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# Quantum Mechanics - PHY00072H

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• Department: Physics
• Credit value: 20 credits
• Credit level: H
• Academic year of delivery: 2024-25
• See module specification for other years: 2023-24

## Module summary

This course introduces advanced topics and techniques in quantum mechanics, followed by relativistic quantum mechanics, which will provide initial understanding of the principles and concepts in high energy physics through study of pertinent aspects of special relativity applied to quantum mechanics (i.e. relativistic wave equations, but not QFT).

## Related modules

Pre-requisites:  Mathematics, Professional Skills and Laboratories and Stage 2 Quantum, Nuclear and Particle Physics, or the appropriate equivalent modules.

## Module will run

Occurrence Teaching period
A Semester 1 2024-25

## Module aims

The specific module aims follow.

For the Quantum Mechanics component, to:

• introduce quantum mechanical commutators and their significance for the compatibility of measurements.

• introduce the quantum mechanical treatment of angular momentum and demonstrate co-ordinate transformation from Cartesian to spherical polar co-ordinates. Apply this treatment to the angular momentum operators and time-independent Schrödinger equation and find solutions of the time-independent Schrödinger equation for a spherically symmetrical potential with applications to the hydrogen atom.

• introduce matrix mechanics, with particular application to spin – spin operators, Pauli spin matrices, Dirac notation and discuss the theory of measurement as illustrated by the Stern-Gerlach measurement of spin.

• develop approximate methods for solving the Schrödinger equation when no analytic solutions exist, such as time-independent perturbation theory and the variational principle.

In terms of relativistic quantum mechanics, this module will consider relativistic generalisations to key equations of motion for both bosonic and fermionic particles, with continued emphasis on angular momentum. In particular, it will provide introductory material on:

• Spin-0 relativistic particles, Klein–Gordon equation.

• Relativistic kinematics.

• Pauli equation (non-relativistic spin-1/2 particles) and the Dirac equation (relativistic spin-1/2 particles).

• Spin-1 relativistic particles (massive and massless), photons and W/Z bosons, and

• A glimpse at the Higgs mechanism.

## Module learning outcomes

Quantum Mechanics (QM):

• Understand the physical significance of commutators in terms of compatibility of measurements and perform simple commutator algebra

• Derive operators for the angular momentum components Lx, Ly, Lz, and for L2 in Cartesian and spherical coordinates and use these operators to solve their respective eigenproblems.

• Explain the use of the central force theorem for a spherically symmetric potential within QM and apply this to solve the full analytical eigensolution for the case of the hydrogen atom, thereby providing a physical interpretation of the quantum numbers n, l and ml pertaining to these solutions.

• Apply the matrix formalism of quantum mechanics to solve the spin eigenproblem using the Pauli spin matrices as well as demonstrate applications of this formalism to the generalised Stern-Gerlach experiments.

• Derive the first- and second-order eigenenergy corrections in non-degenerate perturbation theory and apply the formulae to simple problems, e.g. anharmonic oscillators, as well as learn and apply other approximate methods, such as the matrix eigenvalue formalism for degenerate perturbation theory (e.g., the Stark effect in hydrogen), and the variational method.

Relativistic Quantum Mechanics:

• Explain the general procedure of quantising equations of motion in special relativity

• Apply the principles of Lorentz transformation properties in 4-vector formalism to problems involving any spin statistics properties

• Manipulate expressions with commutators and anticommutators of operators

Relate physical phenomena including spin, the gyromagnetic ratio of the electron and the fine structure of the hydrogen atom to key aspects of relativistic quantum mechanics

## Module content

Syllabus

Quantum mechanics:

• An introduction to quantum mechanical commutators and their significance for the compatibility of measurements.

• An introduction to the quantum mechanical treatment of angular momentum.

• The time-independent Schrödinger equation for a spherically symmetrical potential, and application of this via separation of variables to obtain eigensolutions for hydrogen and hydrogen-like atoms.

• Extension of quantum mechanics to incorporate spin.

• Introduction to matrix mechanics, with particular application to spin.

