This is an archival listing of my notes from 2009 (electrodynamics, Lagrangian and Hamiltonian formalisms, Noether's theorem for solids and fields, quantum mechanics)

Dec 21, 2009 Energy and momentum for assumed Fourier transform solutions to the homogeneous Maxwell equation.

Fourier transform instead of series treatment of the previous, determining the Hamiltonian like energy expression for a wave packet.

Dec 16, 2009 Electrodynamic field energy for vacuum.

Apply the previous complex energy momentum tensor results to the calculation that Bohm does in his QM book for vacuum energy of a periodic electromagnetic field. I'd tried to do this a couple times using complex exponentials and never really gotten it right because of attempting to use the pseudoscalar as the imaginary for the phasors, instead of introducing a completely separate commuting imaginary. The end result is an energy expression for the volume element that has the structure of a mechanical Hamiltonian.

Dec 13, 2009 Energy and momentum for Complex electric and magnetic field phasors.

Work out the conservation equations for the energy and Poynting vectors in a complex representation. This fills in some gaps in Jackson, but tackles the problem from a GA starting point.

Dec 6, 2009 Jacobians and spherical polar gradient.

Dec 1, 2009 Polar form for the gradient and Laplacian.

Explore a chain rule derivation of the polar form of the Laplacian, and the validity of my old First year Proffessor's statements about divergence of the gradient being the only way to express the general Laplacian. His insistence that the grad dot grad not being generally valid is reconciled here with reality, and the key is that the action on the unit vectors also has to be considered.

Nov 30, 2009 Two particle center of mass Laplacian change of variables.

Nov 26, 2009 Lagrangian and Euler-Lagrange equation evaluation for the spherical N-pendulum problem

Nov 15, 2009 Force free relativistic motion.

Nov 13, 2009 Linearizing a set of regular differential equations.

Nov 11, 2009 question on elliptic function paper.

Nov 4, 2009 Spherical polar pendulum for one and multiple masses (Take II)

The constraints required to derive the equations of motion from a bivector parameterized Lagrangian for the multiple spherical pendulum make the problem considerably more complicated than would be the case with a plain scalar parameterization. Take the previous multiple spherical pendulum and rework it with only scalar spherical polar angles. I later rework this once more removing all the geometric algebra, which simplifies it further.

Oct 27, 2009 Spherical polar pendulum for one and multiple masses, and multivector Euler-Lagrange formulation.

Derive the multivector Euler-Lagrange relationships. These were given in Doran/Lasenby but I did not understand it there. Apply these to the multiple spherical pendulum with the Lagrangian expressed in terms of a bivector angle containing all the phi dependence a scalar polar angle.

Oct 4, 2009 Linear transformations that retain two by two positive definiteness.

Sept 26, 2009 Hamiltonian notes.

Sept 24, 2009 Electromagnetic Gauge invariance.

Sept 22, 2009 Lorentz force from Lagrangian (non-covariant)

Sept 20, 2009 Spherical Polar unit vectors in exponential form.

Sept 19, 2009 Reader notes for Jackson 12.11, Retarded time solution to the wave equation.

Sept 13, 2009 Relativistic classical proton electron interaction.

Sept 6, 2009 Bivector grades of the squared angular momentum operator.

Sept 5, 2009 Maxwell Lagrangian, rotation of coordinates.

Sept 4, 2009 Translation and rotation Noether field currents.

Aug 31, 2009 Generator of rotations in arbitrary dimensions.

Aug 18, 2009 Spherical harmonic Eigenfunctions by application of the raising operator

Aug 16, 2009 Graphical representation of Spherical Harmonics for $l=1$

Aug 14, 2009 (INCOMPLETE) Geometry of Maxwell radiation solutions

Aug 10, 2009 Covariant Maxwell equation in media

Aug 6, 2009 Comparing phasor and geometric transverse solutions to the Maxwell equation

July 30, 2009 Transverse electric and magnetic fields

July 27, 2009 Bivector form of quantum angular momentum operator

July 21, 2009 Stokes theorem derivation without tensor expansion of the blade

July 17, 2009 Stokes theorem applied to vector and bivector fields

July 2, 2009 Space time algebra solutions of the Maxwell equation for discrete frequencies

July 1, 2009. Poincare incremental transformations.

