Physics 227 (Methods of Theoretical Physics) Home Page, Fall 2012

Physics 227: Methods of Theoretical Physics




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 3
Infinite series

Sept 5: Course Logistics / Intro to Infinite Series

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

Sept 6: 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 7: 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.

Read: Boas, Chap. 1

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 Thursday Sept. 13, 2012]
2. September 10
Series / Complex numbers

Sept 10: Power Series

Introduce conditionally convergent series. Conditionally convergent series can be rearranged to sum to any value (Riemann series theorem). Power series defined. Convergence of power series. Interval of convergence. Allowed manipulations of power series. Taylor series expansions around the origin.

Sept 12: Power series / Defining and representing complex numbers

Taylor series expansions, about the origin and about a general point. Tips on expanding functions in power series. 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.

Sept 13: Complex numbers: algebra, infinite series, power series

Complex conjugate of a complex number. 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. Disc of convergence generalizes the interval of convergence. Rules for manipulating complex power series are similar to those for real power series.

Sept 14: Elementary functions of complex numbers

General discussion of extending functions of a real variable to a complex variable: what properties should (could) the extension preserve. Use power series to extend to functions of complex variable. Exponential function. Euler's formula. Powers of complex numbers. DeMoive's theorem. Roots of complex numbers. Square root of 1.

Read: Boas, Chap. 2

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, Thursday Sept. 20, 2012]
3. September 17
Complex numbers

Sept 17: Elementary functions of complex numbers

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

Sept 19: 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 20: 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 21: 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

PS3 -- Problems: Boas, 3.2.13, 3.2.14, 3.2.18, 3.3.4, 3.3.17, 3.4.20, 3.4.23 [due 11:59 pm, Thursday Sept. 27, 2012]
4. September 24
Linear algebra

Sept 24: Matrices and Gaussian elimination

Matrices, matrix notation, transpose of a matrix. Einstein summation convention. 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 26: 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 27: 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. Kronecker delta function. Cross product: its calculation and properties. convention.

Sept 28: Analytic geometry with vectors

Calculating the cross-product using the Levi-Civita symbol. Equation for a line parallel to a vector. Equation for a line perpendicular to a vector, in 2D. Equation for a plane with a specified normal vector.

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 [due 11:59 pm, Thursday Oct. 4, 2012]
5. October 1
Linearity and linear transformations

Oct 1: Matrix operations Matrix multiplication / Inverse of a matrix

Matrix equations, multiplying a matrix by a number, adding matrices. Multiplying matrices. Conformable matrices. Matrix multiplication is not commutative, but is associative and distributive. Commutator defined. Zero matrix, identity matrix defined. Determinant of product is product of determinants, for square matrices. Inverse of matrix defined.

Oct 3: Inverse of Matrix / Functions of matrices / Linearity

Calculating the inverse of a matrix. Expression for unique solution of n linear equations in n unknowns, in terms of inverse matrix. Powers and polynomials of matrices. Functions of matrices defined in terms of power series. Exponential of a matrix. Note that some of the properties of exponentials of numbers don't hold for exponentials of matrices, unless the matrices commute. BCH formula. Linearity: of vectors, of scalar functions of vectors, of vector functions of vectors (linear functions vs. affine functions). Linear operators defined. Derivative as linear operator on functions.

Oct 5: Transformations in the plane: general linear and orthogonal / Rotations in 2D

Matrices as linear operators transforming vectors to vectors. Active and passive transformation pictures. Orthogonal transformation defined as those that preserve the length of vector. Form of orthogonal matrix in 2D derived. Orthogonal matrix has determinant +1 or -1. For orthogonal M, det M=1 is rotation, det M=-1 is reflection. Derive form of rotation in 2D in active transformation picture, using complex numbers.

Oct 6: Rotations and reflections in 2D and 3D / Linear independence

2D rotation matrix in passive transformation picture. Check that det M=1 for rotation matrix. For reflection, can find line of reflection using Cr=r. Rotations and reflections in 3D: rotations about z axis. Reflections through xy plane. Combination of this is product of the corresponding matrices. Claim any 3D orthogonal matrix with det=1 can be written same way by choosing rotation axis as z-axis. Rotation about y-axis. Rotation axis can be determined by solving Mr=r, reflection plane found by solving Lr=-r. Linear dependent and independent vectors. Row reduction of components of vectors to determine a basis. Rank of component matrix gives number of independent vectors.

Read: Boas, Chap. 3

PS 5 -- Problems: 3.7.33, 3.8.16, 3.8.21, 3.9.15, 3.9.17, 3.10.2, 3.10.4, 3.10.10 [due 11:59 pm, Friday Oct. 12, 2012]
6. October 8
Linear vector spaces

Oct 8: Break

Oct 10: Linear independence / Homogenous equations

Linear (in)dependence of functions defined. Wronskian can be used to determine if a collection of functions is linearly independent. Define homogeneous equations. Sets of homogeneous equations can have a unique solutions (trivial solution), or an infinite set of solutions. A system of n homogeneous equations in n unknowns has nontrivial solutions IFF the determinant of the coefficient matrix is zero. Prove earlier statement about Wronskian. Geometry of non-trivial solutions to sets of (in)homogeneous equations: in an example, we show the solution to a set of inhomogeneous equations is a line passing through a point. The parallel line through the origin is the corresponding homogeneous solution, the point is a solution to the inhomogeneous equation (homogeneous + particular solution).

