Department: PhysicsModule co-ordinator: Dr. Yvette HancockCredit value: 20 creditsCredit level: HAcademic year of delivery: 2022-23

- See module specification for other years: 2021-22

## Pre-requisite modules

## Co-requisite modules

- None

## Prohibited combinations

- None

Occurrence | Teaching period |
---|---|

A | Autumn Term 2022-23 to Spring Term 2022-23 |

This course aims to introduce advanced topics and techniques in quantum mechanics, and links these to relevant applications in nuclear physics and our description of the structure of the atomic nucleus.

On the quantum mechanics side, the course aims to:

- introduce quantum mechanical commutators and their significance for the compatibility of measurements.
- introduce the quantum mechanical treatment of angular momentum
- demonstrate co-ordinate transformation from Cartesian to spherical polar co-ordinates and apply this to the angular momentum operators and time-independent Schrödinger equation find solutions of the time-independent Schrödinger equation for a spherically symmetrical potential
- solve the time-independent Schrödinger equation for the Hydrogen atom (analytical solution) and extend quantum mechanics to incorporate spin
- introduce matrix mechanics, with particular application to spin – spin operators, Pauli spin matrices, Dirac notation
- 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 nuclear physics, this module will focus on advanced topics and begin to examine how these topics are addressed in contemporary nuclear physics research. We will examine the key models that underpin nuclear structure – associated with both “single-particle” and “collective” modes of excitation. The module then aims to develop understanding of the quantum mechanical mechanisms underlying nuclear decays and, hence, to examine what nuclear structure information can be extracted from such measurements. The physics of nuclear fission and fusion will be discussed as well as the principles of operation of fission and fusion reactors. In all of the above, published data will be used regularly to illustrate and test the ideas presented.

**Quantum mechanics:**

**- **understand the physical significance of commutators in terms of compatibility of measurements

- Perform simple commutator algebra, in order to obtain commutators for operators expressible in terms of the position and momentum operators.

- Derive operators for the angular momentum components L_{x}, L_{y}, L_{z}, and for L^{2}, in terms of position and momentum operators in Cartesian coordinates

- Understand how the angular momentum operators are transformed from Cartesian into spherical polar coordinates

- Derive the operators for L_{z} and L^{2} in spherical polar co-ordinates

- Derive and interpret the eigenvalues and eigenvectors of the operators for angular momentum, L_{z}, and L^{2} in terms of possible measurement results.

- Explain the use of the central force theorem for a spherically symmetric potential within the context of the time-independent Schrödinger equation written in spherical polar co-ordinates and applied to hydrogen-like atoms

- Discuss the relationship between the operators L_{z} L^{2} and the above Hamiltonian for a hydrogen-like atom system

- Apply the above to solving the full analytical eigensolution for the case of the Hydrogen atom, as well as sub-components of this eigenproblem

- Reproduce and interpret a labelled diagram showing the energy levels and angular momentum states of the hydrogen atom

- Provide a physical interpretation of the quantum numbers *n, l *and *m** _{l}* and be able to sketch the wavefunction solutions of the hydrogen atom for a given

- Understand the matrix formalism of quantum mechanics and apply this to the case of spin

- Apply the Pauli spin matrices to find the eigenvalues and eigenvectors of spin operators

- Interpret generalised Stern-Gerlach experiments in terms of eigenvector superposition, illustrating the theory of measurement.

- Derive the first and second order energy corrections in non-degenerate perturbation theory and apply the formulae to simple problems e.g. anharmonic oscillators

- Learn of other approximate methods such as the matrix eigenvalue formalism for degenerate perturbation theory and make applications to simple systems e.g. Stark effect in Hydrogen, as well as the variational approach.

**Nuclear physics: **

**-** Describe the significance of nuclear charge and current distributions in regard to nuclear structure and decays

- Discuss the variety of mechanisms that result in the generation of excited states in nuclei.

- Predict angular momentum and parity quantum numbers of excited states in nuclei, based on nucleonic single-particle configurations.

- Interpret aspects of published level schemes in terms of both single-particle and collective models, demonstrating how information on the different types of excitation are extracted from the data.

- Discuss the quantum-mechanical basis for the three modes of nuclear decay.

- Describe the key models and methods used to predict nuclear decay rates.

- Perform sample calculations of alpha, beta and gamma-decay rates, based on the models presented

- Interpret nuclear decay data, through an understanding of these models, in terms of nuclear structure phenomena.

- Discuss the physics of the nuclear fission and fusion processes

- Define what is meant by prompt and delayed neutrons, spontaneous and induced fission and activation energy and be able to predict whether isotopes with fission with thermal neutrons.

-Explain the basics of how thermal fission reactors operate

Please note, students who have not taken the prerequisites listed above must have taken an equivalent version of Quantum Physics II and Mathematics II.

**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.
- 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).

**Nuclear Physics: **

*Nuclear moments:*

- multipole expansion of the nuclear charge and current density
- nuclear magnetic dipole moment and nuclear electric quadrupole moment
- prediction of nuclear moments from shell model

*Nuclear models*:

- review of the extreme single particle model,
- extension to allow particle hole configurations,
- many-particle configurations
- shell-model multiplets and band termination
- properties of even-odd nuclei, properties of even-even nuclei,
- collective properties of nuclei,
- rotational and vibration excitations in nuclei,
- rotational model – even-even and odd-even nuclei
- vibrational model – one-, two- and three-phonon excitations
- single particle states in a deformed potential.
- High spin effects in nuclei – rotational alignments, backbending, and band-termination

*Alpha decay:*

- review of alpha particle decay and systematics,
- theory of barrier penetration,
- role of angular momentum and deformation
- alpha-decay selection rules and fine structure.

*Gamma-ray decay:*

- excited states in nuclei,
- classical electromagnetic radiation and relevance to nuclei
- gamma-ray decay,
- decay rates and the Weisskopf estimates,
- selection rules,
- spectroscopic information from gamma-ray decays,
- internal conversion isomers.

*Beta decay:*

- review of beta particle decay,
- Fermi theory and spectral shape
- decay rates for allowed transitions,
- selection rules
- classification of beta decay, allowed and forbidden transitions,
- electron capture,
- mass of the neutrino, (double beta decay, parity non-conservation).

*Fission:*

- physics of the fission process, prompt and delayed neutrons, fission and the liquid drop model, definitions of spontaneous, induced fission and activation energy. Basic principles of reactor physics

*Fusion:*

- Physics of nuclear fusion

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

Essay/courseworkNuclear Physics Assignment 1 |
N/A | 20 |

Essay/courseworkNuclear Physics Assignment 2 |
N/A | 30 |

Essay/courseworkQuantum Mechanics Assignment 1 |
N/A | 20 |

Essay/courseworkQuantum Mechanics Assignment 2 |
N/A | 30 |

None

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

Essay/courseworkNuclear Physics Assignment 1 |
N/A | 20 |

Essay/courseworkNuclear Physics Assignment 2 |
N/A | 30 |

Essay/courseworkQuantum Mechanics Assignment 1 |
N/A | 20 |

Essay/courseworkQuantum Mechanics Assignment 2 |
N/A | 30 |

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

Krane K S: *Introductory nuclear physics* (Wiley) ****

Heyde K: *Basic ideas and concepts in nuclear physics *(Taylor & Francis/IoP Publishing) **

Burcham W E and Jobes M:* Nuclear and particle physics *(Prentice Hall/Longman) **

Lilley J: *Nuclear physics principles and applications* (Wiley) **