los Alamos, and from the Unversity of Wisconsin, Madison, Department of Physics.

This is a very cool project!

Now that the propsal is in, I can get back to work on the book!

Theoretical Physics Papers #2

2010-10-18T18:22:00.000-07:00

Only two really jumped out at me today:

1) Pablo Arrighi, Jonathan Grattage, A Quantum Game of Life,

I think most of us are familiar with the Game of Life, an example of cellular automata (CA) evolving into structures, some of which seem to be coherent. The hope is that such systems will both demonstrate evolving complexity, and reveal some of the rules for such evolution. This paper describes a universal quantum CA (QCA) in three dimensions that can simulate all other three-dimensional QCAs. One application of this is to study the flow of qubits, another is to simulate quantum gates. This is a very important paper to anyone interested in simulating a quantum computer.

2) Jonathan Hackett, Louis H. Kauffman, Octonions,

This paper is a review of the idea of the octonion, a quaternion with the proprty that a lateral rotation brings about a change in the forward orientation of the quaternion and that this is non-associative. Turning spin up to spin down would be an example of the change in orientation. This is sometimes called the Dirac belt trick.

Some Recent Papers on Theoretical Physics and Mathematical Physics

2010-10-15T08:29:00.000-07:00

1) Gary Horowitz, Surprising Connections Between General Relativity and Condensed Matter,

My only objection to this paper is that it seems to be tied to anti-deSitter space (AdS). Proving things in AdS may be easier than proving them in general relativity (AdS is essentially general relativity with a negative cosmological constant), but it cannot gaurantee applicability to the real world (since we seem to have a positive cosmological constant). The paper seems to rely on an observation that black holes in AdS behave like thermal systems of one dimensional lower, and that the string theory/holographic principle of black holes allows you to extract features from general relativity that look a lot like superconductivity.

2) Geoffrey Lovelace, Mark. A. Scheel, Bela Szilagyi, Simulating merging binary black holes with nearly extremal spins,

This fascinating paper describes the merger of two rapidly rotating black holes.

3) Burkhard Kleihaus, Jutta Kunz, Eugen Radu, Maria J. Rodriguez, New generalized nonspherical black hole solutions,

This paper studies black holes in more than 4 (often more than five!) dimensions. There are some standard solutions described (Minkowski, and Schwarzschild-Tangherlini), and some not so-normal: rod-shaped, ring-shaped, multi-horizoned (see my blog about inner horizons), the black Saturn, concentric black rings, and the singly spinning Myers-Perry solution.

4) Luca Fabbri, On the geometric relativistic foundations of matter field theories and wave solutions as classic concepts,

This paper develops a very interesting connection between general relativity and quantum mechanics. I have only skimmed it, but it deservs careful study!

5) A. S. Fokas, D. Yang, A Novel Approach to Elastodynamics: I. The Two-Dimensional Case,

This is the first of a pair of very technical papers where the principle method of deriving equations involves the application of Fourier transforms followed by substituting in initial and boundary conditions. This method holds out some hope of developing analytical solutions to PDEs similar to those of the Lamb problem in elastodynamics, that is normal line load, tangential line load, and mixed line load.

6) A. S. Fokas, D. Yang, A Novel Approach to Elastodynamics: II. The Three-Dimensional Case,

This is an extension of the previous methods to the three-dimensional case.

7) Christopher Batty, Robert Bridson, A Variational Finite Difference Method for Time-Dependent Stokes Flow on Irregular Domains,

An interesting paper that goes into great detail of developing a numerical 2- and three-dimensional fluid dynamics code based on variational methods. I suspect that the better behavior of the three-dimensional case is due to the unphysical nature of the two-dimensional case. I use the unphysical nature of two-dimensional wave simulations creating atrtifacts that don't exist in three-dimensional models as justification for this opinion.

Book Review: Quantum Mechanics by Landau and Lifshitz

2010-10-15T03:18:00.000-07:00

Time for another book review, this time volume three of the Course in Theoretical Physics by Landau and Lifshitz. This book is available for sale in the bookstore below. This volume is titled, "Quantum Mechanics (Non-relativistic theory)," and covers, in one volume, standard non-relativistic quantum mechanics, atomic and molecular physics, group theory, and nuclear physics.

This book has eighteen chapters and six mathematical appendices on special functions. There are 152 individual sections for a total of 677 pages.

Chapter one covers the basics of quantum mechanics, beginning with the uncertainty principle. This is one of the hardest concepts for people to get used to and it encompasses most of the weirdness of quantum mechanics, this section also develops the idea of the state of a system. These ideas are further developed into the notion of the superposition of states; it is this superposition that prevents the standard idea of the excluded middle from being true. Then the authors begin to develop the mathematical machinery of quantum mechanics by introducing several sections on operators and their spectrum. Then there is a statement that quantum mechanics must reduce to Newtonian mechanics when Planck's constant is small. This leads to a discussion of wave functions and observables.

The second chapter can be thought of as an attempt to extend the results of the first volume (chapter two) to quantum mechanics. The chapter begins by establishing the Hamiltonian operator. This is extended to the derivative of operators with respect to a parameter (in this case, time). The conservation of energy is then extended to stationary states including the ground state, energy levels, degenerate states, and bound states. There are two sections of the mathematics of matrices. Then the authors interoduce the density matrix. This is followed by a section on the momentum operator, extending the idea of conservation of momentum. The chapter ends with a section on the uncertainty relations.

The third chapter introduces the Schrödinger equation and discusses its properties. It then presents the probability current density. Then the chapter shifts gears a little bit and describes how to obtain the Schrödinger equation from a variational principle. The authors then discuss one-dimensional applications, potential wells, the quantum harmonic oscillator, motion in an external field, and reflection and transmission.

