PHYSICS 390: PHYSICAL AND MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS
Fall 2008

Teacher: Alfred Scharff Goldhaber
Yang Institute (Math 6-113, 2-7975, goldhab@insti.physics.sunysb.edu)
Office Hours: By appointment or drop in
Class meetings: TuTh 6:50-8:10 PM
Texts: New course -- hence no textbook
No Written Exams

Motivation

In more than eight decades the teaching of quantum mechanics has been stuck in a pair of patterns which might have been OK when they were fresh and new, but now seem tired and inadequate. The first approach is 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 especially of "modern physics" courses, where "modern" means (at least for a large part of the material) verging on 100 years old! The other method is axiomatic, exemplified by Dirac's great book on the subject. If the former is warm and fuzzy (in the sense of including many interesting bits and pieces, but without an overarching coherent structure), the latter tends to appear as cold and unmotivated. My method 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 path, as indicated in the outline below, goes through Maxwell electrodynamics to one-photon quantum mechanics, and uses that as a base to develop one-electron quantum mechanics. I call this way "stronger, deeper, better" [Stronger: there is an organic connection to electrodynamics, which leads through Einstein's light quanta to photon quantum mechanics. Deeper: The two essential notions of quantized energy and photon intensity give a base for everything else. Better: These two notions allow deduction in a natural and straightforward way of most if not all the remarkable and puzzling features of quantum physics, without the abrupt transitions of a modern physics course or the cold start of a standard formal course.] 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 quantum physics just comes along at the end. Eventually, this might be a first-term sophomore course, and strongly motivated students at that level would be welcome already now.

Teacher

Alfred Scharff Goldhaber

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.

Audience

This new course should be appropriate for students preparing for the standard quantum physics sequence, for students currently enrolled in advanced quantum physics (which I also shall be teaching this fall Tu-Th 520-640 PM), and for students interested in learning the principles of quantum physics who have a strong preparation in first-year classical physics and first-year calculus, whether they are physics majors or not. Thus the prerequisites for the course are two semesters of introductory physics (PHY121-2,131-2, or 141-2 or equivalent) and two semesters of introductory calculus (MAT 125-6 or equivalent). The plan is to keep the course self-contained with respect to more advanced topics, e.g., matrices and linear operators more generally.

Course Outline

  • A. We'll begin with an exposé of Maxwell-Lorentz electrodynamics focusing on local conservation of energy and momentum, including energy density, the Poynting vector of electromagnetic field power intensity, and the Thomson momentum density.
  • B. We'll use solutions of Maxwell equations to introduce properties of waves, including interference and also wave packets. Because the equations represent fundamental physics, this may be a better way to become acquainted with waves than the more customary discussion in terms of waves in media, such as water waves, sound, and string vibrations.
  • C. Following Einstein's 1905 paper, one may assign energy hν (where ν is the frequency of the light) to each "light quantum," or photon, and then interpret the Poynting vector as hν multiplied by the intensity of photons, i.e., number of photons per unit area per unit time. This also allows deduction of the Maxwell energy density as hν multiplied by the number of photons per unit volume, and thereby, from the Thomson momentum density, themomentum of an individual photon, h/λ (where λ is the wavelength of the light). Because the Maxwell equations allow calculation of the way that light energy moves in time, this approach tells where photons can be found at various times, and thusgives a complete one-photon quantum mechanics.
  • D. From the above one may relate energy to the time derivative of the wave function (which here is the electromagnetic field) and the components of momentum to the space derivatives of the wave function. This gives all the tools for constructing the Schrödinger equation for an electron in a hydrogen atom, and for relating the wave function to the probability of finding the electron in various locations, i.e., all of one-electron quantum mechanics.
  • E. At this point we shall switch to discussing the mathematically simplest quantum problem, that of a system which can exist only in one of two quantum states, or a linear superposition of the two. Using two-by-two matrices, we can work out examples of nearly all the general principles of quantum mechanics.
  • F. From here we'll go on to discussing larger, even infinite, systems, and how the principles work out in these settings. The generalization of the concept of Hermitian matrix to that of self-adjoint operator, and the deep significance of the latter, may be the most important part of this development.
  • G. With this base, depending on class interest, we can explore applications, including the surprising phenomena associated with collections of many indistinguishable particles.

    • Requirements and grading

    This course will depend on regular and active class participation by all students. There will be weekly written homework, and the other component of the grade will be based on regular class presentations by students, sometimes of solutions to homework problems and sometimes of important topics that go beyond the lectures. I expect to comment in detail on the student presentations, to make sure that people know what I expect and that any obscure points are clarified. There will be no written exams.

    Important Notices

  • Students with disabilities should see Disabilities Support Services for certification of their disabilities and assurance that these will be accommodated appropriately for the class. Also, students should make sure that proper notification and arrangements are in place for help in emergency evacuation taking account of handicaps.

  • 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. Any suspected instance of academic dishonesty will be reported 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/