PHYS2631 Theoretical Physics 2 (2018/19)
19 lectures + 9 workshops in Michaelmas Term
Required: Analytical Mechanics, L.N. Hand and J.D. Finch (CUP, 1998)
The course is defined by material in this book, in particular Chapters 1, 2, 5, 6, 9. Chapters 3, 4, 7, 8, have been covered to some degree in L1 Foundations, but the content is extended, and presented in a more formal and abstract way.
Syllabus: Lagrangian mechanics: dʼAlembertʼs principle, constraints and degrees of freedom, generalized coordinates, velocities and forces, definition of Lagrangian and Hamiltonian, ignorable coordinates . Variational calculus and its application: Euler equation, Hamiltonʼs principle, Lagrange multipliers and constraints. Linear oscillators: stable and unstable equilibrium, SHO and damped SHO, impulsive forces and Greenʼs function, driven oscillators and resonance. One-dimensional systems and central forces: solution by quadrature, central force problem, gravitational attraction. Noetherʼs theorem and Hamiltonian mechanics: angular momentum conservation, Noetherʼs theorem, Hamiltonʼs equations. Theoretical mechanics: canonical transformations, Poisson brackets. Rotating coordinate systems: angular velocity vector, finite and infinitesimal rotations in 3D, rotated and rotating reference frames, centrifugal, Coriolis and Euler forces, Foucault pendulum. Dynamics of rigid bodies: kinetic energy, moment of inertia tensor, angular momentum, Euler equations, Euler angles, motion of torque-free symmetric and asymmetric tops. Theory of small vibrations: two coupled pendulums, normal modes and normal coordinates.
17 lectures + 9 workshops in Epiphany Term
Required: Introduction to Quantum Mechanics, B.H. Bransden and C.J. Joachain (Prentice Hall, 2nd Edition)
The course is defined by material contained in this book and in particular the material defined in the syllabus below where the numbers refer to the sections in the book.
- State of a system and Dirac notation [5.1]
- Linear operators, eigenvalues, etc, Hermitean operators [5.2]
- Expansion of eigenfunctions [5.3]
- Commutation relations, Heisenberg uncertainty [5.4]
- Unitary transforms [5.5]
- Matrix representations [5.6]
- SE and time evolution [5.7]
- Schrödinger, Heisenberg and Interaction pictures [5.8]
- Symmetry principles and conservation [5.10]
- Angular momentum (operator form) [6.1, 6.3]
- Orbital angular momentum (operator form) [6.2]
- examples: Particle on sphere/rigid rotator [6.4]
- General angular momentum (operator form) [6.5]
- Matrix representation of angular momentum operators [6.6]
- Spin angular momentum [6.7]
- Spin 1/2 [6.8]
- Pauli spin matrices [6.8]
- Total angular momentum [6.9]
- Addition of angular momentum [6.10]
2 lectures in Easter Term, one by each lecturer
Lectures: 2 one-hour lectures per week.
Workshops: These provide an opportunity to work through and digest the course material by attempting exercises assisted by direct interaction with the workshop leaders. They also provide opportunity for you to obtain further feedback on the self-assessed formative weekly problems. Students will be divided into four groups, each of which will attend one one-hour class every week. The workshops for this module are not compulsory.
Mid-term progress tests: Two 40-minute compulsory formative progress tests (weeks 6 and 16).
Problem exercises: See https://www.dur.ac.uk/physics/students/problems/