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# Classical & Quantum Waves - PHY00028C

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• Department: Physics
• Module co-ordinator: Dr. Paul Davies
• Credit value: 20 credits
• Credit level: C
• Academic year of delivery: 2024-25
• See module specification for other years: 2023-24

## Module summary

From mechanical oscillations to quantum mechanics, wave equations are a key component in describing the behaviour of many different physical systems. In this module you will learn the key mathematical models required to describe wave-like behaviours. You will start by looking at mechanical systems and conclude with an introduction to quantum physics. Throughout the module the similarities and differences between different wave equations will be explored. This will help develop an intuitive understanding of waves, allowing you to relate more abstract problems in quantum physics back to well understood physical systems.

## Module will run

Occurrence Teaching period
A Semester 1 2024-25

## Module aims

From mechanical oscillations to quantum mechanics, wave equations are a key component in describing the behaviour of many different physical systems. In this module you will learn the key mathematical models required to describe wave-like behaviours. You will start by looking at mechanical systems and conclude with an introduction to quantum physics. Throughout the module the similarities and differences between different wave equations will be explored. This will help develop an intuitive understanding of waves, allowing you to relate more abstract problems in quantum physics back to well understood physical systems.

## Module learning outcomes

• Understand the mathematical formulation of waves in terms of their phase and amplitude using both complex-exponential and trigonometric forms.

• Construct mathematical models that describe effects such as interference, diffraction, refraction and resonance.

• Apply knowledge of waves to real world systems such as oscillators and optical systems.

• Describe examples of the shortcomings of classical wave theory, and explain how quantum waves differ from their classical analogues.

• Understand how a wave theory of particles leads to different behaviour from their classical analogue.

• Solve simple one dimensional, time-independent quantum mechanics problems using the Schrödinger equation.

## Module content

Periodic motion: Simple Harmonic Motion; Unforced Simple Harmonic Motion (SHM); damped forced Linear Harmonic Oscillator; Resonance applied to mechanical and electrical systems; Coupled oscillators and normal modes, phase

Mechanical waves: trigonometric and complex-exponential forms for representing waves; wave number; angular frequency; derivation of the wave equation; travelling waves on a string; harmonics; Travelling waves along a string under tension and the creation of harmonics; Superposition; The Doppler effect; Beats; Dispersion.

Optics: Light as waves, rays and photons; reflection from plane and curved surfaces; refraction of light through a thin lens; the Lens-maker’s equation; theoretical resolving power, Rayleigh’s criterion and focal ratio; Spectacles and Porro prisms; dispersion; polarization including Maul’s law; phase; coherence and optical path difference; diffraction; interference, thin films and Newton’s rings.

Wave-particle Duality: Failure of classical physics; Particle properties of electromagnetic radiation; Wave properties of particles; Experimental evidence for wave particle duality (e.g. Photoelectric effect, Compton scattering, Davisson Germer); de Broglie’s postulate; Heissenberg’s Uncertainty Principle.

The Bohr model: Stability of the atom; experimental evidence for quantisation in atoms (e.g. spectral lines and Franck Hertz experiment); Bohr’s postulates, Bohr’s model, Correction for finite nuclear mass, Correction for nuclear charge.

Introduction to Quantum Mechanics: Postulates of quantum mechanics, The Schrödinger equation, Wavefunction of a free particle; Born's probability interpretation of the wavefunction, probability concepts in classical and quantum mechanics.

Solving the Time-independent Schrödinger equation (TISE) for a static potential: stationary states; ‘boundary conditions to be satisfied by physically acceptable solutions of TISE: single-valuedness; normalisability and continuity; particle in a one-dimensional potential well; energy eigenvalues and normalised eigenfunctions; orthogonality; orthonormalisation; the Kronecker delta; classically inaccessible regions; reflection and transmission at boundaries; quantum-mechanical tunnelling; particle flux; probability density.

Hermitian operators: observables and their operators; the Hamiltonian operator; position and momentum operators; eigenvalues and eigenfunctions; expectation values.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Classical & Quantum Waves exam
3 hours 80
Essay/coursework
Physics Practice Questions
N/A 20

Other

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Classical & Quantum Waves exam
3 hours 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

H D Young and R A Freedman: University Physics with Modern Physics ***

K S Krane: Modern Physics **

A I M Rae: Quantum Mechanics **

R Eisberg and R Resnick: Quantum Physics **

Some additional detail is required beyond University Physics, this is provided by one of the texts indicated by a double asterisk

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring 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 by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.