Delaware State University

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Nov 15, 2013 (3 years and 7 months ago)

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Del aware St at e Uni vers i t y

Department of Applied Mathematics and Theoretical Physics

Dover, DE 19901

Introduction to Many
-
Body Dynamics



60
-
731
-
00,

3 cr.


Text:

A. M. Zagoskin, H. E. Stanley, J. W. Lynn:

Quantum Theory of Many
-
Body Systems

recommended:

R. D. Mattuck:

A Guide to Feynman Diagrams in the Many
-
Body Problem

H.

Bruus, K.

Flensberg:

Many
-
Body Q
uantum Theory in Condensed Matter Physics: An Introduction

The aim of the course is to introduce, develop and discuss various methods developed for the
study of many (10
19

10
23
) interacting particles. Typically, such particles are sufficiently close for
qu
antum effects to play a crucial role, too many for a straightforward extension of the single particle
theory, but still finitely many, preventing a transition into (continuous) field theory. The standard
approach is then to replace the intractable multitud
e of interacting real particles with a relatively small
number of
quasiparticles

the elementary response of the system to external perturbation.

Prerequisite:
Mathematical methods of Physics IV

(26
-
667), or equivalent (vector calculus, linear
algebra, tens
ors, real and complex analysis, multivariate calculus)
,
Classical Mechanics
(26
-
652),
Quantum Mechanics II

(26
-
676). A successful student is expected to gain a working knowledge of the
covered material, so as to be able to (1)

follow the applications in th
e literature, (2)

solve typical
problems in the field, and (3)

discuss adequately the term
-
paper subject.

Topical schedule
:



Basic Concepts

Introduction; Propagation function in a one
-
body quantum theory; Perturbation
theory for the propagator; Second quant
ization



Green’s Functions at Zero Temperature

Green’s function of the many
-
body system; Perturbation theory



More on Green’s Functions, Equilibrium and Otherwise, and Their Application

Analytic properties of equilibrium green’s functions; Matsubara formalis
m;
Linear response theory; Nonequilibrium Green’s functions; Quantum kinetic
equations; Electrical conductivity and quantum point contacts; Method of
tunneling Hamiltonian



Methods of the many
-
Body Superconductivity

Introduction; Instability of the normal s
tate; Pairing (BCS) Hamiltonian;
Green’s functions of a superconductor; Andreev reflection; Tunneling of single
electrons and Cooper pairs

C
URRICULUM
C
OURSE
R
EVIEW
:


Introduction to Many
-
Body Dynamics


1.

Course Title/Number:

Introduction to Many
-
Body Dynamics /
60
-
731
-
00


2.

Number of Credits:

3


3.

Curriculum Prog
ram Title:

Ph.D. in Applied Mathematics and Theoretical Physics


4.

Curriculum/Course is:

[ X ]

New

[

]

Revised

[

]

Required Course

[

X

]

Elective Course


5.

List Prerequisites:

26
-
667
(Mathematical methods of Physics IV)
, or equivalent (vector ca
lculus, linear algebra,
tensors, real and complex analysis, multivariate calculus)

26
-
652
(Classical Mechanics)

26
-
676
(Quantum Mechanics)


6.

List Courses Being Replaced or Changed:

This is a new course.


7. List Courses Being Deleted:

No courses are bein
g deleted.


8. Needs Statement:

This course is needed for students pursuing a Ph.D. in all areas of theoretical physics and
especially for those interested in microscopic (quantum) physics underlying the phenomenological
properties of bulk materials

typica
l of
macroscopic
objects in everyday life. The course also serves as
a bridge between the fundamental, microscopic physics and its collective, macroscopic manifestations,
with an outlook towards
emergent

phenomena that are not reducible in any simple fashi
on.


9. Catalog Description of the Course:

This course introduces, develops and discusses various methods developed for the study of the
collective phenomena
of many (10
19

10
23
) interacting particles: too close neglect quantum effects, too
many for straigh
tforward extensions of single particle theory, but too few for a transition into
(continuous) field theory.