June 27, 2009. Relativistic Doppler shift. Deriving this with a Lorentz boost is much simpler than the time dilation argument in wikipedia.

June 21, 2009. Lorentz invariance of energy momentum four vector.

June 21, 2009. Maxwell wave equations in absence of charge and current.

June 18, 2009. Expand on Pauli's SR based motivation of the Schrodinger equation.

June 17, 2009 Examine two variations of the Lorentz force Lagrangian.

June 7, 2009. Canonical stress energy tensor and Noether's theorem. Figured out the math parts, and now have applications of the math to some specific Lagrangian densities (starting with the relativistic wave equation). Have more to do with the Maxwell Lagrangian result.

May 31, 2009. Assemble most of my GA notes into book like form. This groups most of my standalone GA notes into a somewhat organized compilation.

May 28, 2009 GA Musings on macroscopic Maxwell's equation. Got my "new" second hand 2nd ed. of Jackson's E&M in the mail, and got distracted reading the introduction. Turns out that a trivector "current" term (with basis vectors in the Dirac vector space) to supplement the four-vector current completely summarizes the mess of B,D,H,E,M,P,J,rho variables nicely in a fashion very similar to the \grad F = J variation of Maxwell's equation for the microscopic case.

May 23, 2009 Lorentz boost of Lorentz force equations. My own attempt to walk through the Lorentz transformation of the pair of Lorentz force and power equations, as done in Bohm's 'The Special Theory of Relativity'. Bohm's text left out a number of details (as well as had a number of sign typos and some dropped terms). Try to get it right. Was able to do some of it, but part of the final "the reader can verify bits" have me stumped. How to do those last bits is not obvious to me, which is likely why Bohm left this out of this pseudo-layman book. This set of notes starts off with a large digression (again I think) on how to express and translate from the GA hyperbolic exponential Lorentz boost formulation to the "classical" coordinate and vector representations used in the Bohm text (and other places). My initial reason for writing that up for myself all in one place was that I intended to try the Lorentz force boost procedure of the Bohm text completely in GA form, but I also have not gotten to attempting that (my goal was to finish the details of the "old-fashioned" way first, but the algebra for that way is so messy I don't see how to do it).

May 14, 2009. Rectangular quantum barrier penetration. Work through all the messy algebra, for the wave function calculation (ie: determine the coefficients), the probability densities, and the probability currents, in full gory detail. This was in response to problem 11.4 of Bohm, and an attempt to reconcile a first busted attempt at that problem with other sources (such as wikipedia and QMD).

May 8, 2009. Start of chapter 11 problems from Bohm.

May 3, 2009. A playful description of the V = I R triangle for my dad, who's been complaining for years that these are all cyclicly defined in terms of each other. The point in the end is perhaps not much more than, yes it is possible to define things in a non-cyclic fashion.... but I'm not so sure that my non-cyclic description itself ended up that good.

April 30, 2009. Pi-Meson problem from the calculus of variations problems of Byron and Fuller. Also compute the Noether current for phase invariance, and the attempt to show how a local gauge phase transformation leads to the interaction Lagrangian. I end up with one term having a different sign than in the original, and can't spot an error. Is there a sign error in the problem of the text? May 3: Do the snell's law problem (at least the first part, showing the angle relation by minimizing the path over time).

April 28, 2009. Developing some intuition for Multivector Taylor Series. May 4. Explicit expansion and Hessian matrix connection. Factor out the gradient from the direction derivative for a few different multivector spaces.

April 28, 2009. Simple minded variation of one dimensional wave equation Lagrangian. Getting from the Lagrangian to the wave equation, without using the field form of the Euler-Lagrange equations.

April 27, 2009. Select problems from Chapter 10 of Bohm's Quantum Theory.

April 21, 2009. Maxwell equation from Lagrangian. A tensor only derivation.

April 20, 2009. Fixup tensor only derivation (from Lagrangian) of Lorentz force equation. Handle the off by -1 sign error I left in some of the index gymnastics.

April 20, 2009. Quantum Harmonic Oscillator. First two energy level solutions without using the fancier operator raising or lowering methods.

April 18, 2009. Biot-Savart Derivation. Try this starting from (steady state) Ampere-Maxwell equation using the three vector potential, instead of the full Maxwell equation expressed in the Lorentz gauge (as done last time when I thought this would be helpful to understand magnetic field energy).

