**Physical and Mathematical
Foundations of Quantum Mechanics**

PHY
307 - Fall 2014

Stony
Brook University
Department of Physics and Astronomy,
College of Arts and
Sciences

Course Instructor: **Alfred Scharff
Goldhaber**
goldhab@max2.physics.sunysb.edu, 631 632 7975, fax 631 632 7954. Office Math 6-113. (Twitter)

Lectures:
Tu-Th 2:30-3:50 p.m., Frey 217
Recitations: Sec. 1 W 12:00 p.m.- 12:53 p.m.,
Physics P-123. Sec. 2 W 2:30 p.m.- 3:23 p.m., Physics P-123.
Course
credits: Four credits corresponding to four contact hours.
Office Hours: To be
determined, but principally by appointment at class, on the phone, or on email.

Teaching
Assistant and Grader: To be determined.

**INTENDED
AUDIENCE:**

As
described below, this course is designed to be fully accessible to students who
have completed one year of introductory physics and the associated introductory
calculus [If Modern Physics, PHY 251, is taken at the same time, the two
courses should reinforce each other; the same applies for PHY 300, PHY 301, and
PHY 405]. For such students the intent is that they should feel as if they were
"gliding into quantum mechanics." As with hang gliding, first it is
necessary to climb to a high place. In this case the "high place" is
the properties of electromagnetic waves, well worth mastering in their own
right. For capable sophomores who do well in PHY 307 the way is open to take
PHY 308, Quantum Physics, in Spring 2014, even if they lack pre-requisites for
308, because in the past students in this position also have done well in 308. This would be a one-year ‘speedup’ in
entering quantum physics, putting our program on a par with many other college
physics programs.

Students
who already have more background, including Waves and Optics (PHY 300) and
Electromagnetic Theory I, II (PHY 301, 302) as well as Quantum Physics (PHY
308) and Advanced Quantum Physics (PHY 405) can benefit as well, because
important and subtle concepts they have seen already are approached from a
different perspective, reinforcing the learning. At the end, we’ll
consider what might or might not have been the practical effect if Albert
Einstein already had constructed one-photon quantum mechanics in
1905.

**COURSE
DESCRIPTION:**

Maxwell
waves and their properties: Intensity, energy density, momentum density.
Complementary descriptions of wave super-positions in position and in
"wave number." Planck-Einstein relation between energy and
frequency for light quanta. Einstein-De Broglie relation between momentum and
wavelength. Number density and probability density of photons. One-photon
quantum mechanics, with Maxwell field as the wave function. Diffraction
phenomena. Uncertainty relation between wave number and position, hence between
momentum and position. Schrödinger
Equation. Three lecture hours and one recitation hour per week.

**Course
pre- and co-requisites**

Pre-requisites
-- Introductory classical mechanics and electromagnetism. Introductory
differential and integral calculus. {Advisory co-requisite -- Multivariable
calculus. Because the Maxwell equations involve three space dimensions, and
time, for the vector electric and magnetic fields, they give a wonderful
laboratory for working with multivariable calculus. Having that course
beforehand or concurrently would give extra and possibly complementary
perspective on these properties, but all that is needed will be worked out
fully in PHY 307.} Course numbers: PHY 122, or 127, or 132, or 142, MAT 132 or
142 or 127 or 171 or AMS 161. {Advisory co-requisite: MAT 203, MAT 205, or
equivalent.} Students who took and passed Physical and Mathematical Foundations
of Quantum Mechanics under the previous course numbers PHY 390 and PHY 274 are
ineligible to take PHY 307. PHY 307, while not required for the major like
other courses including PHY 251 and PHY 308, is intended to provide a
"keystone" which stabilizes and advances the entire arc of
undergraduate quantum studies.

**MOTIVATION:**

For
more than eight decades the teaching of quantum mechanics has followed two main
tracks. The first approach is roughly historical. There is some logic to that,
because if people originally found things in a certain sequence then it is at
least possible for students to learn in that sequence. This is the style of
"modern physics" courses, which give a broad survey of phenomena
especially in microscopic physics, including many of the steps in the somewhat
contorted path that led to modern quantum mechanics.

