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DRPS : Course Catalogue : School of Physics and Astronomy : Undergraduate (School of Physics and Astronomy)

Undergraduate Course: Quantum Mechanics (PHYS09053)

Course Outline
SchoolSchool of Physics and Astronomy CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 9 (Year 3 Undergraduate) AvailabilityAvailable to all students
SCQF Credits20 ECTS Credits10
SummaryThis two-semestered course covers fundamentals of quantum mechanics and its applications to atomic and molecular systems.

The first semester covers non-relativistic quantum mechanics, supplying the basic concepts and tools needed to understand the physics of atoms, molecules, and the solid state. One-dimensional wave mechanics is reviewed. The postulates and calculational rules of quantum mechanics are introduced, including Dirac notation. Angular momentum and spin are shown to be quantized, and the corresponding wave-function symmetries are discussed. The Schrodinger equation is solved for a number of important cases, including the harmonic oscillator and the Hydrogen atom. The fundamentals of time independent perturbation theory will be introduced as a means of approximating solutions to complex problems.

The second semester deals principally with atomic structure, the interaction between atoms and fields, and the atom- atom interactions in molecular physics. The course presents a detailed treatment of the hydrogen atom, including spin-orbit coupling, the fine structure, and the hyperfine interactions. Identical particles are reviewed in the context of electron electron interactions; applications include the Helium atom, Coulomb/exchange integrals, and alkali metals energy levels. Atom-field interactions are discussed, leading to dipole transitions, the Zeeman effect, the Lande g-factor, and the Stark effect. Finally atom-atom interactions are presented, leading to the study of molecular binding, molecular degrees of freedom (electronic, vibrational, and rotational), elementary group theory considerations and molecular spectroscopy.
Course description Semester 1
- Wave function, physical states of a quantum system. Dirac notation.
- Measurements, observables, and operators. Hermitian operators and their properties. Commutators and compatible observables. Complete Sets of Commuting Observables. Degeneracy.
- Dynamical evolution of quantum states: Schroedinger equation. Energy and Hamiltonian.Time-independent Schroedinger equation.
- Completeness and orthogonality relations for the eigenfunctions of a continuum spectrum. Ehrenfest's theorem. Some properties of the solutions of the Schroedinger equation. The Heisenberg representation.
- Schroedinger equation in one dimension. Potential step and continuity equations. Reflected and transmitted waves. Tunnelling. Experimental observation of tunnelling. Infinite potential well. Zero point energy. Potentials symmetric under parity.
- Two- and three-dimensional systems. Vectors of operators, momentum in three dimensions. Link with vector calculus. Canonical commutation relations in three dimensions. The harmonic oscillator in three dimensions; solution by separation of variables.
- Angular momentum. Differential operators using Cartesian coordinates, and spherical coordinates. Commutation relations between components of the angular momentum. Square of the angular momentum. Commutations relations with the components. Simultaneous eigenstates of L2 and Lz. Eigenvalue equations; spherical harmonics and eigenvalues. Quantization of angular momentum in QM. Stern-Gerlach experiment.
- Time-independent Schroedinger equation for a system in a central potential. Separation of variables. Reduction to a one-dimensional problem, effective potential & boundary condition. Solutions for the stationary states. Quantum rotator.
- The Hydrogen atom: set-up of the problem, Hamiltonian,separation of variables, reduction to one-dimensional radial equation, boundary condition, solutions for the stationary states, quantization of the energy levels. Discussion of the physical properties.
- Solution of the 1-dimensional harmonic oscillator using creation and annihilation operators. Properties of the creation/annihilation operators. Spectrum of the Hamiltonian. Eigenfunctions of the Hamiltonian.
- Angular momentum using raising and lowering operators. Construction of the raising and lowering operators from the commutation relations. Eigenvalues, quantization of the eigenvalues. Normalisation of the eigenstates. Explicit form of the eigenfunctions.
- Spin as an intrinsic property of a quantum system. Experimental evidence: Stern-Gerlach experiment. Spin 1/2: description of the states of a spin 1/2 system. Space of physical states, choice of a basis.
- Addition of angular momenta. General result (stated, not proven). Coupled and uncoupled basis. Example: system of two spin 1/2 particles.
- Identical particles. Symmetry of the wave function, example of the He atom. Two- electron wave function, combining spin and spatial wave functions. More on the He atom. Pauli exclusion principle.
- Time-independent perturbation theory. Solution by perturbative expansion. Shifted energy levels and wave functions. Examples.

