# ADVANCED QUANTUM MECHANICS - 2024/5

Module code: PHYM072

## Module Overview

Quantum Mechanics topics are essential building blocks for our understanding of many physical systems. The module assumes basic knowledge in quantum mechanics from modules in earlier years, but will provide a review at the beginning. Topics include a review of quantum mechanics, operator methods and applications to the harmonic oscillator, spin & angular momentum, symmetries in quantum mechanics.

The second half of the module then moves beyond isolated quantum systems by addressing the topic of Quantum Entanglement and Quantum Coherence. Here the important concepts of entanglement and decoherence will be studied by developing an understanding of open quantum systems and the density matrix formalism.

### Module provider

Mathematics & Physics

GINOSSAR Eran (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 70

Lecture Hours: 20

Tutorial Hours: 10

Guided Learning: 30

Captured Content: 20

Semester 2

None

## Module content

Indicative content includes

• Review of Quantum Mechanics with Dirac Notation: We will revisiting the principles and mathematical formalism of quantum mechanics. Students will review key concepts such as wavefunctions, operators, and observables, and explore how they are represented in Dirac notation.

• The Ladder Operator Method: The ladder operator method is a powerful technique used to study quantum systems with discrete energy levels, such as harmonic oscillators and angular momentum systems. Students will learn how ladder operators enable the calculation of energy eigenstates, energy spectra, and transition probabilities between states.

• Symmetries in Classical and Quantum Mechanics: Students will examine how symmetries manifest in physical systems and how they are described mathematically. They will learn about symmetries such as translation, rotation, and time reversal, and understand their implications for conservation laws and the behavior of quantum systems.

• Angular Momentum and Spin: Angular momentum is a fundamental property in quantum mechanics that arises from the rotational symmetry of physical systems. Students will explore the quantization of angular momentum and its role in describing the behavior of particles with intrinsic spin, such as electrons. They will learn about the commutation relations and eigenstates associated with angular momentum operators, and how they are connected to observable quantities.

• Introduction to Quantum Coherence. There will be a general introduction and revision of the idea of orthonormal bases and complete sets of eigenstates, the idea of spin, spin operators and eigenstates and the Block sphere.

• Density matrix. Next, the students move on to the idea of entanglement and Bell states and what it means for such states to decohere when they interact with their environment. It is explained that the best way to deal with this is through the use of the so-called density matrix instead of the quantum state as ket (or a wave function). Various derivations and exercises will be covered in the lectures, such as how to take the trace of the density matrix, its relation to expectation values of observables in QM and the derivation of the reduced density matrix.

• The Measurement Problem and decoherence. The famous measurement problem is central in the emerging area of quantum computing and we examine it through the modern lens of the density matrix approach and open quantum systems. Quantum decoherence is examined carefully and contrasted with the classical notions of dissipation and noise.

• Master equations. Students are introduced to different types, such as those using the Born-Markov approximation, the Lindblad form, the quantum Brownian motion model of Caldeira and Leggett and finally non-Markovian approached.

## Assessment pattern

Assessment type Unit of assessment Weighting
Coursework Quantum Mechanics Coursework Assignment 50
Examination End of semester examination (1.5 hour) on quantum coherence 50

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate individual knowledge, skill, and problem-solving abilities.

Thus, the summative assessment for this module consists of

• A Quantum Mechanics coursework assignment. The assignment will consist of a set of problems, representing each of the topics. The problems will require detailed written solutions by which the students will demonstrate their understanding of the material and their problem solving abilities.

• A 1.5-hour end-of-semester examination on the quantum coherence material.

Formative assessment: Formatively assessed problem sheets are issued during the course, and feedback will be given during tutorial sessions to help students prepare for the summative coursework sheets.

Feedback: Verbal feedback is provided by the lecturer during the tutorials (e.g., when exercises are worked out)

## Module aims

• The module aims to develop a deeper understanding of the consequences of the postulates of quantum mechanics, using Dirac notation, operator methods the role of symmetry
• To apply the principles learned to problems that can be solved analytically
• To introduce the concept of quantum entanglement as a general and fundamental feature of the quantum world
• To give a general introduction to open quantum systems and master equations
• To touch on foundational problems in QM such as the notion of measurement and observers

## Learning outcomes

 Attributes Developed 001 Recall the postulates of quantum mechanics, and apply them to simple two level systems PT 002 Be able to use operators and commutation relations in analysing the simple harmonic oscillator, angular momentum, and spin CK 003 Be able to explain how symmetries are used in quantum mechanics C 004 The appreciate the importance of the role of density matrices, entanglement, pure and mixed states and decoherence as it applies in, for example, the theory of quantum computing CK 005 To relate the idea of pure and mixed density matrices to correlated (entangled) and uncorrelated (product) states CK 006 To gain a modern perspective and understanding of the problem of quantum measurement that goes beyond the old textbook notions of collapse of the wave function through the irreversible act of observation. CK

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Enable students to understand the physics concepts involved in Quantum Mechanics, how to use mathematical tools to find analytical solutions

• Equip students with the knowledge to use operator methods to analyse the simple harmonic oscillator, and angular momentum

• Enable students to represent operators as matrices and use standard matrix methods, for example to compute the eigenvalues and expectation values of operators

• Introduce students to the theoretical ideas behind concepts such as quantum entanglement and decoherence

Thus, the learning and teaching methods include

• Interactive lectures backed up with guided study to stimulate uptake of subject knowledge

• Tutorials where problem sets will be issued throughout the course to give practice at problem-solving in quantum mechanics and peer learning and teaching with structured SurreyLearn hosted discussion boards focused on topics in quantum mechanics

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: PHYM072

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Employability, Digital Capabilities, Global and Cultural Capabilities, Sustainability, and Resourcefulness and Resilience. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: In the coherence and entanglement part of this module, students are introduced to the quantum concepts and mathematical formalism necessary to be able to understand the digital capabilities of quantum computers.

Resourcefulness and Resilience: By working independently through the problems set in the tutorials/ problem classes and then checking their solutions against the model answers provided and their peers, students will develop the confidence and resilience to tackle and solve advanced problems in the subject of quantum mechanics.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Physics with Nuclear Astrophysics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Astronomy MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Quantum Computing MPhys 2 Compulsory A weighted aggregate mark of 50% is required to pass the module
Physics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.