The
other method is axiomatic, exemplified by Dirac's great book on the subject, a
pattern followed by many more recent texts. In my opinion there is a missing
piece in this traditional two-tiered approach. The intellectual jump from the
broad survey to the axiomatic treatment is not easily articulated, and can be
disconcerting for students as they try to negotiate the transition. The
approach of the present course could be called "quasi-historical,"
meaning to build on a history that might have been if Einstein in 1905 had
shown even greater audacity than he did in his "photoelectric effect"
paper. The aim is to provide a path that is as short and direct as possible
from 1905 to 1925-6, when Heisenberg and Schrödinger introduced the theory we
still have today.

If
short, this climb can be quite challenging, and requires an intense study of
Maxwell's electromagnetic waves and their properties to provide the foundation
for the ascent. The take-home message, as in Einstein's special theory of
relativity, is that when one tries to fit together Maxwell's theory with other
knowledge, in case of doubt one should defer to Maxwell. This may seem natural
when one recognizes that Maxwell's is the first fundamental field theory in
physics, still unaltered at the classical level nearly 150 years after its
formulation. Thus the new course complements not only the existing one on
modern physics but also other departmental offerings on aspects of
electromagnetism and light. The path, as indicated in the syllabus below, goes
through Maxwell electrodynamics to one-photon quantum mechanics, and uses that
as a base to develop one-electron quantum mechanics.

I
believe this approach could help overcome what seems to me a deficiency in
current physics training, that students are exposed to classical mechanics and
classical electrodynamics twice during their undergraduate years, but a
systematic approach to quantum physics comes along only at the end of junior
and beginning of senior year. My idea is that this should be a first-term
sophomore course, simultaneous with and complementing the Modern Physics
course, allowing students to enter PHY 308, Quantum Physics, in the spring of
sophomore year. In every previous year PHY 307 has been an example of
“inter-age learning” – the students are a mixture of sophomores, juniors, and
seniors, with each cohort adding something valuable to the experience. As
mentioned already, for sophomores going into PHY 308 in the spring it
would be a one-year speed-up compared to the current pattern here, and would
put our program on the same pace (with respect to quantum physics) as many
other colleges. Colleagues at a number of other institutions have
introduced quantum mechanics through light rather than through massive
particles like electrons, but I'm unaware of any textbook that does this.
The goal of PHY 307 then, is after learning about electromagnetic waves
and wave packets to recognize these solutions as quantum-mechanical waves
describing the behavior of photons, or particles of light.

**EXPECTED
OUTCOMES:**

Quantum
Mechanics often is called counterintuitive, or even ‘weird’. The main take-home message from this
course should be that all the apparent weirdness stems from the wave aspects,
which are less familiar to most of us than particle behaviors. Here are some important examples, with which students should gain experience during the course:

1. Superposition. When two classical waves overlap in some
region, the total wave amplitude is simply the sum of the amplitudes of the two
waves. This property carries over
to the wave functions studied in quantum mechanics.

2. The energy density in a classical wave
is proportional to the square of the wave amplitude. In quantum mechanics the same holds for the probability
density for finding a particle— it is proportional to the square of the
corresponding wave function.

3. Unlike observable classical waves, such
as water waves or sound waves or light waves, quantum wave functions are
intrinsically complex, and therefore cannot be observed directly.

4. Classical waves and their quantum
counterparts obey definite equations determining how they change in time. In other words, they are
predictable.

5. Classical and quantum waves both can
show polarization. That is, in
addition to their direction of motion, they can carry intrinsic directional
information, which for the quantum case can be identified with intrinsic
angular momentum, or ‘spin’.

6. The more sharply pinned down in spatial
location is a classical wave, the broader the range of ‘wave number’ associated
with it. In quantum mechanics this
becomes the Uncertainty Principle, forcing a tradeoff between measurement
accuracies at a given time for the position and the momentum of a particle.