Chapter four extends angular momentum into its quantum mechanical incarnation by introducing the rotation operator and its quantization. The first section also includes a fair bit of tensor manipulation, particuarly along the lines of the finite-rotation tensor. This is then extended to finding the eigenvalues and eigenfunctions of the angular momentum. The authors then present the selection rules for angular momentum. A section on parity follows, and the chapter ends with adding angular momenta.

Chapter five covers motion in a central field that is radially symmetrical. The first section turns the two body problem into an equivalent one-body problem and introduces spherical harmonics, and the radial and magnetic quantum numbers. This is expanded into a discussion of spherical waves along with spherical Bessel functions, Hankel functions, asymptotic expansions, and the phase shift. The next section involves resolving a plane wave. The next section asks and answers the question, "What is the critical field where the fall of a particle to the center of the field becomes possible?" The authors then address the issue of motion in a Coulomb field, here they begin with the case of a discrete spectrum and introduce hypergeometric functions, the principal quantum number, and Laguerre polynomials; the continuous spectrum introduces the confluent hypergeometric function, and then they explore Coulomb degeneracy. The chapter ends with another approach to the Coulomb field, that of using parabolic coordinates.

Chapter six deals with one of the principle methods of approximation in quantum mechanics, perturbation theory. The chapter begins with a presentation of time-independent perturbation theory. Following this are sections on the secular equation and time-dependent perturbations. Then there are several examples of transitions under perturbations: for finite time, periodic perturbations, and for continuous spectra. The authors then discuss the energy-time undertainty relations (a subject that generates a lot of confusion). The chapter ends with potential energy treated as a perturbation.

Chapter seven discusses the semi-classical case of quantum mechanics, beginning with the semi-classical wave function. Then the authors present the boundary conditions that allow a semi-classical approach. The next section introduces the Bohr-Sommerfeld rule and connects to the first volume in terms of adiabatic invariants and the sepration of variables method for solving the Hamilton-Jacobi equation, and the second volume in the presentation of the number of characteristic vibrations of a wave. The next section links the discussion of the semi-classical approach to the fifth chapter, by describing semiclassical motion in a central field. The authors then discuss quantum tunneling. The next section is very practical in its discussion of how to calculate matrix elements for the semi-classical approach and introduces the bizarre notion of complex time. The next section deals with transition probabilities and the problem of reflection above the potential barrier. The final section of the chapter asks the quation as to what the transition probabilities are for an adiabatic invariant.

Chapter eight covers an extremely important, and often misunderstood, topic in quantum mechanics—spin. The chapter opens with a length section introducing the idea of the total angular momentum of a particle as the sum of its orbital angular moemntum and its spin. This leads into a discussion of the spin operator (and the famous Pauli spin matrices). Then the authors introduce you to the mathematical objects called spinors. Using these techniques the wave functions of particles of arbitrary spin are calculated and the irreducible tensor formulation is introduced. These methods are extended, and connected to the theory of rigid bodies, through the finite rotation operator. the authors then cover the issue of the polarization of particles. The chapter ends with the subject of time reversal.

Chapter nine covers the important topic of identical particles, that is fermions and bosons. These particles are introduced in the fisrt section along with the Pauli principle. The next section explains some of the limitations of the Schrödinger equation regarding spin and introduces the exchange interaction for fermions. This is followed by a very important section that lays the ground-work for group theory and introduces the so-called Youngs diagram for symmetry. The final two sections of the chapter lay the groundwork for quantum field theory with the second quantization, first of bosons then fermions.

Chapter ten is very long and introduces atomic physics beginning with atomic energy levels, fine structure, and spin-orbit coupling. This is then extended to the electronic states of atoms, including Hund's rule. These ideas are applied to hydrogen-like atomic energy levels (one-electron). The next section covers the self-consistent field method for calculating energies and wave functions of stationary states. The problem of similar calculations for complex atoms (which cause the self-consistent field method to become unweildy) motivates the Thomas-Fermi equation and utilizes the virial theorem (from volume one). The next section covers the problem of the variation of wave functions of the electron in an atom from long to short distances. This prefaces the next section on the fine structure of atomic energy levels, a subject requiring a knowledge of the electron density near the atomic nucleus. Gears shift and the periodic table is the next topic of disucssion. The next section covers the notion that the binding energy of the inner shell is so large that when an electron transitions to an outer shell, the resulting ion is unstable; this section introduces x-ray terms and the Auger effect. This is followed by a discussion of the quantum mechanical treatment of multipole moments where the dipole moment is a vector and the multipole moment is a tensor. Then the authors discuss the Stark effect, where atomic energy levels are altered in an external field. The last section of the chapter deals with the linear Stark effect, where the atomic energy levels of hydrogen are split proportionally to the field, unlike other atoms.

Chapter eleven extends the atomic theory to diatomic molecules, beginning with a discussion of electron terms in such molecules. This is then extended to cover intersection of the terms and again develops the foundations of group theory. Then the chapter covers the relationship between the atomic and molecular terms. Then the important issue of valence groups is developed. The authors then discuss a topic of great interest to physical chemists, the vibrational and rotational structures in the diatomic molecule, first of the singlet, and then the four multiplet cases. The symmetry of the molecular terms is then explored. The next section is a practical one, calculating the matrix terms for a diatomic molecule. This is then extended to the case of the interaction between the electron state and the rotation of the molecule. Then the topic of atomic interactions at long ranges is addressed, the so-called van der Waals force. The chapter ends with a phenomena where a diatomic molecule can disintegrate spontaneously by passing from one state to another, called pre-dissociation, and their relationship to perturbations in the molecular spectra.

Chapter twelve is the introduction to group theory. The chapter begins with a discussion of symmetry transformations and the notion that these transformations can be commutative. This is the underlying idea for transformation groups and then point groups. Then the authors develop the concepts of group representations and then irreducible representations. These ideas are then extended to classifying terms, the so-called character table. These ideas are further extended to selection rules. Then the focus shifts to continuous groups, specifically the rotation group. The chapter ends with a section describing the representations of finite point groups.

