Quantum Mechanics for Mathematicians:
Problem Set 1
Due Monday, September 24
Problem 1:
Consider the group S3 of permutations of 3 objects. This group acts on the
set of 3 elements. Consider the representation (, C3 ) this gives on the vector
space C3 of
Quantum Mechanics for Mathematicians:
Problem Set 6
Due Wednesday, December 12
Problem 1: Fill in the details of the proof of the Groenewold-van Hove theorem
following the outline given in Chapter 5.4 of Rolf Berndts An Introduction to
Symplectic Geometry
Quantum Mechanics for Mathematicians: The
group U (1) and Charge
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
September 17, 2012
In classical electrodynamics, the strength of interaction of an object with
the electromag
Quantum Mechanics for Mathematicians:
Problem Set 5
Due Wednesday, November 29
Problem 1: Consider the action of SU (2) on the tensor product V 1 V 1 of two
spin one-half representations. According to the Clebsch-Gordan decomposition,
this breaks up into
Quantum Mechanics for Mathematicians:
Problem Set 4
Due Wednesday, November 7
Problem 1: Using the denition
< f, g >=
1
2
2
f (z1 , z2 )g (z1 , z2 )e(|z1 |2+|z2 | ) dx1 dy1 dx2 dy2
C2
for an inner product on polynomials on homogeneous polynomials on C2
S
Quantum Mechanics for Mathematicians:
Problem Set 2
Due Monday, October 8
Problem 1: Calculate the exponential etM for
0
0
0
0
0
0
0
by two dierent methods:
Diagonalize the matrix M (i.e. write as P DP 1 , for D diagonal), then
show that
1
etP DP = P etD
Quantum Mechanics for Mathematicians:
Problem Set 3
Due Monday, October 22
Problem 1: On the Lie algebras g = su(2) and g = so(3) one can dene the
Killing form K (, ) by
(X, Y ) g g K (X, Y ) = tr(XY )
1. For both Lie algebras, show that this gives a bili
Quantum Mechanics for Mathematicians:
Two-state systems and spin 1/2
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
October 9, 2012
The simplest truly non-trivial quantum systems have state spaces that are
inherently two-
Quantum Mechanics for Mathematicians: Linear
Algebra Review, Unitary and Orthogonal Groups
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
September 24, 2012
A course in linear algebra is a prerequisite for this course, an
Quantum Mechanics for Mathematicians:
Representations of SU (2) and SO(3)
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
December 5, 2012
The group SO(3) acts by rotations on three-dimensional space, so we expect the stat
Quantum Mechanics for Mathematicians: Tensor
Products, Entanglement, and Addition of Spin
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
November 15, 2012
If one has two independent quantum systems, with state spaces H1 a
Quantum Mechanics for Mathematicians:
Introduction and Overview
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
September 24, 2012
1
Introduction
A famous quote from Richard Feynman goes I think it is safe to say that no
o
Quantum Mechanics for Mathematicians: The
Spin 1 Particle in a Magnetic Field
2
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
October 15, 2012
The existence of a non-trivial double-cover Spin(3) of the three-dimensional
Quantum Mechanics for Mathematicians: Lie
Algebras and Lie Algebra Representations
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
October 1, 2012
For a group G we have dened unitary representations (, V ) for nitedimensio
Quantum Mechanics for Mathematicians: The
Rotation and Spin Groups in 3 and 4 Dimensions
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
October 25, 2012
Among the basic symmetry groups of the physical world is the orthogo