7. Entanglement: Two classical waves can be correlated, for example, when
they arise from scattering of a single wave from a half-silvered mirror. As a result, information about the one
wave gives constraints on where the other wave is more or less intense. This property again carries over into
quantum mechanics.

**COURSE
REQUIREMENTS:**

**Attendance
and Homework Policy**
One-quarter point per
class for attendance after the add-drop period, and one-sixth point for
recitation attendance. [__Recitations begin in the first week of class__.]
Half-credit for late arrival. Up to five points for each homework
assignment. Late homework will be graded down 20% for each day missed. [That
means after 5 days there will be no credit, but the paper may still be graded
to give feedback to the student.]
Both for homework and for attendance excused lateness or absence will
not count against the student. Homework normally will be assigned on a Tuesday
and due Thursday of the following week.

**Required
and recommended books**
Two required textbooks
(#1 and #2), and one recommended book (#3) from the Schaum Outline Series:

1. Electromagnetics, 3rd
Edition, Joseph A. Edminster and Mahmood Nahvi-Dekhordi, McGraw Hill 2011.

2. Advanced Mathematics
for Engineers and Scientists, 2nd Edition, Murray R. Spiegel, McGraw Hill 1971.

3. Quantum Mechanics, 2nd
edition, Yoav Peleg, Reuven Pnini, Elyahu Zaarur and Eugene Hecht, Mc Graw Hill
1998.

There will be no
assignments from these books, but they contain much useful information about
topics discussed in the course. If
we come to a topic not discussed in one of the books, I shall provide
additional references and/or background material, besides what is offered in
lecture.

**Exams**
None.

**Paper
and Presentation**
Each student is asked to
prepare a 5-page paper and make a 5-minute class presentation about the topic
of the paper. If two people work together on a paper and presentation, each
will get the same grade, but then the paper and presentation should be twice as
long, for three people three times as long, etc. The topic of the paper and
presentation should be a biography of someone who appears in the historical
background for quantum mechanics [In other words, NOT one of the people from
Planck on who directly made quantum physics]. A list of possible subjects
will be distributed early in the course. All papers should both describe the
life and the work of the scientist who did the work. Students may choose a
subject from the list (first-come first served), or propose a different subject
for approval. The stages in the process will be submitted in hard copy in class
as indicated:

1. Submission of subject
choice: Tuesday 9 September.

2. One or two paragraph
abstract and detailed (meaning at least 7 items) outline: Tuesday 23 September.

3. Notes filling out each
item in the outline, and possibly adding more: Tuesday 21 October. The notes should have everything you
would need (if you were alone on a desert island) to write a complete final
draft. That means you should have
all your references [A minimum of three references should be to published work,
whether books or articles in refereed journals. Besides these, references to
websites also may be included.], and any pictures you want to show. The idea is that there are two
big pushes in creating your paper.
The first is gathering all the thoughts and information that you need,
and the second is to put this into a coherent, carefully written essay:

4. Final draft -- This
should be a polished paper, created from the informal but complete notes. The
final draft should be submitted not only in hard copy in class, but also
electronically to SafeAssign: Tuesday 11 November.

5. Presentations -- these
will be given during regular lecture class hours beginning Tuesday 25 November.
There will be guidelines to assure that everything can be done during the
available time.

**Journal
Entries** Very shortly after each lecture,
you are asked to go over your notes, then put them aside, and write a paragraph
or two in the appropriate day's Journal on Blackboard, describing what you
learned including anything you felt needed more explanation. This has two
benefits. For students, there is evidence that doing such an exercise fixes the
information in your mind more effectively than, for example, studying hard just
before an exam. That's especially good here because we don't have exams! For
the teacher, it gives a good chance to find out what is getting across and what
is not. Please try to write coherent sentences. __Lists of formulas don't
really show that you understand something__, so formulas only should appear
as illustrations of what is in your text. Journal entries should be completed
by 12 midnight of the night following each class. The “20% rule” also applies
here for each day’s delay.

