PHYSICS 308: QUANTUM PHYSICS
Introduction to Quantum Mechanics - Spring 2005
Teacher: Alfred Scharff Goldhaber
Yang Institute (Math 6-113, 2-7975, goldhab@insti.physics.sunysb.edu)
Office Hours: TBA Math 6-113, or by appointment or drop in
Class meetings MWF 935-1030 am, Physics P-123
Problem session TBA
Texts: `Feynman Lectures in Physics, Volume 3 (Quantum Physics)’ by Richard Phillips Feynman; `Introduction to Quantum Mechanics' by David Jeffrey Griffiths
Midterm Exam: Wednesday 9 March (based on homework problems).
Course Description
The concepts, historical development, and mathematical methods of quantum mechanics. Topics include Schroedinger's equation in time-dependent and time-independent forms; one- and three-dimensional solutions, including the treatment of angular momentum and spin. Applications to simple systems, especially the hydrogen atom, are stressed.
Prerequisites: PHY 300, 301, and 303
Course Plan
The year 2005 has been declared World Year of Physics, in honor of Albert
Einstein's trio of contributions in 1905: Establishing the Molecular
Hypothesis (by detailed analysis of Brownian motion). Special Theory of
Relativity, and Theory of the Photo-Electric Effect. Of these, Einstein
himself believed, and most modern commentators agree, that the quantum
concepts behind the photo-electric effect were the really revolutionary
contribution. In honor of this anniversary, we shall begin the course by
using 20/20 hindsight to derive one-particle quantum mechanics from
Einstein's theory. The actual process took 20 years and involved many
people, with Einstein himself very much on the sidelines. This will
include material that may not be in any textbook. For the course as a
whole, we shall be using two books, Feynman Vol. 3 and Griffiths.
Feynman emphasizes basic physics concepts, while Griffiths has a more
formal, mini-graduate-course approach. I'll indicate what material in
each book is related to each lecture.
Using Feynman's book may be dangerous. He himself said that his famous
introductuctory course was a failure for most students, and others have
said that the book is best suited to experienced PhDs. There are two
reasons why I am hopeful that our adventure may have a happier ending.
First, Feynman said he had no feedback during the course. I'll bring a
feedback box to every class, and people may drop in comments at the end of
class, or think things over and drop in comments at the beginning of the
next class, or both. Last time I taught the course I arranged for
anonymous web feedback, but got none and so don't plan that this time,
but if people want it I can do it again. The second difference is that
students in this class should be mostly second-term juniors, and so more
mature than the entering freshmen at CalTech, and more knowledgeable, with
a year of introductory physics and a semester of modern physics already
behind them. The hope is that you will get clarification of issues which
may have seemed mysterious in the modern physics course, and then build up
to some of the formal (and quite pretty) developments which Griffiths
emphasizes. There will be a midterm exam in class (Wednesday 9 March 2005) (20%), and
the final exam (40%) will be from 8:00-10:30 am on Friday 13 May 2005. The
remaining 40% of the grade will come from weekly homework assignments.
Teacher
Alfred Scharff Goldhaber
Professor
Ph.D., 1964, Princeton University
Fred Goldhaber represents the second of three physics generations in his family. Collaboration with his parents led to what may have been the first mother-son publications in physics. Among his research publications are articles on magnetic monopoles, elementary particles, nuclei, condensed matter, and, recently, cosmology. Themes that help to bind these topics together include the principle of gauge invariance, the use of classical limits and the correspondence principle, and the study of long-distance, low-energy constraints on objects that may have quite high-energy internal structure. He is co-author of three review articles, "Terrestrial and Extraterrestrial Limits on the Photon Mass," "Hypothetical Particles" (with Jack Smith), and "High-Energy Collisions of Nuclei," as well as an annotated bibliography, Magnetic Monopoles. He enjoys hindsight heuristics, asking why people made discoveries later than they might have; understanding this better could aid future discoveries.
Grader:
Ilmo Sung, Office A-105.
isung@grad.physics.sunysb.edu
Office Hours: Tue 130-230 pm, Wed 530-630 pm
Topics
1. Derivation of one-particle quantum mechanics from Einstein’s formulation of the photo-electric effect.
2. Examination of the classical and quantum versions of the bombardment of a wall with a hole by a beam of particles (or a wave). What is particle-like and what is wave-like in the quantum case? F Ch. 1.
3. Motion of a particle in one space dimension. Comparison of classical and quantum descriptions. Wave function and Schrödinger equation.
G Ch. 1, pp. 1-19.
Remainder TBA.
Background and Perspective
Towards the close of the 20th century and up to the present there has been increasing awareness that the standard model of electroweak and strong interactions, though enormously successful in describing the physics of microscopic distance and high energy scales, nevertheless is unsatisfactory in several ways.
1. There are very many parameters which are not deduced from fundamental principles, in particular, the masses of the elementary particles of the theory, including electrons and other leptons, and quarks.
2. There are regular patterns among these masses which are unexplained.
3. The theory is inconsistent or incomplete, in the sense that its extrapolation to very high energies is not defined.
4. There are phenomena which are not accommodated in the theory, namely, the recently demonstrated nonzero values for neutrino masses.
