# Quantum Mechanics III (Physics 125c)

### Sean Carroll, Physics Department, Caltech

### Spring quarter, 2017

**Announcements**

- Problem Set 6 was slightly revised, to correct typos (5/19).

**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.

The final exam will take place at some point, presumably in the June 14-16 time frame.

**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@nullgmail.com.

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

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

Charles Xu, TA. Email cxu3@nullcaltech.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, Reeh-Schlieder Theorem, Bogoliubov transformations, Rindler space, Unruh effect, black hole evaporation and information puzzle.

**Lecture Notes **

I will try to post copies of my (handwritten) lecture notes along the way. Check back here.

**Lecture One**, 4/3: Intro, why entanglement matters, qubits.**Lecture Two**, 4/5: Two qubits, EPR entanglement, spooky action, density operators.**Lecture Three**, 4/10: Properties of density matrices, reduced density matrices, Schmidt decomposition.**Lecture Four**, 4/12: Bloch vectors, time evolution of density matrices, von Neumann entropy.**Lecture Five**, 4/17: Thermal density matrices, mutual information, von Neumann measurement formalism.**Lecture Six**, 4/21: Generalized measurements, Positive Operator-Valued Measures.**Lecture Seven**, 4/24: Quantum channels, superoperators, operator-sum expansion, Kraus operators.**Lecture Eight**, 4/26: Decoherence, loss of purity, irreversibility, elimination of quantum interference.**Lecture Nine**, 5/1: More decoherence, pointer states, classicality.**Lecture Ten**, 5/3: Foundations, Everett (Many-Worlds Interpretation).**Lecture Eleven**, 5/8: Structure and Probability in MWI.**Lecture Twelve**, 5/10: Decision-theoretic approach to the Born Rule, EPR, Bell’s Theorem.**Lecture Thirteen**, 5/15: Bohmian Mechanics, GRW, QBism.**Lecture Fourteen**, 5/17: Intro to quantum gates and circuits, No-Cloning Theorem.**Lecture Fifteen**, 5/22: Quantum teleportation, quantum adder, Deutsch’s algorithm.**Lecture Sixteen**, 5/24: Complexity classes, Grover’s search algorithm.

**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.

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

**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.

- B. Schumacher and M. Westmoreland,
*Quantum Processes, Systems, and Information** - M.A. Nielsen and I.L. Chuang,
*Quantum Computation and Quantum Information* - M. A. Schlosshauer,
*Decoherence and the Quantum-to-Classical Transition** - Y. Aharonov and D. Rohrlich,
*Quantum Paradoxes: Quantum Theory for the Perplexed** - D. Wallace,
*The Emergent Multiverse: Quantum Theory According to the Everett Interpretation** - J. Preskill, Lecture Notes on Quantum Computation
- J. Preskill, Lecture Notes on Quantum Field Theory
- Birrell and Davies,
*Quantum Fields in Curved Space*

Some additional articles on foundations of QM:

- Cotler et al. on emergent locality.
- David Deutsch’s original article on decision theory and the Born Rule.
- Zurek on envariance and the Born Rule.
- Sebens and Carroll on self-locating uncertainty and the Born Rule.
- A very brief introduction to GRW (collapse) theory.
- Dürr et al., an introduction to Bohmian mechanics.
- Caves, Fuchs and Schack on Quantum Bayesianism.
- Fuchs and Stacey on QBism.