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# Introduction to Thermal & Quantum Physics - PHY00022C

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• Department: Physics
• Module co-ordinator: Dr. Laurence Wilson
• Credit value: 20 credits
• Credit level: C
• Academic year of delivery: 2022-23

• None

• None

## Module will run

Occurrence Teaching cycle
A Autumn Term 2022-23 to Spring Term 2022-23

## Module aims

To explore first the macroscopic and then the microscopic contents of heat, temperature and work. This will include the study of heat transfer by radiation, convection and conduction; the zeroth and first Laws of thermodynamics including cyclic transformations and adiabatic systems. Kinetic theory will be used to account for the bulk properties and behaviour of gases in terms of the motions of their constituent molecules.

To study fluid mechanics from a purely macroscopic viewpoint and at a level using only A-level mathematics background.

To determine the basic components in the quantitative classification of the structure of crystalline solids.

To consider the interatomic forces in solids and to classify solids in terms of bonding.

To introduce students to the need for concepts beyond classical mechanics when dealing with phenomena on the atomic and sub-atomic scales. This will be done via discussion of a few key experiments performed at the beginning of the 20th century and the discussion of evidence for quantisation in atoms. The module will discuss why classical approaches do not work for such small systems and will culminate with an introduction to the basics of the quantum mechanical approach and applications in one dimension.

## Module learning outcomes

Heat and Kinetic Theory:

• Be familiar with the model system of an ideal gas and the principles of kinetic theory.
• Understand the physical concepts of radiation, convection and conduction
• Apply knowledge of thermal transport and phase changes to calculate heat flow due to radiation, convection, thermal conduction
• Understand the components of the 1st Law of Thermodynamics and apply it to simple systems such as cyclic transformations.
• Use kinetic theory to derive expressions for the behaviour of gases such as the pressure of a gas in a container, kinetic energies, root-mean- square speeds of gas molecules, the mean-free path of a molecule and its collision cross-section.
• Be familiar with the principles of equipartition of energy and use these ideas to calculate the specific heat capacities of monatomic and polyatomic gases.
• Understand the concept of a distribution function and apply this to the speeds of molecules

Fluids:

• Use Archimedes Principle and pressure variation in fluids in elementary hydrostatic problems.
• Apply Bernoulli's equation to simple applications, e.g. the Venturi metre.

The Solid State:

• Describe crystal structures in terms of the space lattice + basis and be familiar with cubic and hexagonal crystal structures.
• Determine the nearest neighbour distance and packing fraction for the cubic systems.
• Describe the different types of defects and dislocations that result from disorder in a crystal.
• Understand the origin of the interatomic forces in solids, and the distinction between the different bonding types: ionic, covalent, molecular (van der Waals), metallic and hydrogen.

Quantum Physics:

• Describe experimental evidence to demonstrate the particle properties of electromagnetic radiation (photoelectric effect and Compton’s experiments) and the wave properties of particles (Davisson and Germer experiment).
• Derive the Compton scattering formula and know the photoelectric effect formula and be able to use these to solve problems
• State, explain and utilise the energy and momentum forms of the Uncertainty Principle.
• Explain what is meant by isotopes, isotones and isobars in nuclei and be able to identify examples of these.
• Describe the Rutherford scattering experiment and perform simple calculations for the distance of closest approach between nuclei in a head-on collision.
• Explain the origin of atomic emission and absorption spectra.
• Discuss evidence for quantisation in atoms (eg line spectra, Franck-Hertz experiment).
• Explain the origins of the various series of lines in the hydrogen atom
• Describe the Bohr model and the Bohr postulates.
• Derive the Rydberg equation using the Bohr model and use it to determine the wavelength of transitions between levels of an atom.
• Use the Bohr postulates to calculate the radius and energy of levels in hydrogen –like atoms.
• Explain the consequence of the finite nuclear mass on the Bohr model predictions and perform simple calculations using the modified Rydberg constant.
• Discuss the problems of the Bohr model in relation to the stability of the atom and the problems of applying the model to multi-electron atoms.
• Outline the Schrödinger quantum approach and state two principle postulates of quantum mechanics.
• Quote and interpret the time-independent (TISE) Schrodinger equation.
• Solve the TISE for simple 1-dimensional physical systems, applying appropriate boundary conditions.
• Understand the how energy quantisation arise from the boundary conditions and be able to normalise1-dimensional wave- functions.
• Solve the one-dimensional time- independent Schrödinger equation for simple potential steps and barriers and to be able to calculate their transmission and reflection coefficients.
• State the significance and importance of operators in representing observable quantities.
• Be able to quote operators for position, momentum and energy and describe the quantum mechanical interpretation of a measurement and the significance of eigenvalues as possible results of measurements.
• Be able to ustilise the operator for the hydrogen atom to demonstrate that certain solutions are eigenfunctions.

