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 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
- Applications of time-independent perturbation theory. He atom, spin-orbit interaction, hydrogen atom fine structure.
- Multi-electron systems, Pauli's principle, exchange interactions. Helium energy levels, Coulomb/exchange integrals, Degeneracy. Alkali metals.
- Hartree-Fock, variational methods; focus on applications.
- Phonons.
- Atom-field interactions. Dipole transitions. Normal and Anomalous Zeeman Effect. Lande g-factor. Spectral consequences of applied fields. Stark Effect.
- Atom-atom Interactions. Bonding: Van der Waals, covalency. New degrees of freedom rotations and vibrations. Molecular electronic spectra. Selection rules. Applications of symmetry and group theory: definitions/properties, representations applications to selection rules.
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Information for Visiting Students
| Pre-requisites | None |
| High Demand Course? |
Yes |
Course Delivery Information
|
| Academic year 2016/17, Available to all students (SV1)
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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) | Quantum Mechanics (PHYS09053) | 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
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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 |
| Keywords | QMech |
Contacts
| Course organiser | Dr Christopher Stock
Tel: (0131 6)50 7066
Email: C.Stock@ed.ac.uk |
Course secretary | Miss Yolanda Zapata-Perez
Tel: (0131 6)51 7067
Email: yolanda.zapata@ed.ac.uk |
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