Physics 227 (Methods of Theoretical Physics) Home Page, Fall 2011

Physics 227: Methods of Theoretical Physics


Welcome to the new semester!


Course Information

Course Catalog Description:

The course will present the mathematical methods frequently used in theoretical physics. The physical context and interpretation will be emphasized. Topics covered will include vector calculus, complex numbers, ordinary differential equations (including series solutions), partial differential equations, functions of a complex variable, and linear algebra. Four class hours per week.


Times and places:


Mathematics 12 and Physics 17/24 or consent of the instructor

Course requirements

Statement of Intellectual Responsibility: particulars for this course

How to get the most from this class:



Required (I've ordered these from Amherst books): Additional useful references (NOT required):

Physics: Math books:

Mathematica Tutorials

We may use Mathematica in the homework, to obtain numerical solutions to problems that are not analytically solvable and to simplify plotting of results. If you've never used Mathematica before, or haven't used it much, the tutorials will help you get started. They were written by Professor Emeritus Bob Hilborn and revised by Rebecca Erwin '02. If you download the file and save it to the desktop with a .nb suffix in the name, your computer will recognize it as a Mathematica notebook and will start up Mathematica automatically when you double-click on the icon, provided you have Mathematica installed. Mathematica is installed on lots of the college's public machines, including on the computers in the Physics Department computer lab. Alternately, you can pay the $140 or so to buy the student version.

Lecture Schedule
Week Notes Hmwk Other
1. September 5
Infinite series

Sept 7: Course Logistics / Intro to Infinite Series

Geometric series (finite and infinite). Some useful series.

Sept 8: Convergence (positive series)

Convergent and divergent series defined. Convergence defined via a limit of partial sums. Test for convergence: Preliminary test. Absolutely convergent series defined. Tests for convergence of series of positive terms: (1) Comparison test (2) Integral test

Sept 9: Convergence (positive and alternating series)

Tests for convergence of series of positive terms: (2) Integral test, (3) Ratio test, (4) "Special" comparison test. Alternating series test. Introduce conditionally convergent series. Conditionally convergent series can be rearranged to sum to any value (Riemann series theorem).

Read: Boas, Chap. 1

Problems: none this week (no grader yet!)
2. September 12
Series / Complex numbers

Sept 12: Power Series

Power series defined. Convergence of power series. Interval of convergence. Allowed manipulations of power series. Taylor series expansions around the origin.

Sept 14: Power series / Defining and representing complex numbers

Taylor series expansions around a general point. Complex numbers from solutions to the quadratic equation. The imaginary number i. General complex number as real part + imaginary part. Complex numbers as points in the Argand diagram. Polar representation. Complex conjugate of a complex number.

Sept 15: Complex numbers: algebra, infinite series

Addition, subtraction, multiplication, and division of complex numbers. Modulus of a complex number. Complex equations. Partial sums of complex series. Convergence, absolute convergence of a complex series defined. An absolutely convergent series is convergent. Tests for convergence. Complex power series.

Sept 16: Complex power series

Disc of convergence generalizes the interval of convergence. Rules for manipulating complex power series are similar to those for real power series. Euler's formula.

Read: Boas, Chap. 2

PS1 -- Problems: 1.2.6, 1.4.6, 1.5.4, 1.6.30, 1.9.22, 1.15.30, 1.15.31, 1.15.32, 1.16.2, 1.16.10, 1.16.14, 1.16.18 [due 11:59 pm Tuesday Sept. 20, 2011]
3. September 19
Complex numbers

Sept 19: Elementary functions of complex numbers

Powers and roots of complex numbers. Exponential functions, trig functions, and hyperbolic trig functions of complex numbers.

Sept 21: Elementary functions of complex numbers

Logs of complex numbers. Complex roots and powers of complex numbers. Inverse trig and inverse hyperbolic trig functions of complex numbers. Application: Simple harmonic oscillator using complex numbers.

Sept 22: Complex numbers in physics applications

Simple harmonic oscillator using complex numbers. Why is the harmonic oscillator so important and ubiquitous in physics? Because for a potential with a stable equilibrium point, for sufficiently small excursions around the equilibrium point the potential the potential looks like a harmonic oscillator. Show this explicitly with Taylor series expansion of potential about stable equilibrium point. Set up the damped, sinusodally driven (AC) LRC series circuit problem.

Sept 23: AC circuits using complex numbers

Damped, sinusodally driven (AC) LRC series circuit using complex numbers. Talk about the importance of resonance phenomena generally in physics.

Read: Boas, finish Chap. 2, start Chap. 3

PS2 -- Problems: Boas, 2.5.21, 2.5.48, 2.5.60, 2.6.13, 2.7.15, 2.10.25, 2.11.18, 2.16.9, 2.16.10, 2.16.12 [due 11:59 pm Tuesday Sept. 27, 2011]
4. September 26
Linear algebra

Sept 26: Matrices and Gaussian elimination

Matrices, matrix notation, transpose of a matrix. Start to talk about solving systems of linear equations using row reduction Gaussian elimination. Express systems of linear equations in matrix form. Solving systems of linear equations using Gaussian elimination. Possible outcomes: no solutions, unique solution, infinitely many solutions.

