Physics 356F
Introduction to Quantum Physics.
(last updated 14 Dec 2011)
Lecturer:
Aephraim M. Steinberg
(rm 1103, tel 978-0713, my last name at physics.utoronto.ca)
Office hours: W 2-3
Teaching Assistants:
Jonathan Braden (email jbraden at physics; office 1416; tel. 8-5759)
William Witczak-Krempa (email wwitczak at physics; office 1023A; tel. 8-5444)
Times and Venues
Lectures: Tu 14-16 in MP102
Discussion section 0101-A (William) - W 11 in LM161
Discussion section 0101-B (Jonathan) - W 11 in MP606
Discussion section 0201-A (Jonathan) - W 12 in MP408
Discussion section 0201-B (William) - W 14 in MP1115
FIRST TUTORIAL Wednesday 14 September 2011 --
if you are in TUT0101, then if your last name begins with A-M go to LM161 and if it begins with N-Z go to MP606;
if you are in TUT0201, then go to MP408 to determine which time and room you will be in.
Overview |
Grading |
Syllabus |
Announcements |
Readings |
Postings |
Tutorials |
Problem Sets |
Quantum physics is one of the major scientific and intellectual
developments of the 20th century. Not only has it revolutionized
Man's understanding of the structure of matter,
but it underpins a broad cross-section of modern technology, from
the transistors in your computer to the laser in your CD player.
More than that, however, it has led to a radical change in the
basic structure of how we understand the world. This change is
not limited to atoms, or even to the microscopic world in general.
It applies whether we are discussing electric forces, angular
momentum, light, matter, or the nuclear forces.
In this course, we will study the structure and postulates of
quantum mechanics and will then develop the basic mathematical
and conceptual tools to deal with important topics
such as angular momentum, the Hydrogen atom, the harmonic
oscillator, entanglement, indistinguishable particles, and spin.
In parallel, you will be
challenged to develop your intuition about the quantum world,
with discussions based largely on two-level systems such as the
Stern-Gerlach experiment, photon polarization, and two-slit (or
Mach-Zehnder) interferometers.
The required text is Townsend's A Modern Approach to Quantum Mechanics.
For further reading, I strongly recommend
Richard P. Feynman's wonderful
QED: the strange theory of light and matter, as well as
the following alternate textbooks:
Shankar's Principles of Quantum Mechanics
Feynman's Lectures on Physics vol. III
| |
Fraction of Grade |
| Problem Sets
| 20% |
| Mid Term
| 30% |
| Final Exam
| 50% |
There will be 4 or 5 problem sets over the course of the semester.
The problem sets are to be handed in at the tutorial sessions.
Only selected problems will be marked; the purpose of the assignments
is not primarily evaluation of your performance, but rather to give
you the opportunity to work through the material, develop your understanding,
and also prepare for the exams. Solution sets will be posted
after the problem sets are handed in, and obviously this means that no
late assignments can be accepted.
Quantum mechanics is a challenging subject because it forces us
to come to terms with completely new ideas of how reality should
be described and what can be predicted, and not merely because
it involves solving a particular second-order partial differential equation.
Quantum mechanics is not just a theory of electrons in atoms, or
even of microscopic particles in general, but
as some wag (I've forgotten who) put it, "the Operating System under
which all other physics theories run."
While this course will endeavour
to give you the tools to solve many of the important classes of problems
you are likely to encounted as you continue studying physics, it will
also stress the fundamental features that make quantum physics truly
different from classical physics, and attempt to develop your intuitions
about quantum phenomena. We will approach quantum mechanics in an
axiomatic fashion, and do our best to understand what leads us to believe
these axioms, and what their significance is.
I will also introduce, briefly, some of the
modern "applications" of quantum physics such as quantum computation
and quantum teleportation, although along the way we plan to resolve
some more prosaic questions such as why the spectrum of Hydrogen is
what it is and why the periodic table looks the way it does.
Because of the thorny conceptual issues involved in quantum mechanics, and
because I suspect that two straight hours of technical lecture may be
too much for either me or you, I will often use the first hour of lecture
to cover some standard, "technical," "textbook" material; and after a short
break will discuss more "complementary" issues in the second hour. I expect
to jettison this foolhardy plan some time after I realize how far behind
schedule we are.
