MAT 1723HF (APM 421H1F): MATHEMATICAL CONCEPTS
OF QUANTUM MECHANICS AND QUANTUM
INFORMATION
Wednesdays, 5-6 (BA2135) and Thursdays 4-6pm (BA 6183)
I. M. Sigal TA: Li Chen
November, 2016
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Goals
The goal of this course is to explain key concepts of Quantum Mechanics and to arrive quickly to some
topics which are at the forefront of active research. Examples of the latter topics are Bose-Einstein
condensation and quantum information, both of which have witnessed an explosion of research in the last
decade and both involve deep and beautiful mathematics.
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Prerequisites
I will cover all necessary definitions beyond multivariable calculus and linear algebra, but without familiarity with elementary ordinary and partial differential equations, the course will be tough. Knowledge
of elementary theory of functions and operators would be helpful. No physics background is required.
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Syllabus
- Schrödinger equation
- Quantum observables
- Spectrum and evolution
- Important special cases
- Angular momentum and group of rotations
- Spin and statistics
- Atoms and molecules
- Quasiclassical asymptotics
- Adiabatic theory and geometrical phases
- Hartree-Fock theory
- Feynman and Wiener path integrals
- Density matrix
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- Open systems
- Quantum entropy
- Quantum channels
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Texts
[GS] S. Gustafson and I.M. Sigal: Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer,
2011.
For the material which is not in the book, I will either post the lecture notes or refer to on-line
material.
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Tests and marking scheme
2 quizzes (Oct 12 and November 23), midterm test (November 2), final test (December 1).
Location and time: two quizzes and midterm BA2135, 5:10-6:00pm; final BA6183, 4:10-6:00pm
Quizzes and midterm and final exams will be on the material covered in the lectures. For the quizzes
and midterm exam, all the problems will be taken from homework and for the final test, most of the
problems.
Breakup of the grade:
Class participation (20%), two quizzes (10% + 20%), midterm (25%), final test (25%).
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Homework
The problem numbers listed below are from the book [GS], mentioned above.
Weeks of Sept 19 - Oct 24: Problems 2.3, 2.6, 2.17, 2.20, 3.2, 3.3, 6.2, 6.5, 6.7-6.9 and the following
problem given in the class:
• find the generators of translation, rotation and gauge groups, and show that under certain conditions
(which?) these groups are symmetry groups of the Schrödinger equation.
Weeks of Oct 31 - Nov 14: Problems 7.3, 7.4, 7.6, 4.1, 7.9-7.12, 8.8, 8.9, 17.1-17.3, 17.10, 23.58, 23.62
and the following problems given in the class:
• Show that in the spherical coordinates (r, θ, φ), where
x1 = r sin(θ) cos(φ),
x2 = r sin(θ) sin(φ),
x3 = r cos(θ),
0 ≤ θ < π, 0 ≤ φ < 2π, the Laplacian becomes
∆ = ∆r +
1
∆Ω
r2
where ∆r is the “radial Laplacian” given by
∆r =
∂2
2 ∂
+
∂r2
r ∂r
(∆r depends only on the radial variable), and ∆Ω is the Laplace-Beltrami operator on S2 , given in
spherical coordinates (θ, φ), by
∆Ω =
1 ∂
∂
1
∂2
(sin(θ) ) +
.
2
sin(θ) ∂θ
∂θ
sin (θ) ∂φ2
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• Show that the components, Lj , of the quantum-mechanical angular momentum operator L =
(L1 , L2 , L3 ) given by
L=x×p
where p = −i~∇ as usual, commute with its magnitude squared, L2 = L21 + L22 + L23 .
• Show that the operators L± := L1 ± iL2 satisfy (L± )∗ = L∓ , [L+ , L− ] = 2~L3 and [L± , L3 ] =
∓~L± .
• Show that if φk is an eigenfunction of L3 , with the eigenvalue k, i.e. L3 φk = ~kφk , then L3 L± φk =
~(k ± 1)L± φk .
• Show that in the spherical coordinates (r, θ, φ), L2 (f (r)g(θ, φ)) = −f (r)∆Ω g(θ, φ).
Weeks of Nov 21 and Nov 28: Problems 17.4-17.6, 17.11, 17.18
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