Objectives and Goals
Quantum Mechanics is the fundamental theoretical framework for
modern physics. It underlies our approach to microscopic physics
from fundamental particle physics to chemistry, and lies at the
root of technological advances from the laser to the
transistor. This class provides a thorough introduction to the
conceptual and technical foundations of the subject. These are
reinforced by applying them to specific systems in which the
striking differences between quantum mechanics and classical
physics are manifest. At the end of the course the Hydrogen atom
is studied. This simple system was one of the first known
failures of classical physics and its solution one of the first
triumphs of quantum mechanics.
The technique of finding eigenvalues and eigenvectors of a
quantum mechanical Hamiltonian is developed. Operator notation
is stressed. Basic quantum mechanics postulates are introduced
and then applied to a variety of systems, some having classical
analogs and some not. Where classical analogues exist, this
limit is explored.
The course provides exposure to the following topics:
- Motivations for Quantum Mechanics
- Postulates and Techniques
- Spin 1/2 and Two-Level Systems
- The Harmonic Oscillator
- Angular Momentum
- The Hydrogen Atom
This course is for both advanced undergraduates and beginning
graduate students. For undergraduates, the PHY211/PHY212
sequence will provide a solid foundation for graduate study at
the most competitive schools in the nation. For graduate
students, this course will build upon the "Modern Physics"
courses often offered to you as undergraduates, and prepare you
for many-body and relativistic treatments of quantum mechanics
(found in PHY315).
Methods and Approach
- Prerequisites:
-
A modern physics course such as PHY143 or equivalent is
required so that the student has some familiarity with the
motivation behind quantum mechanics and how and where
classical physics fails. Knowledge of linear algebra is
helpful but what is needed will be reviewed. Other
mathematical tools which will be used are differential
equations, eigenvalues and eigenvectors, and Fourier
series. Undergraduate Newtonian mechanics is required so that
the appropriate quantum mechanical limits can be appreciated.
- Format:
-
This course is taught through three 50-minute lectures per
week and weekly problem sets. Lectures generally involve
blackboard presentations by the professor, but student
participation is encouraged. Problem sets generally consist of
7-8 problems designed to take approximately 10 hours of
concentrated effort to complete. Students are encouraged to
discuss the problems with their peers, but are required to
write solutions independently. Students are responsible for
all material covered in the lectures and in the problem
sets. As with any course in physics, it is impossible to
overemphasize the importance of working problems.
- Texts:
-
The primary text for the course is Cohen-Tannoudji, Diu, and
Laloe "Quantum Mechanics". Students are encouraged to consult
additional texts:
- Townsend's "A Modern Approach to Quantum Mechanics"
- Liboff's "Introductory Quantum Mechanics"
- Exams and Grades:
-
There is a final exam for the course and two in-class
exams. Exams are designed to test each student's grasp of the
fundamental concepts. Grades for the course are determined by
homework (35%), two midterm (15% each), and the final exam
(35%).
Sample Syllabus
(Each bullet represents one week.)
-
Lecture 1: The breakdown of classical mechanics
-
Lecture 2: Spectral decomposition; Schrodinger
equation
- Lecture 3: Separation of variables; quantum numbers
- Lecture 4: Step potentials
-
Lecture 5: Tunneling; reflection; lens-coating
problem
- Lecture 6: Fourier transforms and Heisenberg
Uncertainy Principle; basis changes
-
Lecture 7: scalar product; linear operators;
orthonormality and closure
-
Lecture 8: projection operator; observables
- Lecture 9: matrix elements; basis changes
- Lecture 10: CSCO; degeneracy; examples; 3d
-
Lecture 11: Measurement and collapse of the
wavefunction
- Lecture 12: Expectation values
- Lecture 13: Time evolution; Ehrenfest's
theorem
-
Lecture 14: Summary of postulates; examples
- Lecture 15: Evolution operator or Schrodinger vs
Heisenberg picture; review for exam
- Lecture 16: Exam 1
-
Lecture 17: Spin 1/2 particle in a B field;
Stern-Gerlach; anomalous Zeeman effect
- Lecture 18: The spectrum; raising and lowering
operators
- Lecture 19: Spin operator in arbitrary
direction
-
Lecture 20: Bohr frequencies; expectation
values
- Lecture 21: Two spin-1/2 neutrinos; partial
measurements
- Lecture 22: General two-dimensional treatment
-
Lecture 23: Double well or ammonia
- Lecture 24: Harmonic oscillator; parity
classification
- Lecture 25: H.O. Hamiltonian; rescaled x and p;
number operator
-
Lecture 26: H.O. spectrum; raising and lowering
operators
- Lecture 27: Form of some of the lowest lying levels;
expectation values; time evolution
- Lecture 28: HUP; 2-d H.O.
-
Lecture 29: Classical limit; spectroscopy of diatomic
molecules; selection rules
- Lecture 30: H.O. in electric field
- Lecture 31: Representation of a radiation
field
-
Lecture 32: Two coupled H.O.
- Lecture 33: Review for exam; examples
- Lecture 34: Exam 2
-
Lecture 35: Angular momentum and what it is for;
commutation relations
- Lecture 36: Raising and lowering operators;
spectrum
- Lecture 37: Orbital; spherical harmonics; atomic
application
-
Lecture 38: Addition of angular momentum
- Lecture 39: Rotational levels of ammonia
- Lecture 40: H atom spectrum
-
Lecture 41: H atom in B field
- Lecture 42: Including spin