New version of my Geometric Algebra notes compilation posted.
Posted by peeterjoot on October 31, 2010
Changes since the last posting likely include the incorporation of the following individual notes:
Oct 30, 2010 Multivector commutators and Lorentz boosts.
Use of commutator and anticommutator to find components of a multivector that are effected by a Lorentz boost. Utilize this to boost the electrodynamic field bivector, and show how a small velocity introduction perpendicular to the a electrostatics field results in a specific magnetic field. ie. consider the magnetic field seen by the electron as it orbits a proton.
Oct 23, 2010 PHY356 Problem Set II.
A couple more problems from my QM1 course.
Oct 22, 2010 Classical Electrodynamic gauge interaction.
Momentum and Energy transformation to derive Lorentz force law from a free particle Hamiltonian.
Oct 20, 2010 Derivation of the spherical polar Laplacian
A derivation of the spherical polar Laplacian.
Oct 10, 2010 Notes and problems for Desai chapter IV.
Chapter IV Notes and problems for Desai’s “Quantum Mechanics and Introductory Field Theory” text.
Oct 7, 2010 PHY356 Problem Set I.
A couple problems from my QM1 course.
Oct 1, 2010 Notes and problems for Desai chapter III.
Chapter III Notes and problems for Desai’s “Quantum Mechanics and Introductory Field Theory” text.
Sept 27, 2010 Unitary exponential sandwich
Unitary transformation using anticommutators.
Sept 19, 2010 Desai Chapter II notes and problems.
Chapter II Notes and problems for Desai’s “Quantum Mechanics and Introductory Field Theory” text.
July 27, 2010 Rotations using matrix exponentials
Calculating the exponential form for a unitary operator. A unitary operator can be expressed as the exponential of a Hermitian operator. Show how this can be calculated for the matrix representation of an operator. Explicitly calculate this matrix for a plane rotionation yields one of the Pauli spin matrices. While not unitary, the same procedure can be used to calculate such a rotation like angle for a Lorentz boost, and we also find that the result can be expressed in terms of one of the Pauli spin matrices.
July 23, 2010 Dirac Notation Ponderings.
Chapter 1 solutions and some associated notes.
June 25, 2010 More problems from Liboff chapter 4
Liboff problems 4.11, 4.12, 4.14
June 19, 2010 Hoop and spring oscillator problem.
A linear appromation to a hoop and spring problem.
May 31, 2010 Infinite square well wavefunction.
A QM problem from Liboff chapter 4.
May 30, 2010 On commutation of exponentials
Show that commutation of exponentials occurs if exponentiated terms also commute.
May 29, 2010 Fourier transformation of the Pauli QED wave equation (Take I).
Unsuccessful attempt to find a solution to the Pauli QM Hamiltonian using Fourier transforms. Also try to figure out the notation from the Feynman book where I saw this.
May 28, 2010 Errata for Feynman’s Quantum Electrodynamics (Addison-Wesley)?
My collection of errata notes for some Feynman lecture notes on QED compiled by a student.
May 23, 2010 Effect of sinusoid operators
Liboff, problem 3.19.
May 23, 2010 Time evolution of some wave functions
Liboff, problem 3.14.
May 15, 2010 Center of mass of a toroidal segment.
Calculate the volume element for a toroidal segment, and then the center of mass. This is a nice application of bivector rotation exponentials.
Mar 7, 2010 Newton’s method for intersection of curves in a plane.
Refresh my memory on Newton’s method. Then take the same idea and apply it to finding the intersection of two arbitrary curves in a plane. This provides a nice example for the use of the wedge product in linear system solutions. Curiously, the more general result for the iteration of an intersection estimate is tidier and prettier than that of a curve with a line.
Mar 3, 2010 Notes on Goldstein’s Routh’s procedure.
Puzzle through Routh’s procedure as outlined in Goldstein.
Feb 19, 2010 1D forced harmonic oscillator. Quick solution of non-homogeneous problem.
Solve the one dimensional harmonic oscillator problem using matrix methods.
Jan 1, 2010 Integrating the equation of motion for a one dimensional problem.
Solve for time for an arbitary one dimensional potential.
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 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.
Sept 26, 2009 Hamiltonian notes.
Sept 24, 2009 Electromagnetic Gauge invariance.
Show the gauge invariance of the Lorentz force equations. Start with the four vector representation since these transformation relations are simpler there and then show the invariance in the explicit space and time representation.
Sept 22, 2009 Lorentz force from Lagrangian (non-covariant)
Show that the non-covariant Lagrangian from Jackson does produce the Lorentz force law (an exersize for the reader).
Sept 20, 2009 Spherical Polar unit vectors in exponential form.
An exponential representation of spherical polar unit vectors. This was observed when considering the bivector form of the angular momentum operator, and is reiterated here independent of any quantum mechanical context.
Sept 13, 2009 Relativistic classical proton electron interaction.
An attempt to setup (but not yet solve) the equations for relativistically correct proton electron interaction.
Sept 10, 2009 Decoding the Merced Florez article.
Sept 6, 2009 bivectorSelectWrong
Sept 6, 2009 Bivector grades of the squared angular momentum operator.
The squared angular momentum operator can potentially have scalar, bivector, and (four) pseudoscalar components (depending on the dimension of the space). Here just the bivector grades of that product are calculated. With this the complete factorization of the Laplacian can be obtained.
Sept 5, 2009 Maxwell Lagrangian, rotation of coordinates.
Sept 4, 2009 Translation and rotation Noether field currents.
Review Lagrangian field concepts. Derive the field versions of the Euler-Lagrange equations. Calculate the conserved current and conservation law, a divergence, for a Lagrangian with a single parameter symmetry (such as rotation or boost by a scalar angle or rapidity). Next, spacetime symmetries are considered, starting with the question of the symmetry existance, then a calculation of the canonical energy momentum tensor and its associated divergence relation. Next an attempt to use a similar procedure to calculate a conserved current for an incremental spacetime translation. A divergence relation is found, but it is not a conservation relationship having a nonzero difference of energy momentum tensors.
Aug 31, 2009 Generator of rotations in arbitrary dimensions.
Similar to the exponential translation operator, the exponential operator that generates rotations is derived. Geometric Algebra is used (with an attempt to make this somewhat understandable without a lot of GA background). Explicit coordinate expansion is also covered, as well as a comparison to how the same derivation technique could be done with matrix only methods. The results obtained apply to Euclidean and other metrics and also to all dimensions, both 2D and greater or equal to 3D (unlike the cross product form).
Aug 20, 2009 Introduction to Geometric Algebra.
Aug 16, 2009 Graphical representation of Spherical Harmonics for $l=1$
Observations that the first set of spherical harmonic associated Legendre eigenfunctions have a natural representation as projections from rotated spherical polar rotation points.
Aug 14, 2009 (INCOMPLETE) Geometry of Maxwell radiation solutions
After having some trouble with pseudoscalar phasor representations of the wave equation, step back and examine the geometry that these require. Find that the use of $I\zcap$ for the imaginary means that only transverse solutions can be encoded.
Aug 11, 2009 Dot product of vector and bivector
Aug 11, 2009 Dot product of vector and blade
Aug 10, 2009 Covariant Maxwell equation in media
Formulate the Maxwell equation in media (from Jackson) without an explicit spacetime split.