Undergraduate Course: Quantum Mechanics (PHYS09053)
Course Outline
School  School of Physics and Astronomy 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 9 (Year 3 Undergraduate) 
Availability  Available to all students 
SCQF Credits  20 
ECTS Credits  10 
Summary  This twosemestered course covers fundamentals of quantum mechanics and its applications to atomic and molecular systems.
The first semester covers nonrelativistic quantum mechanics, supplying the basic concepts and tools needed to understand the physics of atoms, molecules, and the solid state. Onedimensional 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 wavefunction 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 spinorbit 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. Atomfield interactions are discussed, leading to dipole transitions, the Zeeman effect, the Lande gfactor, and the Stark effect. Finally atomatom 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.Timeindependent 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 threedimensional 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. SternGerlach experiment.
 Timeindependent Schroedinger equation for a system in a central potential. Separation of variables. Reduction to a onedimensional problem, effective potential & boundary condition. Solutions for the stationary states. Quantum rotator.
 The Hydrogen atom: setup of the problem, Hamiltonian,separation of variables, reduction to onedimensional radial equation, boundary condition, solutions for the stationary states, quantization of the energy levels. Discussion of the physical properties.
 Solution of the 1dimensional 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: SternGerlach 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.
 Timeindependent 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 Hatom wavefunctions.
 Example C: Polarisability of the Hatom, 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 Hatom, spinorbit interaction and Thomas precession described.
 'Good' quantum numbers, angular momentum addition (revised), ClebschGordon coeffs for j.
 Other relativistic Corrections to Hatom (massvelocity, Darwin, Lamb, & Hyperfine) described.
 Two electron states, exchange interaction introduced. Example for 1D potentials.
 RayleighRitz variational method; 1D example.
 Application of the previous two lectures to Elevels of the ground and excited states of helium.
 Multielectron atoms, Aufbau principle, Hund's rules for the ground state (only), examples.
 Hamiltonian including magnetic fields, atomic diamagnetism (Lamor).
 Paramagnetic atomic moments, Lande gfactor.
 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.

Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2024/25, Available to all students (SV1)

Quota: None 
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:
 State in precise terms the foundational principles of quantum mechanics and how they relate to broader physical principles
 Devise and implement a systematic strategy for solving a complex problem in quan tum mechanics by breaking it down into its constituent parts
 Apply the necessary mathematical techniques to build up the solution to a complex physical problem
 Use experience and intuition gained from solving physics problems to predict the likely range of reasonable solutions to an unseen problem
 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: NonRelativistic Theory" 
Additional Information
Graduate Attributes and Skills 
Not entered 
Special Arrangements 
None 
Keywords  QMech 
Contacts
Course organiser  Dr Miguel MartinezCanales
Tel: (0131 6)51 7742
Email: miguel.martinez@ed.ac.uk 
Course secretary  Ms Nicole Ross
Tel:
Email: nicole.ross@ed.ac.uk 