Oct 10: Exam 1

7-10 pm
location: Merrill 3
Covers through end of Oct. 6.

Oct 11: Homogeneous equations / Matrix Trivia

Eigenvalues/eigenvector problems. Find eigenvectors and eigenvalues of a 2x2 real symmetric matrix. Note that the geometry of solutions corresponds to orthogonal line through the origin. Matrix trivia: make observations about transpose of products and inverse of products of matrices, trace of matrix and trace of product, define hermitian conjugation, hermitian and unitary matrices, state that U=Exp[iH] is unitary if H is hermitian.

Oct 12: Linear vector spaces

Extend ideas from 2D and 3D vector space to n-dim Euclidean vector spaces. Vector space as all linear combinations of some set of vectors. Span, basis, and dimension of a vector space. Inner product, norm, orthogonality. Schwarz inequality. Gram-Schmidt procedure for obtaining an orthonormal basis. Define analogs on complex Euclidean space.

Read: Boas, Chap. 3

PS 6 -- Problems: Boas 3.11.16, 3.11.19, 3.11.30, 3.11.35, 3.11.43, 3.11.46, 3.11.51, 3.11.60, 3.11.62
7. October 15
Applications of similarity transformations

Oct 15: Eigenvectors, Eigenvalues, Diagonalization

Eigenvectors of a transformation are rescaled but not rotated by the transformation. In the 2x2 case, the two eigenvalues can be real and distinct, real and degenerate, or complex conjugates. Work out the eigenvectors and eigenvalues of a 2x2 real symmetric matrix M. This particular example has distinct eigenvalues and orthogonal eigenvectors. Can diagonalize the matrix M with a similarity transformation the columns of which are the normalized eigenvectors of M. The resulting diagonal matrix has eigenvalues as the diagonal entries.

Oct 17: Geometrical significance of similarity transformations

Note that similar matrices have same trace and determinant. Work out a 2x2 case in which similarity transformation has interpretation as a change of coordinates. The similar matrices represent that same transformation expressed in different coordinate systems. The columns of similarity transformation C are the new coordinate axes (expressed in the original coordinate system). When a similarity transformation C diagonalizes a matrix M, it amounts to transforming to a coordinate system in which the action of the transformation represented by M (in the original coordinate system) has a particularly simple form. If the eigenvectors of C are coordinates of orthonormal vectors, C is an orthogonal matrix (its transpose is its inverse) and can thus be interpreted at a rotation matrix (or rotation + reflection). The columns of C are the rotated coordinate axes (expressed in the original coordinate system). Claim this is possible IFF real matrix M is symmetric.

Oct 18: More on similarity transformations

Similarity transformations by orthogonal matrices are rotations. If columns of transformation matrix are not components of orthogonal vectors, transformation is to a new coord system whose axes are not orthogonal.
If symmetric matrix has repeated eigenvalues, can choose eigenvectors of eigenspace to be orthonormal using Gram-Schmidt procedure.
A matrix has real eigenvalues and can be diagonalized by a unitary similarity transformation IFF it is hermitian.

Oct 19: Orthogonal transformations in 3D

Read: Boas, Chap. 4

PS 7 -- Problems: see Problem set 7
8. October 22
Multivariable calculus: differential calculus

Oct 22: No Class

Class cancelled: Hurricane Sandy

Oct 24: Title

Oct 25: Title

Oct 26: No Class

Class Cancelled: Day of Dialogue

Read: Boas, Chap. 4

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 11:59 pm, Thursday November 1, 2012]
9. October 29
Multivariable calculus: differential calculus

Oct 29: Title

Oct 31: Title

Nov 1: Title

Nov 2: Title

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

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 [due 11:59 pm, Thursday November 8, 2012]
10. November 5
Multivariable calculus: differential and integral calculus

Nov 5: Title

Nov 7: Title

Nov 8: Title

Nov 9: Title

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

PS 10 -- Problems: 5.2.6, 5.2.10, 5.2.22, 5.2.40, 5.2.48, 5.3.30, 5.4.13, 5.5.10, 5.6.11 [due 11:59 pm, Thursday November 15, 2011]
11. November 12
Multivariable Calculus: integral calculus

Nov 12: Title

Nov 12 (7-10 pm): Exam 2
Covering through the end of Boas, Chapter 4.
Nov 14: Title

Nov 15: Title

Nov 16: Title

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

Problems: PS 11 -- Problems: Boas, 6.3.18, 6.4.6, 6.6.3, 6.6.13, 6.7.8, 6.8.16, 6.8.18, 6.8.19, 6.8.20 [due Friday, November 30, 2012, 11:59 pm]

12. November 26
Vector calculus

Nov 26: Title

Nov 28: Title

Nov 29: Title

Nov 30: Title

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

Problems: PS 12 -- Problems: Boas, 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 Thursday, December 6, 2012, 11:59 pm]
13. December 3
Vector Calculus / ODEs

Dec 3: Title

Dec 5: Title

Dec 7: Title

Dec 8: Title

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

PS 11 -- Problems:
14. December 10
2nd order differential equations with constant coefficients

Dec 10: Title

Dec 12: Title

Dec 13: Title