Chapter thirteen returns to molecules, this time polyatomic molecules. The chapter begins with the application of group theory to classify molecular vibrations, and then vibrational energy levels. Then the issue of the stability of symmetrical molecular configurations is taken up, including the Jahn-Teller theorem. Then the authors disuss the quantization of the rigid rotator, beginning with a spherical rotator, then the asymmetrical rotator. These results are then extended to interactions between vibrations and rotations in molecules. The last section of the chapter takes up the task of classifying the molecular terms of the wave function in total, the sum of electron, nuclear vibrations, and rotations.

Chapter fourteen completes the discussion of angular momentum, beginning with the 3j symbols, and the Clebsch-Gordon coefficients. Then the authors use these to form a spherical tensor. Then there is a section of 6-j symbols (which have a bizarre application in black hole physics). Then they discuss the matrix elements for adding angular momenta (the plural form of angular momentum). The chapter ends with an extension of this to axially symmetric systems.

Chapter fifteen begins what I like to think of as something of a cheat, it is the behavior of quantum systems in fields. Of course in quantum theory fields are mediated by force carriers (in the case of electromagnetism, by photons) and are thus the realm of quantum electrodynamics and quantum field theory. This can be thought of as a semi-classical case of quantum mechanics. The chapter begins with a section on the Schrödinger equation in a magnetic field, introducing the Bohr magneton, and depends heavily on volume two of the series. Then the emphasis shifts to motion in a uniform magnetic field and introduces Landau levels (yes, the same Landau). This is then extended to atoms in a magnetic field, the Landé factor, Langevin's formula, and the quantum mechanical origin of magnetism. The chapter ends with the current dnesity in a magnetic field.

Chapter sixteen forms a core of basic nuclear physics. This begins with a presentation of the similarities between neutrons and protons called isotopic invariance, and brinign some elements of symmetry anf group theory into the mix. This is followed by a section on nuclear forces and their tensor character. There is a lengthy presentation of the shell model of nuclear structure. This is then extended to the case of nonspherical nuclei allowing for energy levels of a fixed nucleus and a rotational energy. There are several nuclear effects on atoms, the isotopic shift of energy levels due to finite mass or finite radius, and the hyperfine structure of the atom. The hyperfine structure can even be applied to molecules.

Chapter seventeen and eighteen constitute a course in the quantum theory of scattering in their own right. This begins with the general theory of scattering, including calculating the cross-section and partial scattering amplitudes, and then the general properties of scattering are explored. This is followed by an introduction to the S-matrix. Then the Born approximation and the transport scattering cross section are presented. These ideas are extended to the semi-calssical case. The next section explores the scattering amplitude for complex energies and introduces some ideas of the theory of Riemann surfaces. These ideas are then simplified by dispersion relations. The next section explores the momentum representation formulation of the scattering problem. Then the problem of high energy scattering is addressed. This is followed scattering at slow speeds and resonance scattering at low energies. This result is then extended to quasi-statioanry states. This is followed by Rutherford scattering. This notion is in turn extended to a system of wave functions in a continuous Coulomb field. Then there is a section on the collision of identical particles. These ideas are extended to nuclear particles in resoance scattering of charged particles. The next section describes elastic collisions between fast electrons and atoms. These ideas are generalized to include spin-orbit coupling-dependent scattering. The final section of chapter seventeen is another application of complex analysis, the so-called Regge-pole theory.

The final chapter of the book is a lengthy treatment of inelastic collisions, beginning with elastic collision in the presence of inelastic processes, introducing the reaction cross section. Then it addresses full-inelastic scattering of slow particles. Then the authors address the S-matrix where reactions occur. These ideas are then applied to the problem of quasi-stationary states. The focus then shifts to the problem of interactions between particles formed as a result of a previous interaction. Then the problem of calculating cross sections near the reaction threshold. The authors then address the problem of inelastic scattering between fast electrons and atoms. the next section introduces the effective retardation, the energy loss due to an interaction. Then the issue of interactions between high mass particles and atoms. This is then applied to neutron scattering. The last section is then about high-energy inelastic scattering.

New developments at MAST

2010-10-13T10:43:00.000-07:00

MAST (Madison Area Science and Technology) will be seeking funding from the NSF to open a computational science center and, if funded, it will allow us to offer internships for High School Seniors, Citizen Scientists, Undergraduate Students, Graduate Students, and even Post-docs who are interested in computational science, including, but not limited to: astrophysical fluid dynamics, compact objects, accretion disks, relativity, cosmology, mathematical physics, numerical methods, fluid dynamics, atmospheric modeling, shock wave phenomena, chaotic dynamical systems, scattering, quantum information theory, solution methods for ODEs and PDEs, and bioinformatics. We will handle all necessary educational requirements to bring the students up to speed (including training in Mathematica).

We will even be accepting a number of foreign internships.

The proposal will be submitted to the NSF by 27 October of this year.

Book Project Milestone!

2010-10-11T22:48:00.001-07:00

The classical mechanics book I am writing with Leonard Susskind has reached a milestone. As of now the page count stands at just over 100 pages! That is four chapters completed, the first two lectures, an introduction to calculus, and an introduction to Newtonian mechanics.

The Inner Horizon of a Black Hole

2010-10-09T08:45:00.000-07:00

Everyone has, by now, heard of black holes. Astronomically, they are likely to have been supermassive stars that exploded, whose remnant gravitationally collapses to a size the is smaller than a function of its mass called the Schwarzschild radius,

where M is the mass of the star. Once collapse passes this point the star is gone and it becomes a new thing, a black hole. Within the Schwarschild radius nothing can escape from the black hole, the escape velocity grows to be more than the speed of light. There is a boundary that we call the event horizon at the Schwarzschild radius, this is what separates the normal spacetime from the weirdness that happens near the black hole.

