PH561: Mathematical Physics

In Fall 2012 the course is being taught by Prof. Ethan Minot.

Calendar

Week 1

MNo ClassComprehensive Exam
1WDiff. Eqns.day1.pdf: Classifying differential equations. Quick analysis of v' = -g + αv². Overview of Diff EqnsReading Assignment 1
2FPythonday2.pdf: Numerical solution of v' = -g + αv². Meet in Weniger 412. hw1, hw1solnb.pdf

Week 2

3MApplication of Complex Numbersday3.pdf: Using complex numbers for the damped-driven HO. Cartoon intro to Green's functions.Butkov §2.1 - 2.5
4WGreen's function methodday4.pdf: Standard format for ODEs: Differential operator on left, source term on the right. Example: solving the damped driven particle problem. Formal definition of Green's function. Example: string under tension with perpendicular forces. Butkov §12.1
5FGreen's function methodday5.pdf: Recipe for applying Green's function technique. Extend to PDEs. hw2, hw2soln.pdf

Week 3

6MGreen's functions, Homogeneous ODEsday6.pdf: Green's function for the Laplacian differential operator. General techniques for solving homogeneous ODEs.
7WHomogeneous ODEsday7.pdf: Power series technique for solving homogeneous ODEs.Butkov §3.4
8FHomogeneous ODEsday8.pdf: Power series technique for the Legendre Eqn and the Bessel EquationButkov §3.5hw3, hw3soln.pdf

Week 4

9MPartial Diff. Eqns.day9.pdf: Separation of variables: Pick apart the PDE for the H atom wavefunctionButkov §8.1 - 8.4
10WSeries/Contour Integrals on Argand Planeday10.pdf: Quick discussion of series convergence. Introduce contour integration on the Argand plane and the Residue theorem - applications for calculating definite integrals.Butkov §2.7-2.12, Mandelbrot set
11FThe Residue Theoremday11.pdf: Applying the residue theorem to evaluate definite integralsButkov §2.12hw4, hw4soln.pdf

Week 5

12MFourier seriesday12.pdf: Finding the Fourier coefficients.Butkov §4.1 - 4.3
13WFourier series / transformday13.pdf: Application of Fourier series - bending a rigid beam. Transition from a Fourier cosine series to a Fourier cosine transform. Application of Fourier cosine transform: Diffraction of light passing through a narrow slit.Butkov §4.8
14FFourier series / transformday14.pdf: Fourier transform derived (heuristically) from the complex form of the Fourier series. Butkov §7.1hw5, hw5soln.pdf

Week 6

15MFourier Transform Applicationsday15.pdf: Review the “plucking a guitar” homework question. Fourier transforms used to calculate diffraction patterns in optics. The “uncertainty principle for wave packets”. Fourier transform of a Gaussian and a delta function. Fourier transforms used to analyze noisey experimental data.Butkov §7.2
16WFourier Transform: Solve ODEday16.pdf: Transforming an ODE. Top hat force function acts on a harmonic oscillator.Butkov §7.7
17FFourier Transform: Solve ODEday17.pdf: Top hat force function acts on a harmonic oscillator. Practice using the residue theorem to finish off the formal solution. Check answer using easy cases.Butkov §7.7, Mahajan §2

Week 7

18MQuantifying heat transportday18.pdf: Temperature distribution over a strip of sheet metal. Practice separation of variables to solve a PDE, putting constraints on the separation constant, solving ODEs, using Fourier series. Heat eqn
19WQuantifying QM scatteringday19.pdf: Review HW#5. Review plane waves. Set up the “Integral Schrodinger Equation” to approximate a scattered wavefunction, this gives practice with Green's functions and Fourier Transforms.
20FQuantifying QM scatteringday20.pdf: Apply a Fourier transform to the k-space representation of the Green's function. This leads to a discussion about integrals that go straight through simple poles. Discuss the meaning of the final answer (spherical waves). hw6, hw6soln.pdf

Week 8

21MPictorial Proofsday21.pdf: Review QM scattering (use a picture). Discuss the use of pictures in solving mathematical problems and demonstrating “why” something is true. Example 1: Finite series of odd numbers. Example 2: The Pythagorean Theorem. Example 3: The Stirling Approximation.Mahajan §4
22WTaking out the big partday22.pdf: Multiply numbers with small fractional corrections, add the fractional corrections (used in fast mental arithmetic, error analysis in experiments). Physics question: Measure the depth of a well by timing a stone's fall time plus a travel time for sound. Pros and cons of the precise approach vs. the “first things first” approach.Mahajan §4
23FEigenvalue equationsday23.pdf: Define a vector space for functions. Define an inner product space for functions. Define a Hilbert space for functions. Method to convert a set of basis functions to an orthonormal set of basis functions.Butkov §9.3 - 9.4hw7, hw7soln.pdf

Week 9

24MEigenvalue equationsday24.pdf: Hermitian operators, limits on boundary conditions, Sturm-Louiville form.Butkov §9.3 - 9.4
25WEigenvalue equationsday25.pdf: PDEs can be separated into to ODEs, the unspecified separation constant can always be thought of as an eigenvalue, and the solutions to the ODE can be thought of as eigenfunctions.Butkov §9.1 - 9.4
FNo classThanksgiving

Week 10

26MReview weekday26.pdf: Review HW#7. Example problems for final, covering Days 1 - 9.
27WReview weekday27.pdf: Example problems for final, covering Days 10 - 18. hw8, hw8soln.pdf
28FReview weekday28.pdf: Example problems for final, covering Days 19 - 25.

Syllabus

Complex numbers

1. Circuit analysis
2. Deriving trig identities, tables of integrals
3. Switching to complex numbers to solve DEs
4. Kramer's Kronig relations

Differential equations

1. Driven damped harmonic oscillator
2. Stretched string
3. Shapes of 2 body orbits

Green's Functions

1. Driven motion of a sphere in water
2. Driven damped harmonic oscillator
3. Stretch string
4. Electrostatics
5. Diffusion equation

Matrices

1. Eigenstates of three coupled 1d quantum wells

Power series

1. Simplifying problems by neglecting higher order powers (e.g. Lennard-Jones becomes harmonic potential)
2. Summing infinite series (e.g. light reflection from a thin film)

Fourier analysis

1. Calculating diffraction patterns in optics