Quantum Mechanics I - MAT00024H

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• Department: Mathematics
• Module co-ordinator: Prof. Atsushi Higuchi
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2019-20

Related modules

• None

Prohibited combinations

• None

Pre-Requisites for Natural Sciences Students:

• MAT00036I Applied Mathematics Option I
• MAT00019I Mathematics for the Sciences II

Module will run

Occurrence Teaching cycle
A Autumn Term 2019-20

Module aims

This module aims to deepen students’ understanding of quantum mechanics, which they started learning at the Second Stage. In particular, it shows how classical mechanics can be understood as a limit of quantum mechanics and introduces students to the mathematical foundation of quantum mechanics.

Module learning outcomes

• Understand the wave-mechanics description of quantum mechanics and its classical limit.

• Understand the abstract operator formalism of quantum mechanics and its application to simple harmonic oscillator and angular momentum.

• Appreciate features of quantum mechanics distinguishing it from classical mechanics, such as tunnelling and Heisenberg’s uncertainty relation.

Module content

[Pre-requisite modules for Natural Science students: Applied Maths Option 1 (MAT00036I), and Maths for the Sciences 3 (MAT00019I).]

Syllabus

• The time-dependent Schrödinger equation: the general solution in terms of the energy eigenstates; continuity equation for the probability.

• The space of wave functions: the position, momentum and energy as Hermitian operators; commutation relations; the Fourier transform of the wave function as the momentum representation; measurement postulates for energy, position and momentum; Heisenberg’s uncertainty relation between the position and momentum without proof.
• Free quantum particle on a line: momentum eigenstates; the propagator and the evolution of the Gaussian wave packet.

• Ehrenfest’s theorem and the classical limit.
• Scattering problem in one dimension; discussion of tunnelling..

• Dirac's bra-ket notation; the simple harmonic oscillator with ladder operators.

• The angular momentum: the representation of the angular-momentum algebra; spherical harmonics in the context of angular-momentum representation.

• Academic skills: students will learn a fundamental theory describing the physical world through combining their mathematical skills learned in earlier Stages.

• Graduate skills: through lectures, problems classes and seminars, students will develop their ability to assimilate, process and engage with new material quickly and efficiently.They develop problem-solving skills and learn how to apply techniques to unseen problems.

Assessment

Task Length % of module mark
University - closed examination
Quantum Mechanics I
2 hours 100

None

Reassessment

Task Length % of module mark
University - closed examination
Quantum Mechanics I
2 hours 100

Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.