There are circumstances that can form another surface inside the event horizon. A charged black hole (Riessner-Nordstrøm) can form an inner horizon caused by a charge barrier. A spinning black hole (Kerr) can form an inner horizon due to transfer of angular momentum.

All horizons are places where, in effect, time stops with respect to an outside observer. The inner horizon must also be traversed by anything that falls into the black hole on its way to the singularity predicted to be at its center. The inner horizon gives the possibility of avoiding the singularity, but there seems no way to gaurantee a trajectory through the black hole to avoid the inner horizon prior to crossing the event horizon (you could still cross the inner horizon).

Even if you avoided the inner horizon, it is unlikely that anything other than subatomic particles will survive the experience.

Book Project

2010-10-07T15:07:00.001-07:00

I have just finished lecture 2 of Classical Mechanics, that completes three chapters total. This chapter develops the ideas of conservation of energy, conservation of momentum, and the principle of least action.

Musings on the Initial Singularity

2010-10-05T09:43:00.000-07:00

For those of you who do not know, the so-called initial singularity is an artifact of Big Bang cosmology implied by the expansion of the universe. If all points in spacetime are moving away from each other (not the matter within spacetime, just the spacetime envirnment-sort of like a river flowing, things in the river flow with it, but are not pulled apart), then it seems reasonable that everything had to start at a point. This time reversal leading to something that looks like a black hole was the subject of Stephen Hawking's thesis around 40 years ago. Other people have studied this time reversal process and have speculated about the dynamics of this singularity. The only problem is that they treat it as a spacetime singularity, like that of a static black hole.

The problem is, it could not have been a spacetime singularity! Spacetime did not yet exist before the Big Bang happened, so there could not have been a spacetime singularity.

This is, of course, one of the reasons why we need a truly quantum theory of gravity. This realization destroys any real chance for unification of the forces, since without spacetime curvature there can be no gravitation; and radiation pressure becomes dominant during the inflationary period. At the moment of the Big Bang, the strong, weak, and electromagnetic forces cause a rapid expansion. Spacetime starts to expand, but without a quantum theory of gravity we cannot understand this process since it is entirely in the regime of quantum mechanical distances, even if the energy densities are huge.

A very deep problem, and one that can only be solved by quantum gravity.

Progress Report for the Last Week

2010-10-04T06:26:00.000-07:00

Things I am Working On:

1. Classical Mechanics book: Almost done with lecture 2, should finish today. Thinking about writing a small chapter on dimensional analysis to derive force laws. After lecture 2 I will start on a chapter dealing with vector analysis, and with lecture 3.

2. Book reviews: Quantum Mechanics (Landau and Lifshitz), Advanced University Physics (Rogalski and Palmer).

3. Infinity Computing: Not doing much, though will start to write some code this week.

4. Quantum Computing: Beginning work on a Quantum Computer Simulator.

That's all I have been working on from a theoretical point of view.

Blog Update

2010-09-30T21:27:00.000-07:00

I have not posted for a couple of days because I am hard at work on transcribing Lecture 4 of Classical Mechanics. This is about symmetry and conservation laws.

I am also working on a book review for a book I received for review from CRC Press, Advanced University Physics.

Lots on my plate this week.

Book project update

2010-09-24T21:01:00.000-07:00

I have just completed a 15 page chapter on calculus that covers limits, derivatives, integrals, differential equations, infinite series, and partial derivatives. This chapter is for the proposed classical mechanics book I am working on with Leonard Susskind.

I took a very minimalist approach to the subject. If you want to see how I crammed so much into so little, wait for the book... :-)

Actually, some of this material will likely find its way into my column (and I took some of it from my existing columns).

The Grand Design - The Straight Story

2010-09-22T03:21:00.000-07:00

I have just completed reading, "The Grand Design," by Stephen Hawking and Leonard Mlodinow. It was a good book. I do not agree with everything they write, but then it is for a lay audience and not for specialists. This book is available for sale from the bookstore below.

Let me state one thing right up front and make it completely clear: Nowhere in the book do the authors state that God does not exist! Period! As I suspected (see a previous post) all the authors state is that the current physical theories are such that no appeal to God is necessary to explain the existence of the universe. In other words, the physical laws are rich enough to explain how the universe came into being out of nothing.

The one objection that I have to the book is the reliance on M-theory, this is a theory that I do not understand; so it is difficult for me to critique it. I know just enough to get myself into trouble, so I will refrain.

I enjoyed the book a lot! if you are interested in physics, this is worth getting.

Ignore all the hype, it is nothing but lies! No one who has read the book can truthfully state that the authors said that God does not exist. It is a very nioe history of the development of theories of the universe of all types.

Secret Weapons of The Theoretical Physicist

2010-09-20T20:36:00.000-07:00

Tonight I am thinking about a set of the most powerful techniques of the theoretist, classically called back-of-the-envelope calculations. These consitute two broad categories of methods.

The first is the method of estimation, where you use some basic reasoning abilities to determine a rough estimate of the quantities you are considering. For example, you want to estimate the number of hairs on someone's head. You think about it and decide that if you knew the area of the part of the head covered by hair, then knew the average area of a hair, you could figure it out. Then you make the necessary assumptions about the quantities (either by looking up the information, or by figuring it out for yourself).

The second is the ability to check the correctness of equations and to derive new equations by studying the relevant units of the quantities. All terms of any equation must have the same units. When you chaeck the terms, any that do not have the correct units are just wrong. By including powers of each quantity in each term and making sure all of the units come out the same, you invent a system of algebraic equations that can be solved. This gives you the necessary powers of each variable and shows you the structure of each term within a factor of a constant of proportionality. In this way you can learn what each term must look like. This allows you to invent new equations if you know the units involved. This is called dimensional analysis.

I will be writing about these ideas in my column in the coming weeks. I will also be discussing this in the Classical Mechanics book I am writing with Leonard Susskind.