Class participation is encouraged during all lectures, and particularly
during the concept-based "complements," when if my throat begins to get
sore, students should be prepared to discuss questions in small groups and
then defend their findings.
You will find that active participation in discussions is essential
for developing a real understanding of the material, and this will
be reflected both on the exams and in the writing assignments.
The lectures will break down into 6 units, of approximately two weeks
apiece. The following tentative
syllabus indicates the material I would ideally
like to cover by the end of term, and likely topics for the "complements."
Naturally, some topics may need to be shortened or omitted entirely in order
to finish in 12 weeks, but we begin in a spirit of optimism!
(The most relevant readings are listed along with the subject headings below, although we will not be following the book religiously, and some topics are not treated there at all.)
| UNIT |
"Technical" lectures |
"Conceptual" complements |
| 1 |
Photon polarisation and electron spin;
the phenomena of quantum measurement (ch's 1-2) |
Single-photon and single-electron interference (QED; see also ch. 8 of Townsend);
quantum cryptography |
| 2 |
The structure and postulates of QM;
matrix mechanics and Dirac notation (ch 2; Shankar ch's 4 & 1) |
Interaction-free measurement;
the quantum eraser |
| 3 |
Commutation, conservation, and uncertainty;
time evolution (ch's 3-4) |
Quantum jumps and the Quantum Zeno Effect |
| 4 |
Wave mechanics and the harmonic oscillator (ch's 6-7) |
Measurements and correlations;
the Einstein-Podolsky Rosen "paradox" and Bell's inequalities (ch 5) |
| 5 |
Separation of variables, multi-dimensional problems, and angular momentum (ch's 3 and 9) |
Quantum teleportation;
quantum computation |
| 6 |
The Hydrogen atom;
indistinguishable particles (ch's 9,10, and 12) |
If time permits, some real-world examples |
The sections in Townsend most relevant to each section of
the course are listed along with the syllabus
given above. You will be expected to keep up with the reading.
Sometimes, additional sources may be suggested, especially for the
complements.
For interference and probability amplitudes, I recommend the
following references:
Feynman's QED
Feynman's Lectures on Physics, vol. III, ch's 1-3.
For two-state systems, ch's 5 and 6 of Feynman vol. III may prove
useful.
For the postulates of quantum mechanics, Shankar's Principles of
Quantum Mechanics is a very clear reference.
For philosophical and interpretational aspects, including great detail
on the debates between Bohr and Einstein, I suggest
Wheeler & Zurek, Quantum Theory and Measurement and
Bell, Speakable and Unspeakable in Quantum Mechanics.
The following articles may be accessible introductions to some of
the more modern experiments and issues in quantum mechanics, related
to the complements:
Horgan, "Quantum Philosophy," Sci. Am. 267 (1), 94 (7/92).
Bennett et al., "Quantum Cryptography," Sci. Am. 10/92
Chiao et al., "Faster than light?", Sci. Am. 8/93
Greenberger et al., "Multiparticle Interferometry," Physics Today 46 (8), 22 (8/93)
Kwiat et al., "Quantum Seeing in The Dark," Sci. Am. 275 (5), 52 (11/96).
Scully et al., "Quantum optical tests of complementarity," Nature 351, 111 (1991)
Zurek, "Decoherence and the transition from quantum to classical," Physics Today 44, 36 (1991)
Collins, "Quantum Cryptography," Physics Today 45, 21 (1992).
A very nice page on "Visual Quantum Mechanics," with useful material
for review, can be found here.
Lecture 1, 13 Sep 2011; or in pdf format
For the first two weeks, you should finish reading the first two chapters of Townsend, as indicated on the syllabus. It may be interesting to start with section 2.7, on photon polarisation, since this is the approach we are taking in class (mathematically equivalent to the Stern-Gerlach problem treated in chapter 1).
For those of you not sufficiently comfortable with polarized light, I strongly recommend first reading David Harrison's short introduction, and/or the Wikipedia page on polarizers.
For the material on two-slit interference, I know of no better reference than the Feynman book on "QED: The Strange Theory of Light and Matter."
The Wikipedia page is all right, and links to discussions of "complementarity" and the "quantum eraser," topics which we will continue discussing along with the "Feynman rules." Its discussion of the "path-integral formulation" is the mathematically more general description of what I've been calling Feynman's rules.