A remarkably similar situation prevailed towards the close of the 19th century. The standard model of that time was what today is called classical physics - mechanics, electrodynamics, and statistical mechanics:
1. There were very many parameters not deduced from fundamental principles, in particular, the masses of atoms and molecules (whose very existence, though widely accepted, was denied by some). Other unexplained parameters, which today can be accounted for at least approximately, included properties of materials such as the density and hardness of various substances.
2. There were regular patterns which were unexplained, such as the chemical regularities summarized in Mendeleev's table of elements.
3. Theory was inconsistent or incomplete. In particular, its extrapolation to very high energies was not defined, because it included the notion of point particles carrying electric charge, and according to classical electrodynamics such a particle would have infinite energy. Further, though this was not well understood before Einstein found and resolved the problem with his special theory of relativity, Newtonian mechanics and Maxwell electrodynamics were not consistent with each other.
4. There were important phenomena not accommodated in the theory: The fact that atoms and molecules radiate light with very definite frequencies, in analogy with a tuned musical instrument; the stability of matter against collapse, which could not be understood if the only forces acting were electric and magnetic; the existence of radioactivity; the frequency distribution of light energy radiated by a hot body. All these difficulties are overcome in the 20th century standard model, and a huge piece of that model is quantum mechanics. The first paper on quantum physics, even introducing the term `quantum', was Max Planck's work on radiation from hot bodies in 1900. This was followed by many suggestive experimental and theoretical developments, but it took a quarter century until Heisenberg in 1925 invented matrix mechanics and Schrödinger in 1926 invented wave mechanics. These two schemes soon were recognized as mathematically equivalent forms for what became known as quantum mechanics. Almost immediately this theory was crystallized in `final' form, and in the almost eight decades since has never been successfully challenged or altered. Over time, however, new implications of the theory were recognized. Despite its striking departure from classical physics, quantum mechanics has become one of the most robust and powerful of all scientific tools. Indeed, informal surveys of physicists show a strong belief that if further developments require modification in current understanding, the last thing to be modified will be quantum mechanics.
Lecture Summaries
Lec. 1, 24 January - Three centuries of debate about particle versus
wave nature of light.
In 1800s plenty of evidence that light has wave properties,
culminating in Maxwell equations
with transverse electromagnetic wave solutions. In 1905 Einstein
reopens the whole issue
with his particle description of photons and how they can kick
electrons out of a metal.
Implicit in this discussion is that the number of photons is related to
the intensity of the
corresponding wave.
Lec. 2, 26 January - Consistency also implies that the photons are quite
different from familiar
Newtonian particles, which would speed up upon entering a medium with
high refractive index,
while the waves slow down. The correct particle description involves
something quite different
from Newtonian forces at the boundary between refractive media. Thus
the particle and wave
descriptions have to be linked with each other.
Lec. 3, 28 January - Considering that at a boundary an incident wave is
partly transmitted and
partly reflected, for each photon there is a probability of reflection
and a probability of transmission.
Therefore the classical notion of completely determinate trajectories
for each particle is lost. For
the same reason, if light is diffracted, each photon may end up in
various places with different
probabilities, related to the classical wave intensity at each place.
Each photon 'interferes with
itself. Putting everything together gives a complete one-particle
quantum mechanics for a photon
of definite frequency. All the basic elements of quantum mechanics
are there, and we are ready
to start looking for the correct quantum description of nonrelativistic
massive particles.
Lec. 4, 31 January - The previous lecture brought us the idea of a wave
function whose square gives probability density for finding a photon at
any point. Now we need to learn from this experience to design a wave
function that will do the same thing for ordinary familiar particles,
in particular particles moving at speeds much smaller than the speed of
light. We find a wave equation for a free particle, and then see how
standard Newtonian potential functions can be added to give the effect
of forces on the wave function. This process raises lots of questions
which can at least be stated, even if some of the answers may have to
wait till later in the course. The equation we get is precisely the
one Schrodinger wrote down 21 years after Einstein's work on the
photoelectric effect, and two years after de Broglie's suggestion that
electrons could behave like waves. Therefore we are moving very fast by
comparison with the actual history!
Lec. 5,6,7, 2,4,7 February -- After discussing solutions of Homework 2, we begin
by drawing attention to some questions requiring further discussion
(including some already mentioned last time), but defer those
discussions to later in the course. The new homework gives a chance to
get acquainted with Feynman's first two chapters, and then introduces a
class of problems developed in Griffith's first two chapters. Thus the
course begins to get more conventional, though the combination of the
two books certainly is unconventional. The first technical problem
introduces an extremely important notion in quantum physics, namely
"tunneling" into a classically forbidden region.
Lec. 8, 9 February -- After discussing solutions of Homework 3, we turned
to the solutions for the infinitely high square well in one space
dimension, finding that the solutions are sine functions labeled by an
integer n which gives the number of half-wavelengths in the interval
between the two walls of the well. Functions for different n are
orthogonal in the sense that the integral of the product of two such wave
functions vanishes, unless they have the same n. By suitable choice of
coefficients of these functions, one may optimize the approximation of an
arbitrary function of x in terms of the sine function solutions.