Sept 28: Linear equations and determinants

Relate categories of possible outcomes to relationships among (rank of M, rank of A, number of unknows). Calculate determinant of nxn square matrix, where n=1, n=2, and n general. Some relations to help calculate determinants more quickly.

Sept 29: Vectors

Cramer's rule for solving systems of linear equations using determinants. Algebra of vectors, geometric and in terms of components. Cartesian unit vectors. Dot product: its calculation and its properties. Cross product: its calculation. Einstein summation convention, Kronecker delta function.

Sept 30: Analytic geometry with vectors / Matrix operations

Read: Boas, Chap. 3

PS 3 -- Problems: Boas, 3.2.13, 3.2.14, 3.2.18, 3.3.4, 3.3.17, 3.4.20, 3.4.23
5. October 3
Linearity and linear transformations

Oct 3: Matrix multiplication / Inverse of a matrix

Oct 5: Functions of matrices / linearity

Oct 5: Exam 1

7-10 pm
location TBA

Oct 6: Transformations in the plane: general linear and orthogonal

Oct 7: Rotations and reflections in 2D and 3D / Linear independence

Read: Boas, Chap. 3

PS 4 -- Problems: Boas, 3.5.13, 3.5.37, 3.5.44, 3.6.6, 3.6.17, 3.6.30, 3.7.25, 3.7.33, 3.8.16, 3.8.21
6. October 10

Oct 10: Break

Oct 12: Title

Oct 13: Title

Oct 14: Title

Read: Boas, Chap. 3

PS 5 -- Problems: Boas, 3.9.15, 3.9.17, 3.10.2, 3.10.4, 3.10.10, 3.11.16, 3.11.19, 3.11.30
7. October 17
Applications of similarity transformations

Oct 17: Title

Oct 19: Title

Oct 20: Title

Oct 21: Title

Read: Start Boas, Chap. 4

PS 6 -- Problems: Boas 3.11.35, 3.11.43, 3.11.46, 3.11.51, 3.11.60, 3.11.62
8. October 24
Multivariable calculus: differential calculus

Oct 24: Title

Oct 26: Title

Oct 27: Title

Oct 28: Title

Read: Boas, Chap. 4

PS 7 -- Problems: see Problem set 7
9. October 31
Multivariable calculus: differential calculus

Oct 31: Title

Nov 2: Title

Nov 3: Title

Nov 4: Title

Read: Boas, Chap. 4, start Chap. 5

PS 8 -- Problems: 4.1.5, 4.1.14, 4.1.20, 4.1.22, 4.2.6, 4.4.1, 4.4.9, 4.4.15, 4.5.6, 4.6.9 [due Wednesday, November 9, 11:59 pm]
10. November 7
Multivariable calculus: integral calculus

Nov 7: Title

Nov 9: Title

Nov 10: Title

Nov 11: Title

Read: Boas, Chap. 6, and read "div, grad, curl, and all that"

PS 9 -- Problems: 4.7.6, 4.7.16, 4.7.23, 4.7.25, 4.8.5, 4.9.9, 4.10.5, 4.11.2, 4.11.5, 4.11.10, 4.12.5, 4.12.6, 4.12.16, 5.2.6, 5.2.10 [due 11:59 pm, Tuesday November 29, 2011]
11. November 14
Vector calculus

Nov 14: Title

Nov 16: Title

Nov 16 (evening): Exam 2
Covering through the end of Chap. 4.
Nov 17: Title

Nov 18: Title


Problems: (carried over from last week)

12. November 28
Vector calculus

Nov 28: Title

Nov 30: Title

Dec 1: Title

Dec 2: Title

Read: Boas, Chap. 6; div, grad, curl, and all that

PS 10 -- Problems: 5.2.22, 5.2.40, 5.2.48, 5.3.30, 5.4.13, 5.5.10, 5.6.11, 6.3.18, 6.4.6, 6.6.3, 6.6.13, 6.7.8 [due 11:59 pm, Tuesday December 6, 2011] Problem set 10

13. December 5
Fourier series / ODEs

Dec 5: Title

Dec 7: Title

Dec 8: Title

Dec 9: Title

Read: Boas, Chap 6, and Div, Grad, Curl, and All That

PS 11 -- Problems: 6.8.16, 6.8.18, 6.8.19, 6.8.20, 6.9.3, 6.9.12, 6.10.6, 6.10.9, 6.11.8, 6.11.14, 6.11.21, 6.12.26, 6.12.30 [due 11:59 pm, Tuesday December 13, 2011]
14. December 12
2nd order differential equations with constant coefficients

Dec 12: Title

Dec 14: Title