Contact: Email->bnikolic@physics.udel.edu Phone -> 831-6677. Instructor Information:
Research interest of the QTT group
is in condensed matter theory with focus on quantum transport, mesoscopic
physics, quantum chaos, and quantum information science. My interest
in teaching at the graduate level is to integrate lectures with current
research problems, techniques, and tools (computer algebra packages and
numerics) . This should foster development of fearless attitude and confidence
when attacking such problems in the (near) future.
Lectures: MWF 10:10AM-11:00AM
in 320 Gore Hall (expect 36--38 lectures
per semester).
Office hours: Tuesday, Thursday
1:30-2:30 in 222 (future: 234) Sharp Laboratory, or by appointment
Classes start
on Wednesday, September 4 and end on Wednesday, December 11.
Exams: midterm (Monday,
October 14) and final (Friday, December 20).
Holidays (no classes):
Election Day, November 5; Thanksgiving break, November 27--December
1.
This is the first semester of a two semester sequence on Methods of
Mathematical Physics. This sequence is part of the core curriculum
for graduate students in physics. It is impossible to neither
study nor do research in physics without the use of mathematical methods.
This course is designed
to give an introduction (or review) to some of those methods. It is
not a "mathematical course" in a
sense of rigorousness and completeness. In fact, any of the "topics"
mentioned below can easily take
one or two semesters in the form of a regular mathematical course.
Instead: motivation will be given
for some of the methods showing its necessity to solve problems; (ii)
technicalities of the method
will be introduced; and then applied to solve more problems.
The objectives of the course are:
to introduce physics graduate students to fundamental techniques
of classical and (especially) quantum mechanics at a theoretically sophisticated
level,
to demonstrate usefulness of this techniques in 'real world'
problems, thereby helping students to master other core subjects (classical
mechanics, electrodynamics, quantum mechanics, ...)
to develop problem solving skills and research attitude,
to prepare students for the Ph.D.
qualifying exams at the University
of Delaware.
Course Prerequisites: Familiarity with undergraduate
Linear Algebra and Calculus.
Topics covered in this semester will revolve around the concept of Vector Spaces in Physics.
Introduction: From Pitagora to Supersymmetry---physics,
mathematics, and the meaning of mathematical physics. [1]
Vectors, Operators, Matrices, Tensors:
Vectors and tensors in classical physics. Linear superpositions.
Scalar, vector, and inner products.Vector spaces (finite dimensional).
[2]
Hilbert space. Quantum states as vectors in Hilbert spaces.
Linear functional and dual vector space <bra|ket notation>.
Orthonormal basis and Gram-Schmidt orthogonalization procedure.
[3]
Linear operators. Commutator, anticommutator, Poisson brackets
in classical mechanics. Identity, adjoint, self-adjoint, and Hermitian
operators. Outer products and projection operators. Operator algebra.
[3]
Eigenproblem: eigenvalues
and eigenstates. Discrete and continuous spectra. Delta function (and
other distributions). Inverse operator and Green functions. [3]
Change of basis. Matrices and matrix operations. Matrix diagonalization.
[3]
Orthogonal function sets. Fourier expansion. [2]
Vectors and matrices with Computer Algebra packages and with
LAPACK. [1]
Tensors product of vector spaces. Separable and non-separable
states. Tensor product of operators and Kronecker (direct) product of matrices.
Statistical operator (density matrix), entanglement, von Neumann entropy. Schmidt
decomposition. Quantum information science: entanglement, quantum teleportation,
quantum dense coding. Multilinear mappings and algebra of tensors. [6]
Symmetries in Physics---Group Theory for Physical Problems
:
Symmetries in classical and quantum physics. Group theory concepts
and general theorems. Conjugacy classes. [2]
Condensed Matter Physics:
Discrete symmetry (point, translation, space) groups. Representation
of finite groups in Hilbert spaces. Equivalent representations and Characters.