## Module content

Syllabus

Term 1: Heat and Matter

• Heat:. Temperature scales, heat flow, specific heat capacity and phase changes; the ‘zeroth’ law of thermodynamics; the definition of a temperature scale through the properties of an ideal gas.
• Application of the Wien displacement Law, the role of spectral emissivity in radiative heat Transfer, and Kirchoff’s Law. The implications of Plancks Law and Stefans Law and their application to radiation from the sun and other hot bodies.
• The application of the time-independent heat conduction equation to simple cases including compound bodies and gases.
• The definition of the convective heat transfer coefficient and consideration of the factors that control its magnitude.
• 1st Law of Thermodynamics: relationship between work done on a system and heat including cyclic tansformations.
• Kinetic Theory: basic assumptions, the concept of an ideal gas, collisions of gas molecules with a surface and derivation of an expression for pressure. The equipartition of energy and degrees of freedom; the heat capacity of a gas; breakdown of classical theory. Elementary treatment of Maxwells speed distribution function. Molecular collisions, collision probability and derivation of a mean free paths.
• Equilibrium Properties of Fluids: Archimedes Principle and pressure variation in fluids in elementary hydrostatic problems.
• Fluid flow: Bernoulli's equation and simple applications, e.g. the Venturi metre.
• Bonding: the origin of the interatomic forces in solids and the distinction between the different bonding types: ionic, covalent, molecular (Van der Waals), metallic and hydrogen.
• Crystal structure: the description of crystal structures in terms of the space lattice + unit cell. The common crystal structures: simple cubic, face centred cubic, body centred cubic, hexagonal close packed, diamond, and zincblende. Nearest neighbour distance and packing fraction of the cubic systems.

Term 2: Introduction to Quantum Physics

• Light as a particle: Photoelectric effect/ Experiment; Compton scattering/Experiment
• Summary and comments on the dual nature of light
• Wave properties of matter; de Broglie’s postulate
• Davisson – Germer experiment
• Wave – particle nature of matter
• Uncertainty Principle ; uncertainty in real space and time, and in reciprocal (momentum and energy) space; examples.
• Properties and constituents of the atom; Size, mass and constituents
• Rutherford scattering and experiment
• Constituents of the nucleus ( including concept of isotopes)
• Stability of the atom; Quantisation in atoms
• Atomic spectra (emission and absorption); the hydrogen atom emission series
• Franck-Hertz experiment
• The Bohr model, Bohr’s postulates, Bohr’s model
• Correction for finite nuclear mass
• Failure of classical mechanics to explain spectral lines, atomic bonding, stability of the atom and the ultraviolet catastrophe
• Quantum approach - The Schrödinger equation
• Postulates of quantum mechanics (main ones only)
• The time independent Schrödinger equationExamining the terms of the Schrödinger equationWavefunction of a free particle; Born's probability interpretation of the wavefunction.
• Wave-particle duality; Youngs two slit experiment; probability concepts in classical and quantum mechanics
• Waves and wavevectors
• Time-independent Schrödinger equation (TISE); static potential; stationary states; ‘boundary conditions to be satisfied by physically acceptable solutions of TISE: single-valuedness; normalisability and continuity.
• Introduction to operators; observables and their operators; the Hamiltonian operator; position and momentum operators; eigenvalues and eigenfunctions; expectation values; examples.
• Particle in an infinite one-dimensional potential well; solutions of TISE, energy eigenvalues and normalised eigenfunctions; orthogonality; orthonormalisation; the Kronecker delta; parity; the finite square well; classically inaccessible regions
• Reflection and transmission at steps, barriers, and wells; reflection and transmission coefficients; quantum-mechanical tunnelling; particle flux, probability density and probability current density.
• Particle subject to a 1D Coulomb potential; comparison to the spherically symmetric hydrogenic wave functions; hydrogenic energy eigenvalues.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Introduction to Thermal and Quantum Thermal Physics Exam
3 hours 80
Essay/coursework
Quantum Physics Assignment
N/A 10
Essay/coursework
Thermal Physics Assignment
N/A 10

None

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Introduction to Thermal and Quantum Thermal Physics Exam
3 hours 50
Closed/in-person Exam (Centrally scheduled)
Quantum Physics Exam
N/A 50

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