Foundations of Quantum Mechanics
01:640:325
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, nohidden-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.
Syllabus
Lecture
Topics
1
The Schrödinger equation, configuration space,
Laplace operator.
2
The Born rule, the continuity equation, probability
density. Hilbert space.
Reading assignment due
3
Unitary operators, rotation matrices. The double-slit
experiment, interference. Classical mechanics.
4
Bohmian mechanics. Determinism. Reductionism.
5
Transport of probability, equivariance. Conditional
wave function, decoherence.
6
Fourier transformation. Momentum measurements, the
Born rule for momentum, momentum operators, noncommuting 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.
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.
10
The Ghirardi-Rimini-Weber (GRW) theory of wave
function collapse. The Poisson process.
11
GRW theory, primitive ontology, empirical tests.
12
Complementarity. The Copenhagen interpretation of
quantum mechanics.
13
Problems of the Copenhagen interpretation. The Bohr- A. Einstein: Reply to Criticisms
Einstein debate.
(1949)
14
Catch-up and review.
15
Midterm exam, 10/24/2011.
16
Schrödinger's many-worlds theory, Everett's manyworlds theory.
17
Many-worlds theories.
18
Predictions of Bohmian mechanics, GRW theory,
many-worlds theories.
19
The Einstein-Podolsky-Rosen paradox.
R. Feynman: Lecture on the
double-slit experiment (1963)
S. Goldstein: Bohmian
mechanics (2001)
J. Bell: De Broglie-Bohm,
delayed-choice double-slit
experiment... (1980)
T. Maudlin: Three
Measurement Problems (1995)
J. Bell: Are there quantum
jumps? (1987)
J. Bell: Six possible worlds of
quantum mechanics (1986)
A. Einstein, B. Podolsky, N.
Rosen: Can QuantumMechanical 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.
22
Misconceptions about nonlocality.
23
Bohr's reply to EPR. Generalized observables
(POVMs). The main theorem about POVMs.
Limitations to knowledge.
24
Density matrix and mixed quantum state. Trace of an
operator, positive operator.
25
No-hidden-variables theorems.
26
Reduced density matrix. Partial trace. Decoherence.
27
Density matrix and the measurement problem.
28
Catch up and review.
Maintained by R. Tumulka and last modified 19 April 2012
J. Bell: Bertlmann's Socks and
the Nature of Reality (1987)
N. Bohr: Can QuantumMechanical Description of
Physical Reality Be Considered
Complete? (1935)
J. Bell: On the problem of
hidden variables in quantum
mechanics (1966)
J. Bell: Against "Measurement"
(1990)