# ACM30210 Foundations of Quantum Theory

This module introduces Quantum Mechanics in its modern mathematical setting. Several canonical, exactly-solvable models are studied, including: a class of potential wells, the harmonic oscillator, angular momentum, and addition of angular momentum.

[Motivation] The postulates of Quantum Mechanics.

[Mathematical background] Complex vector spaces and scalar products, linear forms and duality, the natural scalar product derived from linear forms, Hilbert spaces, linear operators, commutation relations, expectation values, uncertainty.

[Time evolution and the Schrodinger equation] Derivation of the Schrodinger equation for time-independent Hamiltonians, the position and momentum representations, the probability current, the free particle.

[Angular momentum] Motivation: angular momentum in the hydrogen atom, as derived from spherical harmonics, angular momentum in the abstract setting, intrinsic angular momentum, addition of angular momenta, Clebsch-Gordan coefficients.

[Piecewise constant one-dimensional potentials] Bound and unbound states, wells and barriers, scattering, transmission coefficients, tunnelling.

[The harmonic oscillator] Creation and annihilation operators, coherent states.

Further topics may include:
Spin coherent states.
How to build a microwave laser.
One-dimensional Dirac potentials.
[Introduction to Quantum Information*] Qubits and quantum logic gates.

* If time permits.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module students should be able to:

1. Perform standard linear-algebra calculations as they relate to the mathematical foundations of Quantum Mechanics.

2. Solve standard problems for systems with finite-dimensional Hilbert spaces, e.g. the two-level system.

3. Solve standard one-dimensional models including the Harmonic oscillator.

4. Use of creation and annihilation operators, including the characterisation of coherent states.

5. Explain the quantum theory of angular momentum and compute expectation values for appropriate observables. These computations will involve both the matrix representation of intrinsic angular momentum, and the spherical-harmonic representation of orbital angular momentum.

6. Add independent angular momenta in the quantum-mechanical fashion.

7.* Understand the foundations of quantum logic.

* If time permits.

Student Effort Hours:
Student Effort Type Hours
Lectures

31

Specified Learning Activities

24

Autonomous Student Learning

45

Total

100

Approaches to Teaching and Learning:
Lectures and problem-based learning
Requirements, Exclusions and Recommendations
Learning Recommendations:

Students should have followed
ACM30010 Analytical Mechanics
or equivalent.

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Exam (In-person): End of semester exam n/a Standard conversion grade scale 40% No

70

Exam (In-person): In-class test n/a Standard conversion grade scale 40% No

10

Assignment(Including Essay): Take-home assignments n/a Standard conversion grade scale 40% No

20

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour