## Pre-requisite modules

## Co-requisite modules

## Prohibited combinations

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

A | Autumn Term 2021-22 |

Building on the Stage 1 Introduction to Quantum Physics module, Quantum Physics II 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.

**In Quantum Mechanics, to: **

- 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, to: **

- 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.

*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.

**Lecture notes**

Students are expected to make their own notes from lectures. In addition, handouts are provided covering background material and material that is primarily complicated mathematics which takes time to write on the board and simply help the understanding of the physics.

**Suggested preparation**

The first year lecture material on Quantum Physics is sufficient preparation. The summer preparation material on Mathematics, as well as the booklet “Quantum Mechanics Primer” by Warner & Cheung (available from the Student Administration Office) is highly recommended.

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

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

Essay/courseworkAssignment |
N/A | 20 |

Online ExamQuantum & Atomic Physics II |
N/A | 80 |

None

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

Essay/courseworkAssignment |
N/A | 20 |

Online ExamQuantum & Atomic Physics II |
N/A | 80 |

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

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

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