An Apology and a Reminder

2010-09-19T13:24:00.000-07:00

I just deleted a comment on a previous post, I will not tell who sent it, or exactly what it was about. It had the broad topic of mysticism and religion. While I am happy to discuss such things, this is not the forum for such discussions.

Please do not send comments about such topics to me. I will not publish them.

If you want to discuss physics, that is fine. If you think you have proved Newton, Galileo, Einstein, Bohr, Planck, Feynman, Hawking, etc. wrong, tell the New York Times, not me! I sincerely doubt that your theory will meet my standards...

If you have interesting ideas about physics, that is fine. I have an open mind, but it is not so open that I am liable to fall in... I also am completely skeptical of wild claims (ask anyone that has been a speaker at a physics colloqium or seminar that I have attended...).

This blog is about theoretical physics education and research. Let's limit it to that.

George

Downplaying the Hawking-Mlodnow "Grand Design" Controversy

2010-09-17T15:22:00.000-07:00

People, mostly reporters, have been driving me crazy again. Specifically, they are making statements about the new book, "Grand Design," by Stephen Hawking and Leonard Mlodnow without quoting the book directly. Now, let me say up front that I have not yet read the book. I have heard the statements that Hawking has made in the past along the lines of, "God is not necessary to the creation of the universe." At no point, including in this book, has either author made that statement that God does not exist, nor that God did not create the universe.

Despite this, reports are that they have said these things in their book. Having seen statements made by the authors and listening to interviews with Mlodnow, I cannot believe that people are ignoring the authors!

Come on people! Stop making an issue out of nothing!

Pythagorean Triples

2010-09-16T12:55:00.000-07:00

Pythagorean triples are groupings of three integers that each form the sides of a right triangle. I have been playing with their distributions, and have made some plots in Mathematica. I think these are pretty cool.

The Mind of a Theorist - The Relaunch of the Column

2010-09-15T13:22:00.000-07:00

You can find the rewritten and reformated column at www.madscitech.org
I will be placing a new column up every week.

New Version of Mathematica Announced

2010-09-14T20:48:00.000-07:00

I was just informed by Wolfram Research that a new version of Mathematica will be coming out soon. As I understand it, this version is going to be a big step forward. This version will have built-in GPU support for CUDA, built-in control systems and wavelets, advanced data visualization tools, and a huge raft of other things I have heard about, but have not seen officially so will keep my mouth (fingers?) shut about.

This is exciting news since there is a personal version of Mathematica for only about $300, with a $500 upgrade to the most current professional version. There is no difference between the personal and profession versions.

Book Review: The Classical Theory of Fields

2010-09-13T21:42:00.000-07:00

Time for another book review. You can purchase this book at my book store below.

This time I am going for volume 2 of the Course of Theoretical Physics by Landau and Lifshitz. This volume is titled, "The Classical Theory of Fields," and covers, in one volume, special and general relativity, electrodynamics in a vacuum, and optics in a vacuum. The authors assume you are familiar with vector analysis and the electromagnetic phenomena equivalent to that covered in a general physics course (charges, electric and magnetic fields, and induction).

The first chapter is on special relativity and it begins by establishing the principle of relativity, the fact that there must be a maximum velocity for the transfer of information, and the requirement that the principle of relativity apply in all inertial reference frames. Next the authors define the concept of the event, and then show that the spacetime interval for an event is invariant of the inertial reference frame. This idea is then extended to multiple events and in this way the idea of spacetime is built-up. Next the idea of proper time is developed. The authors then derive the Lorentz transformations, the idea of proper length and length contraction, proper volume, and time dilation. This is followed by a discussion of the transformation of velocities, and the relativistic aberration of light as a consequence. Then there is a lengthy section on the mathematics of four-vectors and four-tensors. The chapter ends with a presentation of four-velocity.

The second chapter can be thought of as a quick survey of how you might apply special relativity to volume one of the Course. This begins with deriving the relativistic version of the least action principle. Then the ideas of relativistic momentum and energy, including the rest energy (you know, E=mc^2), the momentum four-vector, and the relativistic Hamilton-Jacobi equation, are presented. Then there is a very interesting section on the distribution in phase space of particles having given components of momentum. An application of special relativity is to the decay of particles, the next section covers that in detail. This is followed by a section on the invariant cross-section, and that leads into a section on elastic collisions. The chapter ends with a section on relativistic angular momentum.

The third chapter is where electrodynamics begins with a discussion of fields, the impossibility of rigid bodies in relativity, and the conclusion that this leads to the notion that particles must be treated as points. The book then establishes the four-potential with spatial components corresponding to the components of the vector potential and the time component being the scalar potential, this leads to the Hamiltonian. Then the equations of motion of a charged particle in a field are presented, along with the definitions of the electric and magnetic fields, and the Lorentz force. The next section is on gauge invariance, where the equations are invariant under transformations of the potentials. Standard electrostatics and magnetostatics are presented as aspects of a constant electromagnetic field. Then motion in a constant uniform electric field, a constant uniform magnetic field, and then in constant uniform electric and magnetic fields are presented. This is followed by a discussion of the electromagnetic field tensor based on the least action principle. Then the authors derive the Lorentz transformations of the field. Then they wrap the chapter up with the invariants of the field.

The fourth chapter covers Maxwell's equations. These are presented in their tensor form and in their more traditional vector forms. There is a nice section that establishes the action functional of the electroamgnetic field. Other nicely handled topics are the four-current vector, the electromagnetic equation of continuity, energy density and energy flux (including the Poynting vector), the stress-energy tensor (called the energy-momentum tensor) and its application to the electromagnetic field and to systems of particles. The chapter ends with the relativistic virial theorem.