April 15, 2009. A small experiment with the Lorentz force Lagrangian in terms of the conjugate momentum. The Lagrangian can be expressed in a QM like form in terms of a sum of mechanical and electromagnetic momentum, mv + qA/c. The end result is the same and it works out to just be a factorization of the original Lorentz force (covariant) Lagrangian.

April 13, 2009. Proving that the i-commutator (imaginary scaled) and anticommutator operators are Hermitian (given Hermitian operators). Notes from reading of Pauli's wave mechanics.

April 10, 2009. Acceleration four vector notes. Also from reading Pauli.

April 10, 2009. Reconcile the covariant velocity addition formula (via Lorentz boost), with the traditional introductory differential division presentation. Further reading of Pauli's 'Theory of relativity' text prompted this bit of self clarification.

March 29, 2009. Klein Gordon equation notes. To get the feel for global and local guage transformations, work through this for the KG equation, as Susskind did in his relativity lecture. As context, examined the equation and its solution a bit as well.

March 26, 2009. Proving the binomial theorem for integer powers the super easy and dumb way.

March 24, 2009. Revisit Noether's conserved current for boost and rotation symmetries of Maxwell Lagragian. Write out things explicitly for the rotation as well as the boost, and put in E and B three vector form. Reason for revisiting this is to compare to PF thread on QM spin that Lut pointed me to.

March 22, 2009. Start the calculus of variations problems from Byron and Fuller. Shortest line using polar coordinates. Spherical geodesics (unfinished).

March 20, 2009. Work through the rotor bivector treatment of the Lorentz force by Baylis. Time evolution of a particle in a field as a bivector differential equation, solving for the active Lorentz transformation on the rest frame worldline. Work it out at my own pace in both the GA and tensor formalism.

March 14, 2009. Some notes on an a Levi-Civita summation identity. A summation identity given in Byron and Fuller, ch 1. Initial proof with a perl script, then note equivalence to bivector dot product.

March 13, 2009. Dot product linearity using cosine form. Also demonstrate the law of cosines.

March 9, 2009. Some notes on Pauli's 'The Completeness Relation' of chapter II, in "Wave Mechanics". Explore the delta function representation given in Pauli's book, and express some of Pauli's content in my own words to build understanding, and relate fourier series and transforms under the general umbrella of orthonormal inner product spaces.

March 8, 2009. Some of the problems Chapter 9 of from Bohm's Quantum Theory book. Have done all but one of these problems, and now written up solutions to a subset of these.

March 5, 2009. A test application using tempered distribution theory for Fourier analysis. Solve the homogeneous first order advanced wave equation in one variable using classical Fourier techniques and also the distribution formalism to compare the two, and get a feel for the latter. This applies the distribution techniques from Prof Brad Osgood's Fourier lectures (Stanford on iTunesU). After listening to the distribution lectures, I wasn't convinced that this method would be practical, but the proof is in the application. I was surprised that it is actually simpler, with no "so many words" requirements to pull delta functions out of magic hats from PV sinc evaluations of the exponential integral.

February 28, 2009. N-sphere and N-hypersphere vector parameterization and volume elements. Volume calculations for 1-sphere (circle), 2-sphere (sphere), 3-sphere (hypersphere). Followup with a calculation of the differential volume element for the hypersphere (ie: Minkowski spaces of signature (+,-,-,-). Plan to use these results in an attempt to reduce the 4D hyperbolic Green's functions that we get from Fourier transforming Maxwell's equation.

February 22, 2009. Revisit Solutions to Goldstein Mechanics problems from chapter I and II. Problem 8 from Chapter I was never really completed in my first pass (and it looks like I missed the Kinetic term in the Lagrangian too). The question of if angular momentum is conserved in that problem is considered in more detail, and a Noether's derivation that is specific to the calculation of the conserved ``current'' for a rotational symmetry is performed. I'd be curious what attack on that question Goldstein was originally thinking of. Although I believe this Noether's current treatment answers the question in full detail, since it wasn't covered yet in the text, is there an easier way to get at the result?

