Course Outline

Quantum theory is a foundation of physics, providing a general scheme for understanding a vast range of physical phenomena. Every physicist needs to be familiar with the main ideas and results of quantum theory and many use it routinely. This course aims to give you a solid understanding of the framework and applications of quantum theory.

Quantum theory is tremendously important in our modern lifestyle; it explains the operations of semiconductors, lasers, magnetic resonance and other technologies that were inconceivable before the development of the subject. It has also profoundly changed our view of the physical world. Nearly 100 years after quantum theory was formalized, experts are still discover apparent paradoxes within the theory and dispute their implications.

In Physics 321, the general rules of quantum theory will be described using the state representation framework rather than the more limited wavefunction approach. This will be illustrated with two-state quantum systems, which have no classical counterparts but capture the essential ideas of the subject without the distracting mathematical complications associated with single particles with a position degree of freedom. The general rules will then be applied to systems with a position degree of freedom.

The course will cover the following topics subject to minor modifications.

  1. Spin half systems and Stern-Gerlach type experiments.
  2. State representation, measurements and time evolution for spin half and analogous systems.
  3. General framework of quantum mechanics.
  4. Particles in one dimension: wave mechanics.
  5. One dimensional harmonic oscillator.
  6. Rotations and angular momentum.
  7. Particles in central potentials, hydrogen atom.

Course Notes

My notes for the course, in book format, will be update regularly and can be found at Phys 321: Notes.

Homework Assignments

Due: 27 January 2023 Homework 1
Due: 31 January 2023 Homework 2
Due: 3 February 2023 Homework 3
Due: 7 February 2023 Homework 4
Due: 10 February 2023 Homework 5
Due: 14 February 2023 Homework 6
Due: 17 February 2023 Homework 7
Due: 21 February 2023 Homework 8
Due: 24 February 2023 Homework 9
Due: 28 February 2023 Homework 10
Due: 10 March 2023 Homework 11
Due: 14 March 2023 Homework 12
Due: 17 March 2023 Homework 13
Due: 28 March 2023 Homework 14
Due: 31 March 2023 Homework 15
Due: 4 April 2023 Homework 16
Due: 7 April 2023 Homework 17
Due: 11 April 2023 Homework 18
Due: 14 April 2023 Homework 19
Due: 18 April 2023 Homework 20
Due: 28 April 2023 Homework 21
Due: 2 May 2023 Homework 22
Due: 5 May 2023 Homework 23
Due: 9 May 2023 Homework 24

Exams

There will be two hour long exams during class on the following dates: Thursday 2 March 2023 and Thursday 20 April 2023. There will be a comprehensive final exam on Tuesday 16 May 2023.

Exams and solutions from previous semesters.

Fall 2007 Class exam 1
Fall 2007 Class exam 1: Solutions
Fall 2007 Class exam 2
Fall 2007 Class exam 2: Solutions
Fall 2007 Final exam
Fall 2007 Final exam: Solutions
Spring 2022 Class exam 1
Spring 2022 Class exam 1: Solutions
Spring 2022 Class exam 2
Spring 2022 Class exam 2: Solutions
Spring 2022 Final exam
Spring 2022 Final exam: Solutions

Exams and solutions from this semester.

Solutions will be posted after each exam has been graded.

Spring 2023 Class exam 1
Spring 2023 Class exam 1: Solutions
Spring 2023 Class exam 2
Spring 2023 Class exam 2: Solutions

Supplementary Reading

There are many additional texts which are potentially suitable for this course. The following is a selection.

  1. General Texts

    1. R. P. Feynman, R. B. Leighton and M. Sands, Lectures on Physics, Vol III, Addison-Wesley (1965).

      The first chapter of Vol III of Feynman's lectures still contains one of the best introductions to quantum mechanics. This is essential reading for anyone interested in the subject. The are several chapters which include comprehensive discussions of spin half and other discrete quantum systems at a level suitable for this course.

    2. J. S. Townsend, A Modern Approach to Quantum Mechanics, University Science (2000).

      A "spins first" approach to undergraduate quantum theory.