The fifth chapter picks up one of the themes of chapter three, that of constant fields. This begins with Coulomb's law, the Poisson equation, and the Laplace equation. This is then extended to the idea of electrostatic energy. This is then applied to the field of a uniformly moving charge and then to motion in a Coulomb field. Another important situation covered is the dipole moment, and then the multipole moment (specifically the quadrupole moment where Legendre polynomials and spherical harmonics are introduced). This is then extended to a system of charges in an external electric field. Then the focus shifts to the constant magnetic field, then to magnetic moments, ending in a discussion of Larmor's theorem.

The sixth chapter introduces waves. The chapter begins with the wave equation and the Lorentz gauge. Then there is a section on transverse plane waves, energy flux, and momentum flux. The authors then define monochromatic waves (waves whose field is a periodic function of time), the wave vector, and explore the idea of polarization. There is also a section on Fourier analysis. Then there is a discussion of partially polarized waves (waves whose polarization changes with time), and the polarization tensor and the Stokes parameters make their appearance. The next section describes how the field produced by charges can be expanded in plane waves by Fourier transforms, and introduces longitudinal waves. The idea of Fourier series is then extended to the problem of the energy and the momentum of a field.

The seventh chapter is about optics. This begins with geometrical optics, the eikonal equation, and Fermat's principle of least time. This is extended to caustics and the intensity of light. The next three sections cover the ray theory of light. This is followed by a section on resolving power. The emphasis then shifts to wave optics with diffraction, Frensel diffraction, and Fraunhofer diffraction. Teh treatment is very abstract since it deals only with light propagation in the absence of matter.

The eighth chapter describes the field of moving charges. This begins with the so-called retarder potentials. This is then extended to the Lienard-Weichert potentials. Then Fourier analysis is applied to expand the field of moving charges into monochromatic waves. The final section deals with the issue of a finite interaction velocity and its ramifications for the Lagrangian and Hamiltonian.

The ninth chapter is long and wraps up electrodynamics in a vacuum. This begins by exploring the field of a system of charges at long distances by applying Fourier analysis. This is applied to the dipole radiator. This result is then extended into the dipole radiation due to colliding charges. This in turn is extended to the idea of bremsstrahlung. Bessel functions are discussed in the context of Coulomb interactions. Then dipole, quadrupole, and magentic dipole radiation are discussed. Then the focus shifts to fields close to a radiator. Then the focus shifts again to the fields produced by relativistic motion and synchrotron radiation. The authors then discuss radiation damping, the relativistic, and ultra-relativistic cases (where the particle energy is large when compared to the rest energy). The chapter ends with three sections on scattering.

Beginning with the tenth chapter the book switches over the general relativity. This chapter begins with a description of gravitational fields and the Lagrangian of the gravitational potential. Then the book discusses the metric and the idea of general covariance. The authors then present tensor analysis. This is followed by the motion of a particle in a gravitational field, the constant gravitational field, and a uniformly rotating frame of reference. The final section of this chapter deals with the electromagnetic field in a gravitational field.

The eleventh chapter presents Einstein's relativistic field equations. This begins with the Riemann curvature tensor and its properties, this is essentially a primer on differential geometry. Then they introduce the action functional of the gravitational field and its stress-energy tensor. This leads to the Einstein equations of general relativity. Then some consequences of the stress-energy tensor are introduced. After some more differential geometry the authors present the famous tetrad formulation, which ends the chapter.

The twelfth chapter explores the theory of gravitational fields beginning with a Newtonian treatment in tensorial language. Moving into the full relativistic treatment is the section on the centrally symmetric field including the famous Schwarzschild solution, leading to the prediction of black holes. Then they discuss motion in the Schwarzschild geometry. Leading into black hole theory is the section on the gravitational collapse of a spherical body, it is here that the idea of the event horizon is presented. These results are then extended to studying the internal condition of a collapsing body. The idea of a black hole is extended to a body in rotation described by the Kerr geometry. The authors then discuss gravitational fields at long distances from their sources. The chapter ends with the post-Newtonian approximation.

The thirteenth chapter is extremely relevent in the days of LIGO and LISA; gravitational waves. These are oscillations in the structure of spacetime itself. Beginning with weak and then strong gravitational waves, the authors discuss gravitational waves in curved space and the radiation of gravitational waves from sources.

The book wraps up with a chapter on cosmology. This begins with a discussion of cosmology of an isotropic space (the Friedmann model), then a closed isotropic space, and then an open isotropic space. This is followed by a discussion of the red shift, including the Hubble constant. Then the issue of the stability of an isotropic universe is addressed. Then the authors discuss homogeneous spaces. Then there is a section on the flat anisotropic model. This is followed by two sections on the initial singularity.

Bear in mind that this is not the normal treatment of relativity found in Misner, Thorne, and Wheeler or in Wald. This is full-blown relativity in a Lagrangian and Hamiltonian mechanical world, not for the faint of heart; but well worth the effort to work through.

The Wonders of Physics

2010-09-12T20:23:00.000-07:00

Earlier this year, I participated in a public demonstration lecture called the Wonders of Physics. In this year's show I demonstrated the physics of board-breaking.

I had a lot of fun, and you can find a video from a friend of mine, showing my participation in one of the earlier shows: http://www.youtube.com/watch?v=4eQWRLZG1rU

The official web site for the Wonders of Physics, where you can see all of the shows over the years, is here: http://sprott.physics.wisc.edu/wop.htm

I had a lot of fun, and it is a nice to give such presentations to the public, who hopefully will become interested in physics.

Book Review: Mechanics, by Landau and Lifshitz

2010-09-11T14:21:00.000-07:00

It is my intent to review every book in my store, below, so I will start with the first book in the store. You can purchase this book at my book store below.

This is the first volume of the classic Course of Theoretical Physics due to Lev Landau and the Landau Institiute of Theoretical Physics in Russia. These ten volumes were the subject of what Landau belived to be the required preparation in physics for any theoretical physicist.