February 21, 2009. Reduce the Fourier derived Green's function for the Poisson and wave equations. Work through the details of how to derive the Poisson integral kernel starting with the Fourier transform derived Green's function. Do the same thing with the wave equation, and produce the retarded and advanced form solutions. A few years in the works since seeing them in Feynman and wondering where they came from. Feb 25. Did a reduction of the 1D forced wave equation's Green function to a difference of unit step functions. Have to compute derivatives to see if this really works.

February 18, 2009. Work the energy momentum Lorentz force (density) relation in tensor form.

February 16, 2009. Perform some of the wave packet integrals from Bohm's book. Just calculus.

February 16, 2009. Revisit: Some vector differential identities. Make a note of the last two identities that I wanted to work through (incomplete attempt at them ... trickier than I expected, and probably why they were omitted from Feynman's text.)

February 14, 2009. Energy momentum tensor and its relationship to the Lorentz force. Express the energy momentum tensor in terms of the four vector Lorentz force. This builds on the previous observation that the T(\gamma_0) is related to the work done against the Lorentz force.

February 13, 2009. Revisit field energy momentum and stress tensor notes from last year. Relate the energy-density-rate + Poynting divergence equation to the Lorentz force and discuss. Also relates the various terms of the stress energy tensor to the Lorentz force. See now how the covariant Lorentz force and the stress energy tensor is related, and also have some intuitive justification now for why we call E^2 + B^2 the field energy density (ie: justify in terms of work done against Lorentz force).

February 8, 2009. Revisit Fourier series (plane wave) solutions of vacuum equation. My first attempt is getting confusing, especially after seeing after the fact that plane wave constraints on the solution are required for the solution to maintain a grade two form. Summarizes results from the first attempt in a more coherent (albeit denser) form.

February 7, 2009. Fourier potential solution of the vacuum Maxwell equation. Split from the first order treatment below.

February 4, 2009. Fourier series solution of vacuum Maxwell equation. Go through Bohm's treatment that preps for the Rayleigh-Jeans result in his quantum book in a more natural way (use complex exponentials, with the STA pseudoscalar for i, and use the much simpler STA maxwell vacuum equation as the base).

February 1, 2009. 4D Fourier transform solutions of Maxwell's equation. Wow, using a spacetime Fourier transform for a Maxwell's solution is much simpler. This is a neat result.

February 1, 2009. A closed form double sided Green's function solution of Maxwell's equation using Fourier methods. Application of the Fourier transform to the spacetime split of the gradient term of Maxwell's equation allows for a complete solution of both the vacuum and current forced fields without requiring any computation with four vector potentials. Presuming I got all the math right, this is a beautiful application of both Fourier theory and the STA algebra. Note that the Rigor police are thoroughly away on vacation in this particular set of notes!

January 31, 2009. Apply my new favorite tool, the Fourier transform to the forced wave equation and Maxwell's equation. Work out a Green's function solution of sorts for the non-homogenious Maxwell's equation.

January 26, 2009. One dimensional wave equation solution using Fourier transform. Produces the f(x,t) = g( x - vt ) solution quite nicely! This works in a fashion for the 2 and 3D cases too, but there the Green's function doesn't reduce nicely to a delta function as in the 1D case.

January 25, 2009. Play with vacuum solutions to the Maxwell equation. Carry the separation of variables to a reasonable point of completion, deriving a tidy relativistic solution for F_{\mu\nu}. After this try generalizing that a bit with some intuition that turned out to be busted. Left my dead ends as a marker pointing where not to go in the future.

January 24, 2009. Relativity equations of motion from a Hamiltonian treatment. A walk through of Pauli's relativity intro from his "Wave Mechanics" book at my own pace.

January 23, 2009. A dumbed down derivation of the one dimensional Ehrenfest theorem. This theorem expresses the classical limit of QM in terms of expectation values of the position and momentum operators. Rather than use the slick and fancy operator commutator formalism, which I haven't taken the time to learn yet, do this with just integration by parts.

January 21, 2009. Cheat sheet for Fourier transform convention variations.

January 19, 2009. Heat and wave equation solutions using Fourier series and Fourier transforms. Apply the series technique to solve for the general time evolution of a wave function for a free (no potential) particle constrained to a circle, and the transform method for a one dimensional linear (non-periodic) scenerio.