    3. D. J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall (1995).

      Possibly the most accessible quantum mechanics text at this level. Generally very well written and easy to read. However, this text focuses almost exclusively on wave mechanics and there is little actual use of the Dirac formalism. Somewhat limited in it's discussion of time evolution of quantum systems.

    4. J. J. Sakurai, Modern Quantum Mechanics, Prentice-Hall (1995).

      An excellent text which uses two state systems to introduce many key features of quantum mechanics. This may be a little challenging to read but it develops the subject in a manner similar to that of this course.

    5. R. Shankar, Principles of Quantum Mechanics, Plenum (1980).

      An excellent comprehensive text which is very systematic and covers many topics. Although it emphasizes the physics of particles moving in one or more dimensions, there is adequate coverage of spin half and other discrete quantum systems. The presentation of the main axioms is sufficiently general. This may also be a little challenging to read - I don't expect you to be able to read the chapter on Langrangian Mechanics (Ch 2) but the book is written so that you can skip this.

    6. C. Cohen-Tannoudji, B. Diu and F. Laloe, Quantum Mechanics, John Wiley (1977).

      Encyclopedic compendium of all things quantum mechanical, circa 1977. This two volume set is dense, often quite heavy mathematically and has an initially bewildering indexing system, but it covers a vast array of quantum mechanical topics in great details and features numerous interesting examples. One of the few texts to discuss topics like two-state systems, tensor products, etc..., in adequate detail. If you are going to do more than one course in quantum mechanics in your graduate career, this is an essential text.

    7. D. Bohm, Quantum Theory, Dover (1979).

      "Old school" classic written in 1951 by one of the leading experts in quantum mechanics. Comprehensive coverage of wave mechanics, albeit with an archaic notation. This contains many important examples worked out in great detail as well as thorough discussions of the key experiments that led to the development of quantum mechanics. Added bonus: it's a Dover publication and you should be able to buy it for less than $20.

    8. T. F. Jordan, Quantum Mechanics in Simple Matrix Form, Wiley (1986).

      This is one of the few introductory level texts which describes spin-half systems in detail. However, its treatment of the subject is quite mathematical and idiosyncratic; the state vector is never introduced and the statistics of measurement outcomes are described entirely in terms of algebraic properties of appropriate matrices. By doing this one can avoid the pitfalls of associating a reality with or interpreting the state vector and have the measurements occupy center stage. For example, for spin-half systems, matrices associated with spin components square to a positive scalar multiple of the identity. Evidently one can then conclude that the spin component has values plus or minus the square root of the scalar multiple. Following this one can demonstrate the usual features of measurements associated with these systems. The connection between these matrices and the results of measurements is never really precisely spelled out. A more conventional formalism that is like this would use the notion of density operators and eigenvalues of observables.

      If you are still confused on the basics of complex numbers, linear algebra or probabilities, the first six (short) chapters should be a great help.

Links and Animations

  1. Reference Sources
    1. Physlink. Reference information and data, including decimal system notation, physical constants, math constants, astro-physical constants, etc,....
    2. Eric Weinstein's World of Physics. Encyclopedia of Physics maintained by Wolfram Research. Entries at a variety of technical levels.
    3. Periodic Table of Elements. WebElements site.
    4. NIST Standard Reference Databases. Administered by the National Institute for Standards and Technology. The final word in physical data. Intended for professionals.
  2. Animations
    1. PhET. From the University of Colorado.
    2. oPhysics Physics simulations provided by Tom Walsh.
    3. LTU Applets. Collection of simulations provided by Scott Schneider, Lawrence Technological University.
    4. Animations for Physics and Astronomy. Collection of simulations from Dr. Michael R. Gallis, Penn State University, Schuylkill. Youtube channel
    5. Physclips. Collection of simulations from the University of New South Wales, Australia.
    6. The Quantum Mechanics Visualisation Project (QuVis). Collection of simulations from the University of St. Andrews, United Kingdom.
  3. Experimental Investigations of the Foundations of Quantum Physics

      The basic ideas of quantum mechanics are frequently presented in the form of thought experiments. In this course we have used sequences of imaginary Stern-Gerlach experiments on spin 1/2 systems to illustrate the fundamental physical and mathematical concepts of quantum mechanics. Such sequences of Stern-Gerlach experiments have never actually been performed. So what is the evidence for the physics we describe in the classroom? Mostly it emanates from equivalent experiments in quantum optics. Here are some references.

