## Pre-requisite modules

## Co-requisite modules

## Prohibited combinations

Occurrence | Teaching cycle |
---|---|

A | Autumn Term 2022-23 |

Thermodynamics is a branch of physics that can be applied to any system in which thermal processes are important, although we will concentrate on systems in thermal equilibrium. It is based on four laws (derived from experimental observation) and makes no assumptions about the microscopic character of the system. It is therefore very powerful and general. We will introduce these laws, consider their consequences and apply them to some simple systems.

This part of the module will prepare you for applications in different branches of physics, and provide a foundation for the model-dependent statistical mechanics approach.

Building on the Stage 1 Introduction to Quantum Physics module, the Quantum Physics part of the module extends understanding of both quantum mechanics and atomic physics. Through this, concepts of quantization, quantum states, and quantum interactions will be introduced.

The quantum mechanics component moves on from the initial description in Quantum Physics I, introducing the time dependent Schrödinger equation and the relationship between this and the time-independent Schrödinger equation. Simple 1-, 2- and 3- dimensional physical systems are developed using Schrödinger's equation. It is shown how observable quantities such as position and momentum are represented by Hermitian operators. The properties of these operators are studied. The expansion theorem is introduced and its interpretation in relation to the theory of measurement. The theory is related to observations whenever possible.

The module continues with atomic physics where the principal aim is to impart a basic knowledge of atomic structure, and to illustrate how atomic structure is interpreted from the measurement of spectra. The classical Bohr and Bohr-Sommerfeld theories and semi-classical vector model of atomic structure are applied to the hydrogen atom. The discussion moves to interpretation of the Stern-Gerlach experiment and introduces electron spin and fine structure. Methods for measuring optical spectra, and the observation and interpretation of the Zeeman effect are outlined.

On completion of this course the student will be able to –

**In Thermodynamics:**

- Define and explain fundamental concepts such as system, state function, quasistatic reversible process, thermodynamic equilibrium and equation of state.

- State the Zeroth Law of Thermodynamics; explain how this leads to the definition of empirical temperature, describe how the International Temperature Scale is realised and perform calculations related to empirical temperature scales.

- State the First Law of Thermodynamics and show how this leads to a definition of the internal energy, U, as a state function and to the conservation law dU = dW + dQ.

- Define bulk parameters, such as the principal heat capacities, and perform calculations requiring application of the First Law.

- Explain the concept of an ideal reversible heat engine, describe a Carnot cycle and derive the efficiency of a Carnot engine.

- State the Kelvin-Planck and Clausius forms of the Second Law of Thermodynamics and show they are equivalent. Use this law to prove Carnot’s theorem and its corollary.

- Show how thermodynamic temperature may be defined from the Second Law. Perform calculations relating to ideal engines, refrigerators and heat pumps.

- Derive Clausius’ theorem from the Second Law and show how this theorem leads to the definition of entropy, S. Prove that S is a state function. Derive the entropy form of the First Law. Calculate entropy changes for simple irreversible processes.

- Define the Helmholtz and Gibbs functions and show how these are related to conditions of thermodynamic equilibrium.

- Derive the four Maxwell relations for systems with two degrees of freedom and use them in calculations.

- Define the order of a phase transition in terms of derivatives of the Gibbs function.

- Derive the Clausius-Clapeyron equation for a first order phase transition and apply it to solid-liquid, liquid-vapour and solid-vapour phase transitions. Obtain Ehrenfest’s equations for second order transition.

- State the Third Law of Thermodynamics and describe some of the consequences for the behaviour of systems at low temperatures.

- Discuss and use the fundamental ideas of thermodynamics in a range of systems such as (i) showing that U is independent of T for an ideal gas; (ii) deriving the TdS equations and using them to describe the behaviour of heat capacities; (iii) applying a thermodynamic approach to the elastic deformation of a rod; (iv) deriving the equations for the Joule and Joule-Kelvin coefficients and explaining how the Joule-Kelvin effect is used in the liquefaction of gases; (v) the thermodynamic analysis of black body radiation etc.

**In Quantum Mechanics: **

- Quote and interpret the time-dependent (TDSE) and time-independent (TISE) Schrödinger equations.
- Understand the relationship between the TDSE and the TISE.
- Solve the TISE for simple 1-, 2- and 3-dimensional physical systems, applying appropriate boundary conditions.
- Normalise 1-, 2- and 3-dimensional wave-functions in Cartesian, polar and spherical polar coordinates.
- State the significance and importance of Hermitian operators in representing observable quantities. Be able to quote and apply operators for position, momentum, energy, and angular momentum.
- Prove simple theorems relating to the properties of the eigenfunctions and eigenvalues of Hermitian operators.
- Expand a wave function in terms of a basis set of functions, and interpret the expansion coefficients in terms of measurement probabilities.

**In Atomic Physics: **

- Give brief accounts of the models developed to describe atomic structure, realising their strengths and weaknesses.
- Describe the origin of absorption and emission spectra.
- Define degeneracy, and calculate the degeneracy of atomic systems.
- Understand the origin of quantum numbers describing electronic states.
- Use and interpret spectroscopic notation.
- Illustrate how spectroscopic measurements are made.
- Construct, label, and compare energy level diagrams.
- Apply selection rules to determine allowed transitions.
- Perform calculations for simple atomic systems.

**Thermodynamics**

- Introduction to systems, state functions, quasistatic reversible processes and equations of state.

- The Zeroth Law of Thermodynamics, empirical temperature scales and thermometers. The International Temperature Scale.

- The First Law of Thermodynamics and internal energy U. dU = dQ + dW. Perfect and imperfect differentials. Discussion of whether U, Q and W are state functions.