If you've been reading or listening to the news, you may have heard that a European collaboration appears to have observed particles moving faster than the speed of light, a big no-no according to Einstein. As usual, the BBC News report is reasonably careful and accurate, while Physorg.com has a slightly more detailed story, and this blog discusses the controversy as a case study of how science should be done in general (since there are numerous counter-examples, such as the one of "cold fusion").
Lecture 2b, 20 Sep 2011; or in pdf format
Since a number of people have asked for reading material about the Mach-Zehnder interferometer, I can suggest three links:
(1) The shortest and simplest
(2) The longest, most complete, & most accurate
& (3) Something in between the two
As I announced in class, you should be reading chapter 3, and then I recommend reading at least section 6.1 before continuing through chapter 4. The "complementary" material on EPR & Bell is discussed in sections 5.1-5.4.
I also announced that Nima Arkani-Hamed's colloquium will be Thursday Oct 6 at 4 pm in MP102; more information is available here; in addition to talking about spacetime, string theory, and such things, this will make connection to the "Feynman-path" picture of quantum mechanics which we are using when we discuss interferometers, and which is the basis of the "QED" book.
The informal "quantum tea" will be in the 4th-floor Chemical Physics lounge of Lash-Miller at 3 pm on Wednesday October 5th; more information on this and other CQIQC programmes (including summer fellowships for undergraduates) is available here.
Some sample problems to help you study for the midterm are available at this link.
This one-page write-up derives the properties of Sx used in the practice problems, and may in its own right offer useful practice working with Dirac notation and matrices and understanding the relationship of these states to one another.
NOTE: I was careless in the last sentence I wrote there. It may be instructive to think about what the spin operators would look like if I chose a new coordinate system rather than (x,y,z), and asked about the spin projection on some x' axis that might be 45 degrees between x and y...
NOTE: As explained via email, I also mistakenly claimed that i was the square root of 1. Hopefully you know what I was trying to say...
From Lecture 5B (11 Oct 2011), some material about the EPR "Paradox" and Bell's Inequalities, as well as the no-cloning theorem (even though we didn't get around to discussing that yet); or in pdf format.
Wave-particle duality at the macroscopic scale will be the Physics Department colloquium on Thursday, October 27th at 4:10 pm in MP102. It will present some mind-blow
ing experiments showing that a simple classical system (a water droplet bouncing on a surface of water) can exhibit a kind of "wave-particle duality" and challenge our ideas of what is so striking about quantum mechanics... I strongly encou
rage you all to attend (and you are welcome for tea & cookies at 3:45 in the grad student lounge behind the Burton Tower elevators).
Readings from 24 October...
You have all read chapter 3, much of chapter 5, and the beginning of chapter 6. If you have not yet finished chapter 4, you should do so right away.
Sections 6.2-6.5 formalize the connection we have been drawing between matrix mechanics and wave mechanics, and you should read them now. Sections 6.6-6.11 are essentially a review of wave mechanics you have learned last year: you should make sure you are comfortable with all of that material, and might wish to review your notes and textbook from last year if you are in any doubt, since we will not be going over it in detail again, but we will need it as we move forward.
The new reading for the weeks of 25 October and 1 November is chapter 7, on the harmonic oscillator.
Lecture 7B (1 November) on quantum erasers, EPR, and no-cloning theorems.
This is a short write-up showing how to mathematically connect expressions written in the position basis to expressions written in more general Dirac notation. It could be helpful on the present homework.
During the week of November 7th, you should review the material we have
covered so far, e.g., going over the solutions of the homeworks and the midterm.
Then you should read chapter 9, which will introduce the next topics we are heading towards. Section 10.5 treats the separation of variables, which we will be using (including on the homework).
You may find sections 8.1-8.2 clarify the connection between probability amplitudes as we find them by taking inner products in Dirac notation, and probabilities we calculate in interference experiments. The rest of the chapter is also very useful (providing the mathematics behind the viewpoint of QM I recommended you read Feynman's "QED" for), but we will not cover it in lecture this semester.
Some supplementary reading on cloning et cetera was given in the lecture notes posted, but I can also recommend the following articles for understanding the quantum eraser and its relevant to thinking about issues such as the "quantum-classical boundary":
Horgan, "Quantum Philosophy," Sci. Am. 267 (1), 94 (7/92).