Reducible and irreducible representations. Group projectors. Subduction
and Induction. [6]
Direct products and Clebsch-Gordan expansion. [1]
Symmetries in quantum mechanics. Wigner theorems. Symmetries
in molecular vibrations. [1]
Vector and tensor operators in quantum mechanics. Wigner-Eckhart
theorem and selection rules. Spinors. [1]
High Energy Physics:
Lie Groups and Lie Algebras. Casimir operators. Unitary representations
of Lie Groups. Representations by Young diagrams, irreducible tensors.
[4]
Lorentz and Poincare group. Internal symmetries and gauge theories.
[2]
Coleman-Mandula theorem. A glimpse into Supersymmetry. [1]
Homework: Weekly, with a few exceptions.
Exams: 1 midterm and a final each semester (books open).
There will be approximately one homework assignment
per week, due on the first lecture the following week. As a rule, late
assignments will not be accepted without the prior consent of the instructor.
You may collaborate with others on the problems, but you must make a
note of your collaborators (just as if you were writing a scientific paper).
Noting your collaborators does not in any way detract from your grade. However,
each problem set must be written individually-do not simply copy your collaborator's
solutions verbatim (this will be considered a form of plagiarism). Please have
mercy on your grader and make your solutions neat, concise, and intelligible.
Solutions which are seriously lacking in any of these categories will be
marked down, even if they are ostensibly "correct.''
Usage of Computer Algebra Software (Maple or Mathematica), as well as numerical computation
(if necessary) is strongly encouraged. Maple or Mathematica worksheets
will be accepted as solutions to the homework assignments.
In addition to the homework assignments,
there will be a one hour midterm exam on October 14 (Monday)
and final exam on December 20 (Friday) (two hour exam).
Your final grade is determined using the following
approximate formula: the homework is 40% of your grade, the midterm exam
is 20%, and the final is 40%. Here is a guideline for your final grade,
as a percentage of the total number of points (scaled as above): 80-100,
some type of A; 60-79, some type of B; 59 and below, some type of C. These
numbers may be lowered, depending upon numerous factors, but will not be
raised (i.e., if you have an 80 average you are assured of at least an A-).
The course grades are not curved.
Main Textbook:
0. S. Hassani, Mathematical
Physics: A modern introduction to its foundation (Springer-Verlag, New
York, 1999) [ultramodern, comprehensive, and pedagogical; some chapters
maybe hard for first reading, but any gradaute student should be able to
master chapters 1-11].
1.F. W. Byron
and R. W. Fuller, Mathematics of Classical and Quantum Physics
( Dover Publications, New York, 1992) [old-fashioned, but easy to
read and very affordable].
Supplemental Material:
2. G. Arfken and H. Weber, Mathematical Methods for
Physicists (5th edition, Academic Press, San Diego, 2000). [this book
is probably the bestseller on the subject; it is not good as a textbook,
but contains wide selection of topics and can be useful as a reference, once
you know the subject]
3. Quantum Computation
Lecture Notes
(by J. Preskill) [covers many useful topics of linear algebra: tensor product
of vector spaces, pure and mixed states, entanglement, Schmidt decomposition,
etc.]
4. J. P. Elliot and P. G. Dawber, Symmetry in Physics
Vol. 1 & 2 (Oxford University Press, Oxford, 1985). [nice introduction
to group representation theory with a lot of examples from physics]
5. W. Ludwig and C. Falter, Symmetries in Physics-Group
Theory Applied to Physical Problems (Springer-Verlag, Berlin, 1996).
[a bit more formal than ref. 3]
6. T. Frankel, Geometry of Physics (Cambridge
University Press, Cambridge, 1999). [easy to read
introduction to geometrical methods in physics].
7. R. D. Richtmyer, Principles of Advanced Mathematical
Physics Vol. 1 & 2 (Springer-Verlag, New York, 1978). [covers all
topics at somewhat rigorous level]
8. D. Richards, Advanced Mathematical Methods
with Maple (Cambridge University Press, Cambridge, 2002).
