Quantum Mechanics III (Physics 125c)

Sean Carroll, Physics Department, Caltech

Spring quarter, 2017

Artist's impression of entangled particles, from YouTube/stargazer via Business Insider

Announcements

  • Final exam for seniors and graduate students will be available at noon on Weds 6/7, due at noon on Fri 6/9. For non-senior undergraduates, the exam will be available at noon on Tues 6/13, due at noon on Thurs 6/15.

Description

Physics 125c is the third quarter of the upper-level undergraduate/graduate quantum mechanics sequence. While it is a continuation of 125a and b, 125c is a somewhat stand-alone course, suitable for anyone who has mastered the basic tools of quantum mechanics (wave functions, Schrödinger equation, Hilbert space, operators, etc.).

Rather than being a grab bag of special topics, the course will focus on the theme of subsystems and entanglement. The way that subsystems relate to each other via entanglement is at the heart of what makes quantum mechanics unique. Much of a traditional pre-125c course in quantum mechanics will focus on the nonrelativistic Schrödinger equation and its solutions; this course will have almost none of that. There will be a lot of tensor products and density matrices.

Dates

The course will meet Mondays and Wednesdays from 10:30 to 12:00 in Downs 107. First day of class is Monday April 3; last day is Wednesday June 7.

Policies

The final grade will be based 70% on problem sets and 30% on a take-home final exam. You are encouraged to talk to your fellow students about the problem sets, but make sure that what you hand in is produced by you; the final exam is yours alone.

Problem sets will be handed out Wednesday, due the following Wednesday at 5:00pm in a box in Bridge Annex. Extensions are generally not encouraged, but if you really need one, make sure one of the TAs knows by 24 hours before the set is due.

Personnel

Sean Carroll, Professor. Office Hours: Fridays 10:30-noon, Lauritsen/Downs 401. Email seancarroll@gmail.com.

Thom Bohdanowicz, TA. Email thom@caltech.edu.

Ashmeet Singh, TA. Email ashmeet@caltech.edu.

Charles Xu, TA. Email cxu3@caltech.edu.

TA Office Hours: Tuesdays 7:00-8:30pm, Lauritsen/Downs 4th floor.

Tentative Outline (realistically, probably a subset of this)

  • Basics of Subsystems. Qubits, density matrices, Schmidt decomposition, entropy, mutual information.
  • Measurement. Von Neumann measurement scheme, generalized measurements, Positive Operator-Valued Measures, decoherence, pointer states, Bell's Theorem.
  • Foundations. Everett formulation, Born Rule, alternative approaches.
  • Quantum Information and Computation. No-cloning, teleportation, gates, circuits, complexity classes, Grover's algorithm.
  • Quantum Field Theory. Relativistic scalar fields, Lagrangians, Fock space, Reeh-Schlieder Theorem, Unruh effect, black hole evaporation and information puzzle.

Lecture Notes

These are the handwritten notes from which I lectured. Maybe someday I will write a book.

  1. Lecture One, 4/3: Intro, why entanglement matters, qubits.
  2. Lecture Two, 4/5: Two qubits, EPR entanglement, spooky action, density operators.
  3. Lecture Three, 4/10: Properties of density matrices, reduced density matrices, Schmidt decomposition.
  4. Lecture Four, 4/12: Bloch vectors, time evolution of density matrices, von Neumann entropy.
  5. Lecture Five, 4/17: Thermal density matrices, mutual information, von Neumann measurement formalism.
  6. Lecture Six, 4/21: Generalized measurements, Positive Operator-Valued Measures.
  7. Lecture Seven, 4/24: Quantum channels, superoperators, operator-sum expansion, Kraus operators.
  8. Lecture Eight, 4/26: Decoherence, loss of purity, irreversibility, elimination of quantum interference.
  9. Lecture Nine, 5/1: More decoherence, pointer states, classicality.
  10. Lecture Ten, 5/3: Foundations, Everett (Many-Worlds Interpretation).
  11. Lecture Eleven, 5/8: Structure and Probability in MWI.
  12. Lecture Twelve, 5/10: Decision-theoretic approach to the Born Rule, EPR, Bell's Theorem.
  13. Lecture Thirteen, 5/15: Bohmian Mechanics, GRW, QBism.
  14. Lecture Fourteen, 5/17: Intro to quantum gates and circuits, No-Cloning Theorem.
  15. Lecture Fifteen, 5/22: Quantum teleportation, quantum adder, Deutsch's algorithm.
  16. Lecture Sixteen, 5/24: Complexity classes, Grover's search algorithm.
  17. Lecture Seventeen, 5/31: Moving toward quantum field theory, special-relativity background.
  18. Lecture Eighteen, 6/5: Classical field theory, Lagrangians, gauge symmetries.
  19. Lecture Nineteen, 6/7: Quantum field theory, Fock space, particles.
  20. Lecture Twenty, 6/9: Reeh-Schlieder theorem, Unruh effect, Hawking radiation.

Problem Sets

Problem sets are handed out on Wednesdays, and can be handed in before 5:00pm the following Wednesday, in the 125c box in Bridge Annex.

  1. Problem Set One: handed out 4/12, due 4/19. Solutions.
  2. Problem Set Two: handed out 4/19, due 4/26. Solutions.
  3. Problem Set Three: handed out 4/26, due 5/3. Solutions.
  4. Problem Set Four: handed out 5/3, due 5/10. Solutions.
  5. Problem Set Five: handed out 5/10, due 5/17.
  6. Problem Set Six: handed out 5/17, due 5/24. Solutions.
  7. Problem Set Seven: handed out 5/24, due 5/31. Solutions.

References (optional)

We won't be following any particular text, and no books or articles are required reading, but these might be helpful for background. Asterisks indicate that a book is available electronically for Caltech students, through the Library.

Some additional articles on foundations of QM:

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