Foundations of Quantum Mechanics
01:556:325, Fall 2011

Instructor: Roderich Tumulka
Phone: 732-445-2390 x 1322
Office: Hill 522
Email: instructor's last name, then @, then math.rutgers.edu

Class meets: MTh 10:20-11:40am, ARC 333


Lecture Notes

PDF file

Syllabus

Lecture Topics Reading assignment due
1 The Schrödinger equation, configuration space, Laplace operator.
2 The Born rule, the continuity equation, probability density. Hilbert space.
3 Unitary operators, rotation matrices. The double-slit experiment, interference. Classical mechanics. R. Feynman: Lecture on the double-slit experiment (1963)
4 Bohmian mechanics. Determinism. Reductionism.
5 Transport of probability, equivariance. Conditional wave function, decoherence. S. Goldstein: Bohmian mechanics (2001)
6 Fourier transformation. Momentum measurements, the Born rule for momentum, momentum operators, non-commuting operators, the Heisenberg uncertainty relation.
7 Delayed-choice experiments. Does quantum mechanics imply retro-causation? Observables. The Born rule for arbitrary observables. Linear operators, matrices, projection operators, self-adjoint operators. Orthonormal basis. Eigenvalue and eigenvector, the spectral theorem. J. Bell: De Broglie-Bohm, delayed-choice double-slit experiment... (1980)
8 Spin, the Stern-Gerlach experiment, the Pauli equation, paradoxical two-valuedness of spin. The Pauli matrices, tensor product spaces, entanglement. The projection postulate, wave function collapse.
9 The measurement problem of quantum mechanics. Schrödinger's cat. Positivism and realism. T. Maudlin: Three Measurement Problems (1995)
10 The Ghirardi-Rimini-Weber (GRW) theory of wave function collapse. The Poisson process.
11 GRW theory, primitive ontology, empirical tests. J. Bell: Are there quantum jumps? (1987)
12 Complementarity. The Copenhagen interpretation of quantum mechanics.
13 Problems of the Copenhagen interpretation. The Bohr-Einstein debate. A. Einstein: Reply to Criticisms (1949)
14 Catch-up and review.
15 Midterm exam, 10/24/2011.
16 Schrödinger's many-worlds theory, Everett's many-worlds theory.
17 Many-worlds theories. J. Bell: Six possible worlds of quantum mechanics (1986)
18 Predictions of Bohmian mechanics, GRW theory, many-worlds theories.
19 The Einstein-Podolsky-Rosen paradox. A. Einstein, B. Podolsky, N. Rosen: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? (1935)
20 Nonlocality, Bell's inequality, its relation to the EPR argument.
21 Alternative proofs of nonlocality. Nonlocality in Bohmian mechanics, GRW theory, many-worlds theories. J. Bell: Bertlmann's Socks and the Nature of Reality (1987)
22 Misconceptions about nonlocality.
23 Bohr's reply to EPR. Generalized observables (POVMs). The main theorem about POVMs. Limitations to knowledge. N. Bohr: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? (1935)
24 Density matrix and mixed quantum state. Trace of an operator, positive operator.
25 No-hidden-variables theorems. J. Bell: On the problem of hidden variables in quantum mechanics (1966)
26 Reduced density matrix. Partial trace. Decoherence.
27 Density matrix and the measurement problem. J. Bell: Against "Measurement" (1990)
28 Catch up and review.

Course Description

This course is taught every other year.

Prerequisites: CALC4 or permission of instructor
(As far as mathematics is concerned, students need a working knowledge of complex numbers, eigenvalues of matrices, and partial derivatives. Prior knowledge in physics is not required but helpful.)

Contents: This interdisciplinary course, intended primarily for juniors and seniors majoring in mathematics, physics, or philosophy, deals with what can be concluded from quantum mechanics about the nature of reality. It has been claimed that quantum mechanics entails most radical consequences about the world and our knowledge of it, such as the existence of parallel universes, faster-than-light action-at-a-distance, limitations to what we can know, that reality itself be paradoxical, or that electrons become real only when observed. On the other hand, it has been claimed that quantum mechanics, in its orthodox formulation, be "unprofessional" (J. Bell), "incoherent" (A. Einstein), "incomprehensible" (R. Feynman), and "insane" (E. Schrödinger). We will investigate these claims, their basis and merits. The course will involve advanced mathematics, as appropriate for a serious discussion of quantum mechanics, but will not focus on technical methods of problem-solving.

Topics will include most of the following: The Schrödinger equation, the Born rule, self-adjoint matrices, axioms of the quantum formalism, the double-slit experiment, non-locality, the paradox of Schrödinger's cat, the quantum measurement problem, Heisenberg's uncertainty relation, interpretations of quantum mechanics (Copenhagen, Bohm's trajectories, Everett's many worlds, spontaneous collapse theories, quantum logic, perhaps others), views of Bohr and Einstein, no-hidden-variables theorems, and identical particles.

Text: J. Bell: Speakable and unspeakable in quantum mechanics, Cambridge University Press.

Learning goals: To understand the rules of quantum mechanics; to understand several important views of how the quantum world works; to be familiar with the surprising phenomena and paradoxes of quantum mechanics.


Maintained by R. Tumulka and last modified 19 April 2012