## Mathematical Fundamentals of Quantum Mechanics

 Unit code: PHYS30201 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Physics and Astronomy Available as a free choice unit?: N

Prerequisite

#### Aims

To develop an understanding of quantum mechanics and in particular the mathematical structures underpinning it.

#### Overview

Mathematical Fundamentals of Quantum Mechanics (M)

#### Learning outcomes

On completion of the course, successful students should be able to:
1. Use Dirac notation to represent quantum-mechanical states and manipulate operators in terms of their matrix elements
2. Understand the mathematical underpinnings of quantum mechanics and solve a variety of problems with model and more realistic Hamiltonians
3. Demonstrate familiarity with angular momentum in quantum mechanics at both a qualitative and quantitative level
4.   Use perturbation theory and other methods to find approximate solutions to problems in quantum mechanics, including the fine-structure of energy levels of hydrogen

#### Assessment methods

• Written exam - 100%

#### Syllabus

1. Quantum mechanics and vector spaces   (9 lectures)
Review of vector spaces
Postulates of quantum mechanics
x and p operators and momentum-space wave functions
Time evolution: the SchrÂ¿dinger equation
Stern-Gerlach experiments
Example: Spin precession
Ehrenfest’s theorem and the classical limit
The simple harmonic oscillator: creation and annihilation operators
WKB approximation
Variational methods
Composite systems and entanglement

2. Angular Momentum   (7 lectures)
Angular momentum commutators
Eigenvalues and eigenstates of angular momentum
Orbital angular momentum vs spin
Pauli spin matrices
Example: Magnetic resonance
Clebsch-Gordan coefficients
The Wigner-Eckhart theorem

2. Time independent perturbation theory  (5 lectures)
Examples of perturbation theory
The fine structure of hydrogen
The Zeeman Effect: hydrogen in an external magnetic field
The Stark effect: hydrogen in an external electric field

3. Quantum measurement    (1 lecture)
The Einstein-Podolsky-Rosen “paradox” and Bell’s inequalities

Shankar, R. Principles of Quantum Mechanics 2nd ed. (Plenum 1994)
Gasiorowicz, S. Quantum Physics, 3rd ed. (Wiley, 2003)
Mandl, F. Quantum Mechanics (Wiley, 1992)
Griffths, D. J. Introduction to Quantum Mechanics, 2nd ed (CUP, 2017)

#### Feedback methods

Feedback will be available on students’ solutions to examples sheets through examples classes, and model answers will be issued.

#### Study hours

• Assessment written exam - 1.5 hours
• Lectures - 22 hours
• Independent study hours - 76.5 hours

#### Teaching staff

Judith McGovern - Unit coordinator