10. List of Objectives of the Course:

(
1
)

To provide an introduction to the body knowledge and techniques of many
-
body quantum
dynamics. (
2
)

To see

how these techniques apply to the analysis of the microscopic physics behind the
C
URRICULUM
C
OURSE
R
EVIEW
:


Superstrings and Beyond

macroscopic phenomena in bulk materials. (
3
)

To learn how to identify those phenomena throughout
theoretical physics, which are best described using these methods. (
4
)

To dev
elop the problem
-
solving
skills associated with the application of these methods in theoretical physics, and learn how to extract
experimentally verifiable information from such application.


11. Course Outline:

See the “
Topical schedule
” section in the at
tached brief syllabus.


12. Show how the proposed course fits into the curriculum or course sequence:

This course is an elective within the curriculum of the Ph.D. program in theoretical physics, and
is indispensable for students focusing on the collective

quantum physics of bulk materials. For an
overview of pre
-
requisite dependences and the course’s relation to other courses proposed herein, please
see the attached “
Proposed Course Dependencies
” chart.


13. Are there comparable courses in other department
s?

No.


14. How will the students be affected by this course change?

This course provides the students an opportunity to increase their integration with the research
program of the Department of Applied Mathematics and Theoretical Physics, by understanding

the
mathematical underpinnings of the techniques that are used in contemporary theoretical physics. This
course will improve students’ professional competence, employability in technical fields and ability to
pass professional examinations; the term paper

requirement will foster improving expository skills.
Neither this course nor its prerequisites increase the total number of semester hours in this curriculum or
the number of credit hours required for graduation.


15. What effect will this new course have

on College resource?

None: this course will not require new or additional resources or staffing.


16. How will the course benefit the College?

This course studies materials comprised of many interactive particles: too many to be analyzed
by usual quantum
mechanics, yet too few to be approximated by a continuous field. To this end, one
introduces the concept of
pseudoparticles,

which capture the essence of
collective
behavior but are
simple enough for straightforward analysis. This
paradigm
-
shift

and its su
ccessful quantitative
C
URRICULUM
C
OURSE
R
EVIEW
:


Introduction to Many
-
Body Dynamics

application to otherwise intractable collective phenomena make this course also interesting to students
pursuing a Ph.D. degree in other scientific and even some non
-
science fields.


17. How will the change affect the program?

This co
urse will introduce students to the use of
pseudoparticles

as a quantitatively successful
method of capturing the essential
collective
behavior of many interacting particles. In addition, this
course provides a successful example of a paradigm
-
shift in ana
lysis of physical phenomena, and so also
in the overall philosophy and methodology of theoretical physics. This course will be one of the
electives specific to the Ph.D. program (concentration in theoretical physics) in this department.


18. Evaluation of
Student Performance:

Homework Assignments

40 %

Term
-
paper (take
-
home final)

60 %

Sample homework assignments, in
-
term and final examination question
-
sheets, work sheets,
course notes, review sheets and term papers will be accessible on
-
line.

Course Struct
ure: Three (3) 50
-
minute lectures per week.

References

1.

A. M. Zagoskin, H. E. Stanley, J. W. Lynn:
Quantum Theory of Many
-
Body Systems

(Springer
-
Verlag, 1998, ISBN = 0387983848)

2.

R. D. Mattuck:
A Guide to Feynman Diagrams in the Many
-
Body Problem

(Dover Pub.
, 1992;
ISBN = 0486670473)

3.

H.

Bruus, K.

Flensberg:
Many
-
Body Quantum Theory in Condensed Matter Physics: An
Introduction

(Oxford University Press, 2004; ISBN = 0198566336)


Submitted to Department of Applied Mathematics and Theoretical Physics

by: Tristan

Hubsch, on 27th of November, 2007.