Table of Contents

# Physics Graduate Program

## PH561: Mathematical Physics

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

## Calendar

#### Week 1

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

M | No Class | Comprehensive Exam | |||

1 | W | Diff. Eqns. | day1.pdf: Classifying differential equations. Quick analysis of v' = -g + αv². | Overview of Diff Eqns | Reading Assignment 1 |

2 | F | Python | day2.pdf: Numerical solution of v' = -g + αv². Meet in Weniger 412. | hw1, hw1solnb.pdf |

#### Week 2

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

3 | M | Application of Complex Numbers | day3.pdf: Using complex numbers for the damped-driven HO. Cartoon intro to Green's functions. | Butkov §2.1 - 2.5 | |

4 | W | Green's function method | day4.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 | |

5 | F | Green's function method | day5.pdf: Recipe for applying Green's function technique. Extend to PDEs. | hw2, hw2soln.pdf |

#### Week 3

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

6 | M | Green's functions, Homogeneous ODEs | day6.pdf: Green's function for the Laplacian differential operator. General techniques for solving homogeneous ODEs. | ||

7 | W | Homogeneous ODEs | day7.pdf: Power series technique for solving homogeneous ODEs. | Butkov §3.4 | |

8 | F | Homogeneous ODEs | day8.pdf: Power series technique for the Legendre Eqn and the Bessel Equation | Butkov §3.5 | hw3, hw3soln.pdf |

#### Week 4

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

9 | M | Partial Diff. Eqns. | day9.pdf: Separation of variables: Pick apart the PDE for the H atom wavefunction | Butkov §8.1 - 8.4 | |

10 | W | Series/Contour Integrals on Argand Plane | day10.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 | |

11 | F | The Residue Theorem | day11.pdf: Applying the residue theorem to evaluate definite integrals | Butkov §2.12 | hw4, hw4soln.pdf |

#### Week 5

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

12 | M | Fourier series | day12.pdf: Finding the Fourier coefficients. | Butkov §4.1 - 4.3 | |

13 | W | Fourier series / transform | day13.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 | |

14 | F | Fourier series / transform | day14.pdf: Fourier transform derived (heuristically) from the complex form of the Fourier series. | Butkov §7.1 | hw5, hw5soln.pdf |

#### Week 6

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

15 | M | Fourier Transform Applications | day15.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 | |

16 | W | Fourier Transform: Solve ODE | day16.pdf: Transforming an ODE. Top hat force function acts on a harmonic oscillator. | Butkov §7.7 | |

17 | F | Fourier Transform: Solve ODE | day17.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

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

18 | M | Quantifying heat transport | day18.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 | |

19 | W | Quantifying QM scattering | day19.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. | ||

20 | F | Quantifying QM scattering | day20.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

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

21 | M | Pictorial Proofs | day21.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 | |

22 | W | Taking out the big part | day22.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 | |

23 | F | Eigenvalue equations | day23.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 - 9.4 | hw7, hw7soln.pdf |

#### Week 9

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

24 | M | Eigenvalue equations | day24.pdf: Hermitian operators, limits on boundary conditions, Sturm-Louiville form. | Butkov §9.3 - 9.4 | |

25 | W | Eigenvalue equations | day25.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 | |

F | No class | Thanksgiving |

#### Week 10

Day | Topic | Summary | Additional Reading | Assignments | |
---|---|---|---|---|---|

26 | M | Review week | day26.pdf: Review HW#7. Example problems for final, covering Days 1 - 9. | ||

27 | W | Review week | day27.pdf: Example problems for final, covering Days 10 - 18. | hw8, hw8soln.pdf | |

28 | F | Review week | day28.pdf: Example problems for final, covering Days 19 - 25. |

#### Finals Week

## Syllabus

Complex numbers

- Circuit analysis
- Deriving trig identities, tables of integrals
- Switching to complex numbers to solve DEs
- Kramer's Kronig relations

Differential equations

- Driven damped harmonic oscillator
- Stretched string
- Shapes of 2 body orbits

Green's Functions

- Driven motion of a sphere in water
- Driven damped harmonic oscillator
- Stretch string
- Electrostatics
- Diffusion equation

Matrices

- Eigenstates of three coupled 1d quantum wells

Power series

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

Fourier analysis

- Calculating diffraction patterns in optics