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. Approximate methods of solution are studied, including time-independent perturbation theory, with application to atomic structure. 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.
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| Course description |
Quantum Mechanics (semester 1):
- Wave mechanics in 1D; wavefunctions and probability; eigenvalues and eigenfunctions; the superposition principle.
- Notions of completeness and orthogonality, illustrated by the infinite 1D well.
- Harmonic oscillator. Parity. Hermitian operators and their properties.
- Postulates of quantum mechanics; correspondence between wavefunctions and states; representation of observables by Hermitian operators; prediction of measurement outcomes; the collapse of the wavefunction.
- Compatible observables; the generalised Uncertainty Principle; the Correspondence Principle; time evolution of states.
- Heisenberg's Equation of Motion. Constants of motion in 3D wave mechanics: separability and degeneracy in 2D and 3D.
- Angular momentum. Orbital angular momentum operators in Cartesian and polar coordinates. Commutation relations and compatibility. Simultaneous eigenfunctions and spherical harmonics. The angular momentum quantum number L and the magnetic quantum number mL.
- Spectroscopic notation. Central potentials and the separation of Schrödinger's equation in spherical polars.
- The hydrogen atom: outline of solution, energy eigenvalues, degeneracy.
- Radial distribution functions, energy eigenfunctions and their properties. Dirac notation.
- Applications of operator methods; the harmonic oscillator by raising and lowering operator methods.
- Angular momentum revisited. Raising and lowering operators and the eigenvalue spectra of J2 and Jz.
- Condon-Shortley phase convention. Matrix representations of the angular momentum operators; the Pauli spin matrices and their properties.
- Properties of J=1/2 systems; the Stern-Gerlach experiment and successive measurements. Regeneration.
- Spin and two-component wavefunctions. The Angular Momentum Addition Theorem stated but not proved. Construction of singlet and triplet spin states for a system of 2 spin-1/2 particles.
- Identical particles. Interchange symmetry illustrated by the helium Hamiltonian. The Spin-Statistics Theorem. Two-electron wavefunctions. The helium atom in the central field approximation: ground-state and first excited-state energies and eigenfunctions. The Pauli Principle. Exchange operator.
- Quantum computing, entanglement and decoherence.
- Introduction to Hilbert spaces.
- Basic introduction to linear operators and matrix representation.
- Advanced concepts in addition of angular momenta: the uncoupled and coupled representations. Degeneracy and measurement. Commuting sets of operators. Good quantum numbers and maximal measurements.
Quantum & Atomic Physics (semester 2):
- Time-independent perturbation theory for non-degenerate systems: the first-order formalism for energy eigenvalues and eigenstates. Higher order corrections outlined.
- Applications of perturbation theory: the ground state of helium, spin-orbit effects in hydrogen-like atoms.
- Non-degenerate time-independent perturbation theory: the first-order formulae for energy shifts and wavefunction mixing. Higher-orders.
- Time-independent perturbation theory for degenerate systems: the first-order calculation of energy shifts. Lifting of degeneracy by perturbations. Special cases.
- Hydrogen fine structure. Kinetic energy correction, spin-orbit correction and Darwin correction. Russell-Saunders notation, Zeeman effect, Stark effect, other atomic effects.
- Outline of multi-electron atoms and their treatment in the central field approximation. Slater determinants.
- The Rayleigh-Ritz variational method. Ground state of Hydrogen as an example. Result for Helium atom. Variational bounds for excited states.
- Self consistent field theory (Hartree), density functional theory. Many-particle wavefunction, exchange symmetry and correlations.
- The H2+ ion & molecular bonding. Born-Oppenheimer approximation. Variational estimates of the ground-state energy. Brief discussion of rotational and vibrational modes. Van der Waals force.
- Periodic potentials, reciprocal lattice, softening of band edge.
- Quantization of lattice vibrations (phonons). Linear chains.
- Time-dependent problems: time-dependent Hamiltonians, first-order perturbation theory, transition probabilities.
- Transitions induced by a constant perturbation: Fermi's Golden Rule. Harmonic perturbations and transitions to a group of states.
- Interaction of radiation with quantum systems. Electromagnetic radiation. Interaction with a 1-electron atom. The dipole approximation. Absorption and stimulated emission of radiation.
- Spontaneous emission of radiation. Einstein A and B coefficients. Electric dipole selection rules. Parity selection rules.
- Semiclassical approximation - WKB approximation.
- Quantum scattering theory. Differential and total cross-sections. The Born approximation via Fermi's Golden Rule. Density of states, incident flux, scattered flux. Differential cross-section for elastic scattering. Partial waves.
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Entry Requirements (not applicable to Visiting Students)
| Pre-requisites |
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Co-requisites | |
| Prohibited Combinations | Students MUST NOT also be taking
Principles of Quantum Mechanics (PHYS10094)
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Other requirements | None |
| Additional Costs | None |
Information for Visiting Students
| Pre-requisites | None |
Course Delivery Information
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| Academic year 2015/16, 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 )
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| Assessment (Further Info) |
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %
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| Additional Information (Assessment) |
Coursework 20%
Examination 80% |
| Feedback |
Not entered |
| Exam Information |
| Exam Diet |
Paper Name |
Hours & Minutes |
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| Main Exam Diet S2 (April/May) | Quantum Mechanics (PHYS09053) | 3:00 | |
Learning Outcomes
- 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 quantum mechanics by breaking it down into its constituent parts.
- Apply a wide range of 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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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: |
Course secretary | Yuhua Lei
Tel: (0131 6) 517067
Email: |
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