January 18, 2009. Calculate the time rate of change of the Poynting vector, and the associated conservation law. These notes contain the conservation calculation itself, and verify the end result of Schwartz's tricky relativistic argument (that I have yet to understand) to put the conservation into a divergence form that is volume integrable.

The derivation itself is not too hard. Reconciling all the different notations is actually the tricky bit. Schwartz does this in terms of the dual field tensors F and G, Doran/Lasenby have their GA F \gamma_k F formulation, wikipedia had something different either of than those, and I'd seen in another paper that Jackson used something completely different (but don't have Jackson to see how he did it).

Very interesting here is that we end up with what looks like the Lorentz force law by only looking at conservation requirements based on Maxwell's equation itself. Calling the Poynting vector a field momentum density by analogy (because it showed up in what appeared to be an Energy/momentum (density) four vector) is then seen to be very justifiable. Previously I'd seen that it took two Lagrangians for E&M. One for the fields and one for the interaction term. But now it looks like the interaction term follows from the fields (in a handwaving, fuzzy, not yet fully understood way). Quite interesting, and worth more thought, but seeing how one gets the interaction term from the QM field equation should probably take precedence.

January 17, 2009. Composition of two 90 degree rotations. Work problem from a GA book draft to discuss difference in solution.

January 16, 2009. Work the chapter II quiz problems from Quantum Mechanics Demystified. This includes the problem solutions, and whatever bits of math or QM that was related to the problems that I found interesting (like remembering how to do residue calculus). In the process of working these problems I've also noticed a few typos;)

January 13, 2009. Super simple calculation of polar form velocity and acceleration in a plane. Straight up column matrix vectors and complex number variants of radial motion derivatives.

January 11, 2009. Schrodinger probability density and current conservation equation. Derive the probability current conservation law for myself.

Calculating the rate of change of probability, and using Schrodinger's equation and its conjugate allows for the definition of a probability current, and an electromagnetic like probability-density/current-density conservation law.

What I thought was interesting was that if you put this into a four vector form as a spacetime divergence (ie: the Lorentz gauge of e&m), the resulting "four-component" current vector needs only a \gamma^0 \partial_0 term to be added to it, for that current itself to be the Dirac Lagrangian (omitting the local-gauge term eA). So it looks like taking the spacetime divergence of the Dirac Lagrangian essentially gives you the probability/current conservation equation (except now this would also produce an extra timelike term not there in the original Schrodinger's equation.) There are some notational differences with the wikipedia form of the Dirac Lagrangian, but I believe all the basic content is there once those differences are accounted for. Very suprising to see the Dirac Lagrangian fall so naturally out of the Schrodinger (non-relativistic) equation.

I also observe that the probability wave function is perhaps naturally expressed as a relativistic four vector (with a \gamma_0 term factored out). I still don't understand how Maxwell's equation and QM fit together, but with Maxwell's equation or Lagrangian expressable strictly in terms of four vectors (or the four-gradient and four-curl of such four vectors), there would be a logical cleanliness if one could also express the (relativistic) QM laws strictly in terms of four vectors. Definitely worth playing with.

January 10, 2009. Verify the QM formulation (ie: with hbar's) of the Fourier transform pair seen in the book "Quantum Mechanics Demystified". Very non-rigourous treatment, good only for intuition. Also derive the Rayleigh Energy theorem used (but not proved) in this text.

January 6, 2009. DC Power consumption equation derived from Poynting relationship. Work out P = I V from first principles since I forgot it. Well, from second principles I suppose, since I utilize my recent Poynting derivation.

January 5, 2009. Some vector differential identities. Translate some identities from the Feynman lectures into GA form. These apply in higher dimensions with the GA formalism, and proofs of the generalized idenities are derived.

January 5, 2009. Gaussian bell curve integral evaluation. Nothing fancy, but I couldn't remember how to do it at first. Also added the next two degree Gaussian integrals, and a derivation of the recurrance relations for the higher degree variations.

January 3, 2009. Electrostatic and magnetostatic field energy.
Start working out for myself the electrostatic and magnetostatic energy
relationships. Got the electrostatic part done, and got as far as a
from first principles Biot-Savart derivation using the STA formalism.
Next work out the magnetostatic energy relationship. Also intend to
tackle wave energy and momentum here, but in the end, may split that
into a separate set of notes.