Greenberger et al., "Multiparticle Interferometry," Physics Today 46 (8), 22 (8/93)
Scully et al., "Quantum optical tests of complementarity," Nature 351, 111 (1991)
Zurek, "Decoherence and the transition from quantum to classical," Physics Today 44, 36 (1991)
Kwiat et al., "Three proposed 'quantum erasers'," Phys Rev A 49, 61 (1994)
Kwiat et al., "Observation of a `quantum eraser': A revival of coherence in a two-photon interference experiment," Phys Rev A 45, 7729 (1992)
This is the source for the quantum physics applets I used in class to demonstrate time-dependence. You can probably learn a lot by playing with several of these.
ANNOUNCEMENT: I apologize, but I will be away at a conference and my office hours on Wednesday, 9 November will be cancelled.
For the week of November 14th, I have asked you to reach chapter 10. You have already finished chapter 9, and you may find it helpful to review chapter 3 and sections 5.1-5.3 as we start discussing angular momentum in more depth. Finally, Appendix B gives a brief description of the mathematical formalism behind the addition of angular momentum, a topic of fundamental importance in quantum mechanics.
A recent posting on the arXiv, "The quantum state cannot be interpreted statistically" by Pusey, Barrett, and Rudolph, is in my opinion a major philosophical breakthrough in quantum mechanics -- perhaps the biggest since Bell's 1964 theorem. It is a very well-written paper and you should be able to follow at least the essential argument given what we have already covered in class. I highly recommend reading it. If you want to be lazy, read the Nature News story about it.
Lecture 9B, from 22 November 2011.
Readings for week of 28 November, 2011
Read sections 12.1-12.3.
Lecture 10B, from 29 November 2011: a few notes on quantum computation, specifically the Deutsch-Jozsa algorithm and Grover's search algorithm.
For any one interested in quantum information more broadly, I can highly recommend Quantum Computation and Quantum Information by Nielsen and Chuang, a relatively comprehensive treatment including a nice chapter teaching "everything a computer scientist should understand about quantum mechanics" from scratch. Chapter 1 alone covers many important concepts including some simple algorithms.
I also recommend once more the applets at this web site. In particular, applet 2.3 allows you to play around with the energy and see which energies permit normalizable solutions in a simple potential. This is the basis of the argument we used to calculate the energies of Hydrogen, by insisting that the polynomial expansion should terminate at a finite order. There are also excellent applets about double-wells, two-dimensional states, spherical harmonics, spin-1/2's, et cetera.
Finally, a friend forwarded me this comic about Schrödinger.
Dec 7 ...
By utter coincidence, just after my lectures touching onquantum computers and force-carrying bosons and other disconnected topics, there was an excellent editorial in the New York Times giving a very clear and clear-headed view of what's promising about quantum computing, as well as a fascinating blog entry in the Guardian, compiling limericks and other responses from a dozen or so Nobel laureates and equally respected scientists to a request for their opinion about whether or not the LHC at CERN is likely to have discovered the Higgs, or whether they think it exists at all...
Lecture 11B, from 6 December 2011.
The first problem set can be downloaded here, and is due at the start of your tutorial on Wednesday, October 5.
NOTE: there was an error in the original problem, and the corrected version was posted at 16:05 on 25 Sep.
You should also note that the midterm exam has been scheduled, and will be in class on Tuesday, October 18.
Solution set for problem set 1 posted at 4:30 pm on 7 Oct 11.
The second problem set can be downloaded here, and is due at the start of tutorial on Wednesday, November 2.
The solutions to the midterm problems can be downloaded here.
The histogram of midterm results and some rough indications of breakpoints for letter grades can be found here.
The solutions to the second problem set can be found here.
The third problem set can be downloaded here, and is due at the start of tutorial on Wednesday, November 16.
The solutions to the third problem set can be downloaded here (NOTE: corrected solutions posted 24.11.11).
The fourth problem set can be downloaded here, and is to be returned to your TA by 5pm on the last day of term (Tuesday December 6th).
Draft solutions to the fourth problem set can be downloaded here.
Updated solutions to the fourth problem set posted here 14.12.11
Some people have asked me about additional practice problems. While the material is not exactly identical to what we have covered here, there is a great deal of overlap with a quantum course I taught in 2002, and this is a link to the homework assignments from that year, and some even earlier exams, which you may find useful.
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