    1. J. M. Raimond, M. Brune, and S. Haroche, "Colloquium: Manipulating quantum entanglement with atoms and photons in a cavity," Rev. Mod. Phys. 73 565 (2001).
    2. A. Zeilinger, "Experiment and the foundations of quantum physics", Rev. Mod. Phys. 71 288-297 (1999).
    3. A. Aspect, "Experimental Test of Bell's Inequalities Using Time- Varying Analyzers", Phys. Rev. Lett. 49 1804 (1982).
    4. Photon Quantum Mechanics Undergraduate level laboratories developed at Colgate University. Links to several articles describing experiments.
    5. Modern Quantum Mechanics Experiments for Undergraduates Undergraduate level laboratories developed by Mark Beck at Reed College. Links to several articles describing experiments.

      6. Some of experiments illustrating fundamental aspects of quantum mechanics have also been demonstrated using interference of neutron beams. Such neutron interferometry experiments have been carried out since the 1970s.

    6. Daniel M. Greenberger, "The neutron interferometer as a device for illustrating the strange behavior of quantum systems," Rev. Mod. Phys. 55, 875 (1983).
    7. J.-L. Staudenmann, S. A. Werner, R. Colella, and A. W. Overhauser, "Gravity and inertia in quantum mechanics," Phys. Rev. A 21, 1419 (1980).
    8. S. A. Werner, R. Colella, A. W. Overhauser, and C. F. Eagen, "Observation of the Phase Shift of a Neutron Due to Precession in a Magnetic Field," Phys. Rev. Lett. 35, 1053 (1975).
    9. H. Rauch and W. Treimer, and U. Bonse, "Test of a single crystal neutron interferometer," Phys. Lett. A 47, 369-71 (1974).

      10. There have been several demonstrations of the delayed-choice experiment. These articles are intended for a specialist audience. However, you should be able to understand the description of the conventional delayed choice experiment in the introduction, experimental setup and experimental results sections of Hellmuth, et.al. There is a good discussion of the delayed choice experiment in "The Quantum Challenge" by Greenstein and Zajonc.