**Engagement
in recitations (and the course generally) **
During
recitation, a student or group of students who raise an interesting question
will get one point credit each. One kind of question that is welcome though it
will NOT get credit is "Could you go over a particular problem (or some
item in lecture)?" To get the credit, you need to describe something you
have done. It might be, for example, that you tried to do a problem in a
particular way and ran into some difficulty, and you want help in understanding
that difficulty. Equally well, it might be that you have a proposed solution, and
are presenting it. In lecture, it might be saying that my lecture statements
seem to imply such and such, which doesn’t make sense, or that they seem to
imply so and so, which is an interesting extension of my presentation. I
emphasize engagement for the recitations because they should be very much “live
time” that involves active learning.

**Grading**
Ten homework sets during the term, with a possible 5 points for each
set. This adds up to a possible 50 points for the semester. One-quarter point
for attendance at each lecture, and one-sixth point for attendance at each
recitation, after the end of the add-drop period (i.e., beginning Wednesday 10
September). This adds up to a possible 8 points for the semester. Lateness cuts
the grade for a particular attendance in half. Excused absence or lateness will
not be penalized. Up to 20 points for the paper and presentation. One half
point extra credit for each student on each homework set for which two to four
people have cooperated in discussing the homework, and then written up the
solutions separately (up to 5 points for the semester). To receive this credit,
EACH OF the students cooperating must list the names of OTHER members of the
group on the TOP RIGHT FRONT of the homework paper, taking care to make clear
who is the student submitting that particular paper by writing the SUBMITTER’S
name on the TOP LEFT FRONT. Up to 28 points for Journal entries after each of
the 28 lecture sessions, including the student presentations. Up to 4
points for demonstrating engagement, by asking detailed questions about (or
presenting solutions of) homework or raising interesting questions related to
lecture subjects. This may be done in recitation, by email, in journal
entries, or in lecture. Excluding extra credit, this adds up
to 110 points for the semester.
Letter
grades: >105=A, >100 =A-, >95=B+, >90=B, >85=B-, >80=C+,
>75=C, >70=C-. Actual letter grades will not be lower than implied by
these guidelines, but might be higher.

**MEETING
SCHEDULE**

[NOTE:
FROM PREVIOUS EXPERIENCE, SOME OF THESE ITEMS MAY TAKE MORE THAN ONE
LECTURE -- THERE IS A RESERVE OF AT LEAST TWO LECTURE PERIODS TO
ACCOMMODATE THIS]

Lecture
plan:

