Classical & Quantum Mechanics

Code

MATH2107

School

Mathematical Sciences

Level

2

Credits

20

Semesters

Spring UK

Summary

This course explores the classical and quantum mechanical description of motion. The laws of classical mechanics are investigated both in their original formulation due to Newton and in the mathematically equivalent but more powerful formulations due to Lagrange and Hamilton. 

The formalism of quantum mechanics is introduced via the postulates of quantum mechanics which are developed and applied for finite dimensional Hilbert spaces and for a point particles. 
Applications of the theory to various example problems are covered in classical and quantum mechanics. The course is the foundation for Mathematical Physics modules available at levels 3 and 4. 

Topics include:

• Newton’s laws of motion.
• Lagrange’s description of mechanics – the Lagrangian of a mechanical system, the Euler-Lagrange equations of motion.
• Hamilton’s description of mechanics – transforming from the Lagrangian to the Hamiltonion description of a mechanical system, standard concepts such as phase space and Poisson brackets.
• Basic formalism of quantum mechanics, including properties of state vectors, inner products, Hermitian operators, unitary transformations, the probabilistic interpretation of quantum states and their wave functions (Born’s rule), the Schrödinger equation, Heisenberg’s uncertainty principle and energy eigenstates.
• Quantum mechanical problems, including applications to quantum information in a finite dimensional Hilbert space and the quantum harmonic oscillator. 

Target Students

Single Honours and Joint Honours students from the School of Mathematical Sciences, Mathematical Physics students, Natural Sciences students.

Co-requisites

Modules you must take in the same academic year, or have taken in a previous year, to enrol in this module:

• Vector Calculus and Electromagnetism (MATH2106)

Classes

• Eleven 1-hour lectures each week for 2 weeks
• Eleven 2-hour lectures each week for 2 weeks

Teaching will be through a variety of methods, with delivery tailored to the material on a week-by-week basis.

Assessment

• 40% Coursework 1: Summative assessment based on tasks distributed through the semester.
• 60% Exam 1 (2-hour): Written examination – Spring

Assessed by end of spring semester

Educational Aims

Learning Outcomes

A student who completes this course successfully will be able to:
L1 – Demonstrate knowledge and understanding of the underlying mathematical and physical concepts of classical and quantum mechanics.

L2 – Select and apply suitable mathematical concepts and techniques to solve routine and novel problems in classical and quantum mechanics.

L3 – Develop and justify a mathematical framework, reasoning logically and analytically, to solve problems in classical and quantum mechanics with a high level of accuracy and rigour.

L4 – Summarise complex ideas clearly and concisely, drawing connections between concepts in different descriptions of mechanics and between concepts in other areas of mathematics and transfer their knowledge accordingly.

L5 – Present conclusions, verbally or in writing, using structured and mathematically rigorous arguments and contextually appropriate language.

Conveners

• Dr Sven Gnutzmann
• Dr Tommaso Tufarelli