Semester 2
- Time independent perturbation theory to 1st order in the wavefunction and 2nd order in the energy. Example A - 1D potentials.
- Example B: 1D harmonic oscillator. Revision of the H-atom wavefunctions.
- Example C: Polarisability of the H-atom, approximations for estimating lower and upper bounds.
- Interpretation of C: f sum rule, oscillator strengths.
- Example D: Van der Waals interaction.
- Degenerate Perturbation theory - example A: 2D particle in a square box or similar.
- Example B: Quadratic and Linear Stark effect.
- Relativistic Corrections to H-atom, spin-orbit interaction and Thomas precession described.
- 'Good' quantum numbers, angular momentum addition (revised), Clebsch-Gordon coeffs for j.
- Other relativistic Corrections to H-atom (mass-velocity, Darwin, Lamb, & Hyperfine) described.
- Two electron states, exchange interaction introduced. Example for 1D potentials.
- Rayleigh-Ritz variational method; 1D example.
- Application of the previous two lectures to E-levels of the ground and excited states of helium.
- Multi-electron atoms, Aufbau principle, Hund's rules for the ground state (only), examples.
- Hamiltonian including magnetic fields, atomic diamagnetism (Lamor).
- Paramagnetic atomic moments, Lande g-factor.
- Normal and Anomalous Zeeman splitting, (selection rules given empirically since TDPT not done), Paschen Back effect.
- Examples: calculation of magnetic susceptibility of some materials.
- Overview of approximations used to describe bonding (Born Oppenheimer, LCAO versus Heitler London) omitting detailed calculation.
- Rotations and vibrations of diatomic molecules.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: ( Linear Algebra and Several Variable Calculus (PHYS08042) OR Several Variable Calculus and Differential Equations (MATH08063)) AND Modern Physics (PHYS08045) AND Dynamics and Vector Calculus (PHYS08043)
Co-requisites Students MUST also take: Fourier Analysis (PHYS09054) OR Fourier Analysis and Statistics (PHYS09055)
Prohibited Combinations Students MUST NOT also be taking Principles of Quantum Mechanics (PHYS10094)
Other requirements None
Additional Costs None
Information for Visiting Students
High Demand Course? Yes
Course Delivery Information
Academic year 2024/25, Available to all students (SV1) Quota:  0
Course Start Full Year
Timetable Timetable
Learning and Teaching activities (Further Info) Total Hours: 200 ( Lecture Hours 44, Seminar/Tutorial Hours 44, Formative Assessment Hours 3, Revision Session Hours 1, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 104 )
Assessment (Further Info) Written Exam 80 %, Coursework 20 %, Practical Exam 0 %
Additional Information (Assessment) Coursework 20%
Examination 80%
Feedback Not entered
Exam Information
Exam Diet Paper Name Hours & Minutes
Main Exam Diet S2 (April/May)3:00
Learning Outcomes
On completion of this course, the student will be able to:
  1. State in precise terms the foundational principles of quantum mechanics and how they relate to broader physical principles
  2. Devise and implement a systematic strategy for solving a complex problem in quan- tum mechanics by breaking it down into its constituent parts
  3. Apply the necessary mathematical techniques to build up the solution to a complex physical problem
  4. Use experience and intuition gained from solving physics problems to predict the likely range of reasonable solutions to an unseen problem
  5. Resolve conceptual and technical difficulties by locating and integrating relevant information from a diverse range of sources
Reading List
Albert Messiah, "Quantum Mechanics"
Claude Cohen Tannoudji, "Quantum Mechanics - vol 1"
L. D. Landau ad E. M. Lifshitz, "Quantum Mechanics: Non-Relativistic Theory"
Additional Information
Graduate Attributes and Skills Not entered
Special Arrangements None
Course organiserDr Christopher Stock
Tel: (0131 6)50 7066
Course secretaryMs Nicole Ross
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