- Definition of heat capacity in general terms and expressions for C
_{P}and C_{V}for a compressible fluid. Description of how U varies with P and V for ideal and real gases. Proof that C_{P}- C_{V}= nR for an ideal gas. Quasistatic adiabatic process for an ideal gas. The van der Waals and virial equations for gases and how they attempt to account for the behaviour of real gases.

- Definition of enthalpy and identification of changes in specific enthalpy with specific latent heat and heat of reaction. Proof that C
_{P}= ( H/ T)_{P}. How the work done in a continuous flow process is related to changes in enthalpy and application of this result to ideal continuous flow processes.

- Ideal reversible heat engines. The Carnot cycle and derivation of an expression for the efficiency of an engine operating in a Carnot cycle.

- The Kelvin-Planck and Clausius forms of the Second Law of Thermodynamics and demonstration of their equivalence. Carnot's theorem and its corollary. Definition of thermodynamic temperature from the second law.

- Figures of merit of ideal refrigerators and heat pumps. The Otto and Diesel cycles.

- Clausius' theorem. Entropy S. dU = TdS – PdV. Changes in entropy for some simple irreversible processes.

- Helmholtz and Gibbs functions. Relationship to conditions of thermodynamic equilibrium.

- Maxwell relations for systems with two degrees of freedom.

- The Third Law of Thermodynamics and consequences for behaviour of systems at low temperature.

- Various applications of the fundamental ideas of thermodynamics including at least some of the following: (i) showing that the internal energy of an ideal gas is independent of p and V; (ii) derivation of the two 'TdS Equations' and use of them to describe the behaviour of the principal specific heat capacities; (iii) a thermodynamic approach to the elastic deformation of a rod; (iv) the application of thermodynamics to black body radiation; (v) derivation of the equations for the Joule and Joule-Kelvin coefficients and explanation of how the Joule-Kelvin effect is used in the liquefaction of gases.

- Definition of the order of a phase transition in terms of the derivatives of the Gibbs function.

- Derivation of the Clausius-Clapeyron equation for a first order phase transition and application of this equation to solid-liquid, liquid-vapour and solid-vapour phase transitions.

- Ehrenfest's equation for a second order phase transition.

*Quantum Mechanics*

- Waves and wavevectors; intuitive derivation of the time-dependent Schrödinger equation (TDSE); the Hamiltonian operator; normalisation of the wavefunction.
- Derivation of the Time-independent Schrödinger equation (TISE) from the TDSE; static potentials; stationary states; ‘boundary' conditions to be satisfied by physically acceptable solutions of TISE: single-valuedness; normalisability and continuity.
- Introduction to Hermitian operators and corresponding observables; the Hamiltonian operator; position and momentum operators; the angular momentum operator; commutators and commutation relations; eigenvalues and eigenfunctions; expectation values; root mean square deviations and the uncertainty principle; examples.
- The simple harmonic oscillator (SHO); classical SHO, parabolic potential; the quantum SHO; solutions of the TISE, the Hermite equation; series solution; Hermite polynomials; energy eigenvalues and normalised eigenfunctions for the SHO
- Particle in a two-dimensional box; energy eigenvalues and eigenfunctions; degeneracy table; particle in a three-dimensional box; cubic box; degeneracy table; accidental degeneracy; the three-dimensional harmonic oscillator; isotropic case and degeneracy; degeneracy table; accidental degeneracy.
- Particle in a spherically symmetric potential; the TISE in spherical polar coordinates; the hydrogenic wavefunctions; emphasis on spherically symmetric solutions; hydrogenic energy eigenvalues; radial probability density; expectation value of the radial coordinate. Eigenfunctions of the angular momentum operator.
- Formal basis of quantum mechanics; postulates; observables and Hermitian operators; forms of operators; superposition principle; expansion postulate; superposition states; quantum theory of measurement; conservation of probability; commutators and compatible observables.

*Atomic Physics*

- The spectra of atoms; absorption and emission; the hydrogen spectrum.
- Early models of the atom; Bohr’s postulates; the Bohr model; motion of the nucleus.
- Sommerfeld’s extension of the Bohr model; the Schrödinger equation for the hydrogen atom; origin of the angular momentum and magnetic quantum numbers.
- Summary of the quantum numbers; energy level (Grotrian) diagrams; quantisation and orbital angular momentum; the vector model of angular momentum.
- Magnetic properties of the atom; orbital magnetic dipole moment.
- Stern-Gerlach experiment and electron spin; the spin-orbit interaction; total angular momentum; selection rules.
- Fine structure; term notation; allowed transitions and selection rules; the Zeeman effect.

Please note, in addition to prerequisites listed above, students taking this module should also have taken PHY00022C or PHY00026C,

Task | Length | % of module mark |
---|---|---|

Closed/in-person Exam (Centrally scheduled)Quantum Physics II Exam |
1.5 hours | 40 |

Closed/in-person Exam (Centrally scheduled)Thermodynamics Exam |
1.5 hours | 40 |

Essay/courseworkQuantum Physics Assignment |
N/A | 10 |

Essay/courseworkThermodynamics Assignment |
N/A | 10 |

None

Task | Length | % of module mark |
---|---|---|

Closed/in-person Exam (Centrally scheduled)Quantum Physics Exam |
1.5 hours | 50 |

Closed/in-person Exam (Centrally scheduled)Thermodynamics Exam |
1.5 hours | 50 |

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Adkins CJ: *Equilibrium thermodynamics* (CUP)***

Zemansky MW and Dittman RH: *Heat and thermodynamics* (McGraw-Hill)***

Rae A I M; *Quantum Mechanics* 4th Ed (McGraw-Hill)***

Eisberg R M & Resnick R; *Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles* (Wiley)***