Lec. 9 and 10, 11 and 14 February -- These lectures begin by considering
finite-dimensional vector spaces, obtaining a basis of orthogonal unit
vectors, and studying Hermitian and unitary matrices acting in this
space. This provides a background for returning to the case of the
infinite square well in one space dimension, for which the number of
different energies and wave functions is (countably) infinite.
Lec. 11-15; 16-25 February -- After discussing solutions of homework, we
come to the heart of quantum mechanics. The Schroedinger equation in its
most general form is -i\hbar d\psi/dt = H\psi , where the Hamiltonian H
not only is a Hermitian operator with real expectation values for any
state (meaning it is an observable, the energy) but also has a complete
orthonormal set of eigenvectors each with its real eigenvalue. Thus, if
H is known, and if the eigenvectors and eigenvalues are found, any state
can be expanded in this basis. The problem of prediction in quantum
mechanics then is solved, because given an initial state one may find the
state, and hence the expectation value of any observable, at any later
time. We go on to discuss the dynamics of two-state systems, which
illustrate most properties of quantum systems while remaining quite
tractable, and also of practical use because they describe the spin state
of a spin-1/2 particle. Along the way we introduce unitary operators
and discuss their relationship with Hermitian operators. The special
lecture 14 by Professor Smith introduced basic group theory, as applied to
rotations in 2 space dimensions, and then applied to unitary
transformations on states in a system with just two states. Lecture 15
was spent mostly on solving problems in Homework set 5, discussing how
these solutions illustrate the general principles in this set of lectures.
Lec. 16, 28 February -- This lecture begins with a proof for
finite-dimensional matrices that a unitary matrix is the exponential of an
anti-Hermitian matrix, or equivalently a Hermitian matrix multiplied by i.
This will be followed by discussion of the special nature of Hamiltonian
operators even when not finite-dimensional matrices, and then application
of the whole constellation of ideas to the quantum mechanical simple
harmonic oscillator.
Lec. 17, 2 March -- After solving the homework due this week, we resume
the track of the previous lecture, repairing a flaw in the proof
presented Monday and then going on to more general issues of Hermitian,
unitary, and self-adjoint operators.
Lec. 18, 4 March -- After doing the solutions for Homework 6 (skipped in
Lec. 17), we discuss the transition from Hermitian matrices to Hermitian
operators, where a new distinction arises, and the critical condition for
a Hamiltonian becomes that it is self-adjoint, a stronger condition for
infinite-dimensional operators than just being Hermitian.
Lec. 19, 7 March -- Questions on solutions of previous homework
assignments were addressed, and tended to focus on the most recent one,
plus a few older problems.
Lec. 20, 11 March -- This class began with a review of the course to
date, plus a preview of some important upcoming topics, including the
behavior of wave functions under rotation and their connection with
angular momentum eigenstates, and of course the solution of the
nonrelativistic hydrogen-atom problem. There followed extensive
discussion about self-adjointness versus Hermiticity, with the example
developed in HW 7, the momentum operator acting on the eigenfunctions of
an infinite square well.
Lec. 21, 14 March -- After distributing the graded midterm exams and their
solutions, we explored more intensely the background to the brief answer
for a question in the previous lecture, relating the Heisenberg 'Matrix
Mechanics' to the Schroedinger 'Wave Mechanics'. The key point is that
the time dependence of the expectation value of an operator may be related
either to the expectation value of the specified operator (which has no
explicit time dependence) for a time dependent wave function obeying the
Schroedinger equation, of first order in the time derivative, or to the
expectation value of the time-dependent version of the operator, obeying a
first-order equation in the time derivative, taken as an expectation value
for a wave function with no explicit time dependence.
Lec. 22, 16 March -- After solving HW 7 we go on to discuss the quantum
harmonic oscillator, developing both Heisenberg and Schroedinger
approaches.
Lecture Outline for Next 3 Weeks
-
- Momentum as self adjoint operator: Bead on a circle
- Harmonic oscillator - Schrödinger approach (continued)
, 
, 
Hermite polynomials
Gaussian integral 
- Heisenberg approach - Raising, lowering , and number operators
- (After homework solutions) Heisenberg uncertainty principle - wave and operator analyses
Wave packets
Commutation relations
- Angular momentum and spin, back to SU(2)
Raising and lowering operators, quantization of angular momentum. Integer values of orbital angular momentum
- Addition of angular momenta and conservation of total angular momentum
Half-integer spin
- (After homework solutions) Hydrogen atom spectrum
Radial equation, `accidental degeneracy', quantum numbers n, l, m
- Spin and statistics
Bose-Einstein statistics/ integer spin,
Fermi-Dirac statistics/ half-integer spin
- Shell structure
Breaking of single-particle level degeneracy for atoms with more than 1
electron
- (After homework solutions) Electronic structure of atoms - chemical bonds