1. Perspective on the
course, procedures

2. Particle dynamics and
conservation laws

3. Wave dynamics

4. Complex functions,
Gaussian function, Gaussian integral, Fourier transforms, dual descriptions of
waves

5. Wave packets

6. Gauss law for charge
and electric field, Ampére law for steady current and magnetic field

7. Faraday induction law
for magnetic and electric fields

8. Local conservation of
electric charge

9. Maxwell addition to Ampére
law

10. Differential form of
Maxwell equations, wave solutions

11. Polarization of light,
"magical" behavior of polarizing elements, 2X2 matrices

12. Lorentz force law and
local conservation of energy and momentum

13. Electromagnetic
intensity, energy density, momentum density

14. Photon energy
(Planck-Einstein relation), intensity, number density, momentum

15. Probability density
for a one-photon system

16. Diffraction patterns

17. Relation of energy and
momentum to time and space derivatives

18. Derivation of
Schrödinger equation for non-relativistic particle

Role of potential energy
function, and contrast with refraction of light

19. What would have
happened if Einstein had developed this approach in 1905?

20. Correlations of
indistinguishable particles -- Bose-Einstein and Fermi-Dirac statistics

21. Dirac equation, hole
theory, connection of spin and statistics, CPT theorem

22. Entanglement

Recitations:
The first meetings of the recitations will take place the FIRST WEEK,
and will be for getting acquainted and exploring student hopes for the
course. Already at this stage it should be possible to earn credit for
engagement. After that, in a week with homework due the following day,
questions and discussions about the homework will be the principal business,
though other questions about lecture material, including at the end of the term
presentations by other students, also will be welcome. As mentioned above, it's
expected that students, individually or in groups, will present at least 4
questions (or relevant comments) during the term, giving the possibility of
earning up to 4 points for engagement with the material. Students may
volunteer in advance to present an issue with a problem, or to present a
solution to a problem, or to raise a question relating to lecture. If you
want to check on the validity of your point, you may discuss it with me by
email in advance of the class. I may ask students to share with the class
issues raised in their journal entries, and this also could produce engagement
credit. Besides preparation for solving the homework, which works
best if students have tried it before recitation, these sessions give a chance
to go beyond what has been discussed in lecture. If students have ideas
about how to make the recitations more rewarding, I’m happy to listen (and to
recognize productive suggestions with engagement credit!).

**CLASS
PROTOCOL**

Cell
phone and electronic device statement: Everyone, including the instructor, is
permitted to bring devices to class, but they must be in quiet mode. Under
normal circumstances incoming calls should not be answered during class.
Students are free to make audio or video recordings of the class, and are permitted
to post these recordings on the blackboard site. Our lecture classroom is
equipped for video-recording, and the video-record will be made available
through Blackboard. In the past, students have said that these recordings are
useful not only for a missed class but also for going over difficult points in
lectures.

**CLASS
RESOURCES**

Library
resources: Some texts will be put on reserve in the Math-Physics library.
Blackboard will be used extensively, and students are strongly encouraged to
post comments or questions relating to the class, anonymously if they wish. Of
course, to get engagement credit for an apt comment or question, you do need to
say who you are!

**DISABILITY
SUPPORT SERVICES (DSS) STATEMENT**

If
you have a physical, psychological, medical or learning disability that may
impact your course work, please contact Disability Support Services, ECC
(Educational Communications Center) Building, room128, (631) 632-6748. They
will determine with you what accommodations, if any, are necessary and
appropriate. All information and documentation is confidential. See __http://studentaffairs.stonybrook.edu/dss/index.shtml__

Students
who require assistance during emergency evacuation are encouraged to discuss
their needs with their professors and Disability Support Services. For
procedures and information go to the following website: __http://www.stonybrook.edu/ehs/fire/disabilities__

**ACADEMIC
INTEGRITY STATEMENT**

Each
student must pursue his or her academic goals honestly and be personally accountable
for all submitted work. Representing another person's work as your own is
always wrong. Faculty are required to report any suspected instances of
academic dishonesty to the Academic Judiciary. For more comprehensive
information on academic integrity, including categories of academic dishonesty,
please refer to the academic judiciary website at __http://www.stonybrook.edu/uaa/academicjudiciary/__

For
this course there are some extra aspects associated with two complementary
approaches, cooperation and independence. If you cooperate in discussing the
homework, you should indicate with whom you worked, but you should be
independent in writing up the solutions. That means it is quite possible that
two students who discussed a particular problem could get different grades for
their solutions. On the papers and presentations it is crucial to indicate all
sources for significant statements you make. If you are quoting somebody, you
should put the statement in quotation marks and indicate where you got it. If
your source is a web presentation, you should indicate where it is (i.e., the
URL), and the author if any is given, but you also should check it, as there
are many incorrect statements in web presentations. Something completely
cooperative is group papers and presentations. Something completely individual
is your journal entry, indicating your sense of the content of each lecture.

**CRITICAL
INCIDENT MANAGEMENT**

Stony
Brook University expects students to respect the rights, privileges, and
property of other people. Faculty are required to report to the Office of
Judicial Affairs any disruptive behavior that interrupts their ability to
teach, compromises the safety of the learning environment, or inhibits
students' ability to learn.