    10. T. Hellmuth, H. Walther, A. Zajonc, and W. Schleich, "Delayed-choice experiments in quantum interference," Phys. Rev. A 35, 2532-40 (1987).
    11. B. J. Lawson-Daku, R. Asimov, O. Gorceix, Ch. Miniatura, J. Robert, and J. Baudon, "Delayed choices in atom Stern-Gerlach interferometry," Phys. Rev. A 54, 5042 (1996).
    12. Yoon-Ho Kim, Rong Yu, Sergei P. Kulik, Yanhua Shih, and Marlan O. Scully, "Delayed "Choice" Quantum Eraser," Phys. Rev. Lett. 84, 1-5 (2000).
  4. Stern-Gerlach experiment
    1. "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld," W. Gerlach und O. Stern, Zeitschrift fur Physik 9, 349-352 (1922). "Uber die Richtungsquantelung im Magnetfeld," W. Gerlach und O. Stern, Annalen der Physik, 74, 673-699 (1924). The original articles are in German, but both of these contain prints of the photographic target plate.
    2. Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics Physics Today 56, 53 (Dec 2003). Review of the history of the Stern-Gerlach experiment.
    3. Spin, Quantum Made Simple from Toutestquantique.
    4. Stern-Gerlach Experiment from QuantumVisions.
    5. Stern-Gerlach Experiment Animation from Jillian Kunze, Joint Quantum Institute.
  5. Photons
    1. Mach Zehnder Interferometer Simulation from Quvis, University of St. Andrews.
  6. Particle Diffraction Experiments
    1. Physics Web Excellent summary of experimental efforts to demonstrate interference and diffraction of particles passing through single and multiple slits. From Physics World.
    2. Electron interference patterns from Hitachi, Japan.
    3. Electron interference patterns from IMM Institute of the Italian National Research Council (CNR)
    4. Electron Scattering Davisson-Germer experiment. From PhET, The University of Colorado. Alternative link here.
    5. Single Slit Diffraction of Neutrons, C. G. Shull, Phys. Rev. 179, 252 (1969). Description of a neutron single slit diffraction experiment.
    6. Single and Double Slit Diffraction of Neutrons, A. Zeilinger, R. Gahler, C. G. Shull, W Treimer, W Mampe, Rev. Mod. Phys. 60, 1067 (1988). Description of a neutron single and double slit diffraction experiments.
    7. Optics and interferometry with atoms and molecules, A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, Rev. Mod. Phys. 81, 1051 (2009). Overview of particle interference experiments.
    8. Colloquium: Quantum interference of clusters and molecules, K. Hornberger, et. al., Rev. Mod. Phys. 84, 157 (2012). Description of a particle interference experiments.
    9. Fullerene Diffraction From Anton Zeilinger, University of Innsbruck, Austria,
  7. Uncertainty Principle
    1. O. Nairz, M. Arndt, and A. Zeilinger, "Experimental verification of the Heisenberg uncertainty principle for fullerene molecules," Phys. Rev. A 66, 032109 (2002). Demonstration of the uncertainty principle in large molecules.
  8. One Dimensional Quantum Mechanics
    1. One dimensional quantum system simulator From Paul Falstad.
    2. Quantum Bound States PhET, The University of Colorado. Alternative link here.
    3. 1-d Quantum States From Paul Falstad.
    4. Quantum Dots From Nanohub. Tutorials and applications.
    5. Quantum Dots Available from CD Bioparticles. Describes applications.
    6. QNA Technology Sells quantum dots.
    7. Quantum Dots and Biomedical Assays From LBNL.
    8. Quantum Dot-Based Nanotools for Bioimaging, Diagnostics, and Drug Delivery, Regina Bilan, et.al., Chem. Biochem. 17, 2103 (2016).
    9. Quantum Dots in Diagnostics and Detection: Principles and Paradigms, T. R. Pisanic, et.al., Analyst 139, 2968, (2014).
    10. Gaussian Wavepacket QuVis simulation. From University of St. Andrews.
  9. Trapped Ions
    1. Trapped Ion Quantum Information University of Innsbruck.
    2. Trapped Ion Quantum Information Chris Monroe's group, University of Maryland.
    3. NIST Ion Storage Group.
  10. Harmonic Oscillators
    1. Molecular Vibrations. From ChemTube3D.
    2. Normal Modes PhET, The University of Colorado.
    3. Quantum Optomechanics Aspelmeyer Group, University of Vienna.
    4. Molecular Vibrations from Edwin Scauble, University of California at Los Angeles.
    5. A. Gaidarzhy, G. Zolfagharkhani, R. L. Badzey, and P. Mohanty, "Evidence for Quantized Displacement in Macroscopic Nanomechanical Oscillators," Phys. Rev. Lett. 94 030402 (2005). Possibly the first experimental demonstration of quantum mechanical effects in a macroscopic oscillator - accomplished in 2005.
    6. F. Pistolesi, A. N. Cleland, and A. Bachtold, "Proposal for a Nanomechanical Qubit," Phys. Rev. X 11 031027 (2020). Possible use of a quantum harmonic oscillator for quantum information.
    7. Patricio Arrangoiz-Arriola, et.al. "Resolving the energy levels of a nanomechanical oscillator," Nature Phys. Rev. Lett. 571, 537-540 (2019). Appears to show the three lowest energy modes of a mechanical oscillator. arXiv article.
    8. Bjorn Schrinski, Yu Yang, Uwe von Lupke, Marius Bild, Yiwen Chu, Klaus Hornberger, Stefan Nimmrichter, and Matteo Fadel, "Macroscopic Quantum Test with Bulk Acoustic Wave Resonators," Phys. Rev. Lett. 130, 133604 (2023). Collection of a microgram of atoms oscillating.