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MATH 4810/5810, *Hilbert Spaces and the Mathematics of Quantum (Information) Theory*,
August 2024

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Instructor:

Dr. Markus Pflaum
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Course Contents:

The course will provide an introduction to the theory of Hilbert spaces and their application in Quantum Mechanics.
On the mathematical side, the notions of a hermitian inner product, Hilbert space, bounded linear operator,
Hilbert basis and Fourier expansion, selfadjointness and the spectrum of a linear operator will be explained.
These concepts will then be applied to describe the axioms of Quantum Mechanics
and determine the quantum mechanical spectrum of certain Hamilton operators coming from Quantum Mechanics.
One of the major goals of the course is to understand and prove the so-called spectral theorem
which is crucial to understand Quantum Mechanics.

Syllabus

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Lecture Hours and Venue:

MTWThF 9:00 a.m. - 12:00 p.m., August 5 - 22, 2024, HUMN 1B90.

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Office Hours:

after class and by appointment

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Course Project:

Each student had to write a short paper (around 5 pages) on a
particular topic from Hilbert Space Theory and Quantum Mechanics Topology and give a 10min presentation on this in class.
The papers are due August 22, 2024. A selection of possible topics is provided here,
but you can propose your own project theme.
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Homework:

Homework assignments will be given in class and usually will be discussed the next day at the beginning of class.
Every student should present some homework problem.

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Course Literature:

The course will be based solely on textbooks which are freely available for CU students as eBooks through
CU Libraries or as online lecture notes under an appropriate open document license.

Hall, *Quantum Theory for Mathematicians*, Springer Verlag

Moretti, *Spectral Theory and Quantum Mechanics With an Introduction to the Algebraic Formulation*, Springer Verlag

Teschl, *Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators*, American Mathematical Society

Wilde, *Quantum Information Theory*, 2nd edition, Cambridge University Press, online version available under arXiv:1106.1445

Pflaum et al., The *FANCy* Project