This volume begins with a touching introduction to Lev Landau and his philosophy. It also lays out the prerequisites for this volume. To quote: "...ability to solve any indefinite integral that can be expressed in terms of elementary functions and to solve any ordinary differential equation of the standard type, knowledge of vector analysis and tensor algebra as well as the principles of the theory of functions of a complex variable (theory of residues, Laplace method). If you need this ackground I recommend Hassani, Mathematical Methods for Students of Physics and Related Fields, Second Edition.

The first chapter describes Lagrange's equations of motion from the starting point of generalized coordinates. This lays the ground work for a very nice presentation of the least action principle. Then there is a nice section of the Galilean transformations; it is important to realize that there is no discussion of special relativity in the book. The book then describes the Lagrangian of a free particle in Cartesian, spherical, and cylindrical coordinates. The final section of the first chapter not only discusses the Lagrangian of a system of particles, but nicely links Lagrange's equations to Newtonian theory.

The second chapter presents one of the most important aspects of theoretical physics, the conservation laws. This begins with a unified treatment of integrals of the motion, the conservation of energy, how conservation of energy implies the homogeneity of time, and conservative systems. This leads into the conservation of momentum, how the conservation of momentum implies the homogeneity of space, generalized momenta, and generalized forces. There is a brief discussion of systems of particles, the center of mass, and internal energy. This leads to a section on the conservation of angular momentum, how this implies the isotropy of space, and the idea of the central field. The final section deals with a couple of different topics such as the behavior of the equations of motion under transformations and the virial theorem.

The third chapter is a very practical one about applying the equations of motion to specific situations. This begins with one dimensional motion and a discussion of energy diagrams and oscillations. This leads into a discussion of how to interpret energy diagrams and derive the potential from a specific period of oscillation. The next topic covered is the beginning of the two-body problem in the form of the reduced mass of a system. The two-body problem is then reduced to a one-body problem in a central field in a very clear presentation. This leads to a nice discussion of the Kepler problem for bound and unbounded orbits.

Chapter four is an important discussion of the classical theory of particle scattering. This begins by the idea that particles can break up, this also introduces the lab and center-of-mass frames of reference. Then the authors describe elastic collisions in quite intuitive way. Then the notions of impact parameters and scattering-cross sections are covered. As an example of scattering due to fields, there is a section on Rutherford scattering. The chapter ends with small-angle scattering.

Chapter Five is on the theory of oscillations. Naturally, this begins with free oscillations in one dimension. The book then turns its attention to forced oscillations and resonance. Then the authors treat the idea of oscillations in more than one degree of freedom, including eigenfrequences and normal modes. This is then applied to the classical theory of molecules. Then damped oscillators are covered along with a discussion of dissipative functions in Lagrangian dynamics. The next section introduces dissipation and damped and driven oscillations. At this point the book leaves the realm of simple oscillating systems and goes into parametric oscillations and the Mathieu equation. There is a discussion of nonlinear oscillators, laying the ground-work for the study of chaos, this also includes a section of the resonance of nonlinear oscillators—introducing the Duffing oscillator without calling it that. Finally, there is an interesting section on the motion of a particle in a rapidly oscillating field.

Chapter six is a thorough treatment of rigid body dynamics. This begins with a discussion of the kinematics of rigid bodies. Then the authors introduce the inertia tensor. Indeed, this is one of the few sections in the book with more than a few problems (there are nine here). The section on angular momentum uses the axes of inertia to define angular momentum. Then the equations of motion for a rigid body are derived. This leads to the notion of Euler angles, which in turn leads to the Euler equations. These notions are then applied to the motion of a top. Then they shift gears a little and discuss rigid bodies in contact. Finally, and I think this might have done better at the beginning of the chapter, is a section on motion in noninertial frames, specifically rotating frames. This chapter seems very conventional, though it must be remembered that the first version of it came out in 1960, so a lot of other books are based on material found in here.

The final chapter is an overview of Hamiltonian theory. This begins, reasonably enough, with Hamilton's canonical equations of motion and relates the Hamiltonian and the Lagrangian in the normal way. The Routhian has its own section. Then they address the extremely important topic of the Poisson brackets. The next section establishes the idea of a functional without using those words. Then the authors discuss the principle of Maupertuis. Then they discuss canonical transformations and link them to Poisson brackets, and discuss conjugate variables, laying some of the groundwork for quantum mechanics. The authors next introduct the idea of phase space through a discussion of Liouville's theorem. The next section discusses the Hamilton-Jacobi equation and the idea of general and complete integrals. This is followed by a treatment of the method of separation of variables. Then the authors have an excellent and clear discussion of adiabatic invariants, a subject with applications in plasma physics. Then they discuss action-angle variables. Then the authors turn to the validity of the adiabatic invariant. The chapter, and the book, ends with a section on conditionally periodic motion.

The book, with all problems included, is just 167 pages long and comprises a total of 52 sections. This is something you could easily cover in two months of dedicated study. I think the book could use an update, but it's a classic as it is and its choice of topics are pretty good. I like it a lot, even if it doesn't really cover chaos in any meaningful way.

Conservation of Energy and The Nature of Forces

2010-09-10T22:20:00.000-07:00

Today I have been toying with a bunch of things:

1) Conservation of energy, I have been writing this section for the book project. Nice approach, if I do say so myself. Did you know that we really don't know what energy is other than a calculated number that remains the same in a closed system? We have no better understanding of energy than that. Despite being able to calculate it, predict it, and use it in other kinds of calculations; we have no idea what energy IS!

2) In Newtonian mechanics the entire game is chasing the forces. We have F=ma, and people think this is the definition of force. No! This states that the numerical quantity of force is equal to the product of mass and acceleration. If you do not know what F is prior to doing a calculation you will not be likely to figure it out.

3) I have been messing around, playing with trigonometric and hyperbolic functions.

Been having a lot of fun with these things.