• Discussion of the theory of measurement as illustrated by the Stern-Gerlach measurement of spin.

• Approximate methods for solving the Schrödinger equation when no analytic solutions exist (time-independent); non-degenerate perturbation theory, matrix eigenvalue formalism for degenerate perturbation theory, the variational approach.

Relativistic Quantum Mechanics:

• Vectors: definition of a vector; transformation properties, covariance and contravariance in general, index notation, Einstein convention, 4-vectors, Kronecker delta, Levi-Civita symbol.

• Special relativity: the invariant interval, Lorentz transformations, Lorentz-covariant and -contravariant objects, tensors, vector derivatives, energy-momentum 4-vector.

• Relativistic kinematics: basic use of invariants in 4-vector form, symmetries.

• Electromagnetism (EM): EM in covariant form using 4-potentials, minimal coupling.

• Non-relativistic QM: time-dependent Schrödinger equation and the probability density, probability current and continuity, minimal coupling in QM.

• Klein–Gordon equation statics: quantization of the relativistic E(p) relationship, scalar fields and connection to spin-0 particles, derivation of Yukawa potential.

• Klein–Gordon waves: negative energy solutions and their necessity for a general solution, antiparticles and crossing symmetry.

• Massless spin-1 particles: gauge conditions, photons must be massless and spin-1.

• Massive spin-1 particles: Proca’s equation, forced Lorenz condition (no gauge invariance), illustration of the Higgs mechanism (superconductivity).

• Non-relativistic spin-1/2 particles, the Pauli equation, generators of rotation, eigenstates and eigenvalues of orbital angular momentum, spin, i.e., half-integer eigenvalues of angular momentum and its consequences, magnetic moments and spin.

• Relativistic spin-1/2 particles: Dirac matrix properties, E(p) relationship and the norm of Dirac spinors, representation of spin, non-relativistic limit (Pauli retrieved), prediction of g=2, spin in Dirac is indeed angular momentum.

• Introductory concepts in gauge theories.

## Indicative assessment

Closed/in-person Exam (Centrally scheduled) 80
Essay/coursework 20

Other

### Indicative reassessment

Closed/in-person Exam (Centrally scheduled) 80

## Module feedback

'Feedback’ at a university level can be understood as any part of the learning process which is designed to guide your progress through your degree programme. We aim to help you reflect on your own learning and help you feel more clear about your progress through clarifying what is expected of you in both formative and summative assessments.

A comprehensive guide to feedback and to forms of feedback is available in the Guide to Assessment Standards, Marking and Feedback. This can be found at:

https://www.york.ac.uk/students/studying/assessment-and-examination/guide-to-assessment/

The School of Physics, Engineering & Technology aims to provide some form of feedback on all formative and summative assessments that are carried out during the degree programme. In general, feedback on any written work/assignments undertaken will be sufficient so as to indicate the nature of the changes needed in order to improve the work. Students are provided with their examination results within 25 working days of the end of any given examination period. The School will also endeavour to return all coursework feedback within 25 working days of the submission deadline. The School would normally expect to adhere to the times given, however, it is possible that exceptional circumstances may delay feedback. The School will endeavour to keep such delays to a minimum. Please note that any marks released are subject to ratification by the Board of Examiners and Senate. Meetings at the start/end of each semester provide you with an opportunity to discuss and reflect with your supervisor on your overall performance to date.

Our policy on how you receive feedback for formative and summative purposes is contained in our Physics at York Taught Student Handbook.

A I M Rae: Quantum mechanics (McGraw-Hill)

R C Greenhow: Introductory quantum mechanics (Taylor & Francis/IoP Publishing)

B H Bransden and C J Joachain: Introduction to quantum mechanics (Prentice Hall)

I J R Aitchison: Relativistic Quantum Mechanics (some easy parts thereof)

I J R Aitchison and A J G Hey: Gauge Theories in Particle Physics (some easy parts thereof)

D H Perkins, Introduction to High Energy Physics (some easy parts thereof

The information on this page is indicative of the module that is currently on offer. The University constantly explores ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.