Brief on Calculus and Appreciation to Leonard Susskind for his Lectures

2010-09-09T09:44:00.000-07:00

Hello everyone!

I have been working on the book project with Leonard Susskind. I just completed an 11-page overview of calculus with detailed material on limits, derivatives (including a short table of derivatives), integrals (with another short table of integrals), differential equations (covering mostly separation of variables), and Taylor series.

I am thinking of adding sections on partial derivatives, maxima and minima, and maybe curvature.

Of course, Leonard has not committed to anything yet; I certainly have not produced enough material to do more than whet his appetite for more. It is possible that something could happen to prevent me from finishing, and he probably does not want to stick his neck, or his reputation, out for someone who is not too well known.

I am having great fun writing this material. It is extremely challenging to develop short and clear explanations for these topics. My olf Mind of a Theorist column is good for this, the philosophy that Iadopted for the column is the same the Leonard has in the lectures.

To quote from the introduction, "What is this course about? Who am I teaching to? While undergraduate or graduate students will be able to read this, it is mostly designed for people who are interested in getting into the meat of physics right away. This is not a standard physics course. This is the real deal, theoretical physics at full scale! We use equations, and sometimes hard equations, but we try to use the simplest equations that will do the job. Basically, we try to keep it minimal. The goal here is to get to the basic ideas fast. So we will be telling you what you really, really, need to know to get to the next level. Sometimes the basics can be hard; we will do them anyway, but will only spend the minimum time required to get them right. And here we mean getting them really right, not by metaphors or analogies, but equations when necessary."

I am dazzled by the structure of the video lectures. In lecture one he introduces the ideas of dynamical systems, phase space, and conservation laws with no mathematics; just some diagrams and very clear explanations. Wonderful!

Wonderful Opportunity

2010-09-07T14:20:00.000-07:00

I have begun converting Leonard Susskind's video lecture series on Modern Physics to book form. I have just completed Lecture 1 of the video course Modern Physics: Classical Mechanics and am now working on a chapter on calculus.

So far, so good. These are still early days, but I plan to convert them all in time.

Whether it becomes anything is, at this point, up in the air; we will see how it goes.

I am honored to be working with someone so passionate about providing a good resource on theoretical physics to enthusiasts. I am honored that he didn't just slam the door in my face! I am having a great time in converting these lectures, not only because I get to work with one of the giants of theoretical physics; but also because I am forced to think deeply about the meaning of the lectures and their context. It is a wonderful opportunity.

Thanks, Leonard!

Great Find!

2010-09-01T02:22:00.000-07:00

Dover Publications has produced a version of Richard Feynman and Albert Hibbs' excellent textbook, "Quantum Mechanics and Path Integrals." The text has been cleaned of typographic and grammatical errors, and the notation has been standardized, while maintaining the rough edge to the original. This is probably the best text for the path integral approach to quantum mechanics. Enjoy!

Creeping Featuritis

2010-08-30T21:58:00.000-07:00

In the computer world there is a term used to describe a project, intended to be accomplished in finite time, where the client keeps wanting new things added before the project is completed; called creeping-featureitis. When building a computer project, this is a bad thing.

In theoretical physics, since we know we do not know all of the rules of the game, this is the norm. We are always adding new things: ideas, theorems, examples, counterexamples, contradictions, etc.

It is not surprising that theorists have computer code that is ever expanding; that is how we think!

To that extent, all theorists engage in creeping-featuritis and it is the way it should be.

Weird Fractional Quantum Hall Effect

2010-08-25T10:17:00.000-07:00

In a recent publication to Physical Review Letters, "Optical Probing of the Spin Polarization of the v= 5/2 Quantum Hall State" the authors: M. Stern, P. Plochocka, V. Umansky, D. K. Maude, M. Potemski, and I. Bar-Joseph, describe a phenomena resulting in a pseudoparticle having non-integer charge!

This is all based on the fractional quantum Hall effect.

Recall the classical Hall effect, discovered by Edwin Hall in 1879. This effect occurs when you apply a magnetic field perpendicular to a current in a conductor, producing a voltage difference across the conductor. This voltage difference is sometimes called the Hall Voltage.

If you have a system of electrons in a plane (two-dimensional) or on a surface that are at low temperatures, you can produce Hall voltages of only quantized values by applying a strong magnetic field. This is the integer quantum Hall effect. You can calculate the Hall Resistance,

R= (h/e^2)/v

Where h is Planck's constant, e is the fundamental charge, and v is called the filling factor.

In some cases the electrons behave as if they are a fluid. In this case they behave as a pseudoparticle with fractional charge. Very interesting! This occurs when v is fractional, it is specifically interesting at v = 5/2.

I have not yet finished the letter, but something mysterious occurs at v = 5/2. There is a lot of interesting physics here!

George

Hellow World

2010-08-20T19:29:00.000-07:00

Hello everyone! My name is George, and I used to write a column on The Citizen Scientist (an e-publication put out by the Society for Amateur Scientists) called The Mind of a Theorist. This publication has ended (along with my column), at least temporarily, so I have relaunched it here.

The goal of the column was to provide good content about theoretical physics in an educational way.

I was forced to change the format due to editorial interference.

I want to return to the old way of doing it.

To that end, I will be going over my old columns and resurrecting them for the dead here. In some cases I will not change much, while in some I will trash the old and ring in the new right away.

With each post I will discuss a physics news item, my research, a topic of interest to readers, a comment from a post, or other such. I will also present something educational; something to help you become a theoretical physicist - even an amateur one; and then once a week I will issue a theorist challenge. At the moment, I will not be able to provide any prizes, but if I get enough support that might happen. I will always answer the challenge in the next week.

About me: I am the President of the nonprofit organization Madison Area Science and Technology (MAST). We do science without regard to credentials. You can check out our web site at http://www.madscitech.org/ .

That is all for now, I will now check out how to include mathematical symbols in text.

George