# PHYS2631 Theoretical Physics 2 (2011/12)

## Content

### Classical Mechanics

20 lectures + 3 workshops in Michaelmas Term

*Syllabus:* Lagrangian mechanics: dʼAlambertʼs 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, Hamilton-Jacobi equation, action-angle variables in 1D, integrability. 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, the heavy symmetric top. Theory of small vibrations: two coupled pendulums, normal modes and normal coordinates.

*Textbooks:*

Required: Analytical Mechanics, L.N. Hand and J.D. Finch (CUP, 1998)

### Quantum Theory

18 lectures + 3 workshops in Epiphany Term

*Syllabus:* State of a system and Dirac notation; Linear operators, eigenvalues, Hermitean operators; Expansion of eigenfunctions; Commutation relations, Heisenberg uncertainty; Unitary transforms; Matrix representations; Schrödinger equation and time evolution; Schrödinger, Heisenberg and Interaction pictures; Symmetry principles and conservation; Angular momentum (operator form); Orbital angular momentum (operator form); General angular momentum (operator form); Matrix representation of angular momentum operators; Spin angular momentum; Spin ½; Pauli spin matrices; Total angular momentum; Addition of angular momentum.

*Textbooks:*

Required: Introduction to Quantum Mechanics, B.H. Bransden and C.J. Joachain (Prentice Hall, 2nd Edition)

### Revision

2 lectures in Easter Term, one by each lecturer

## Teaching methods

*Lectures:* 2 one-hour lectures per week.

*Workshops:* These provide an opportunity to work through and digest the course material by attempting exercises and assignments assisted by direct interaction with the lecturers and workshop leaders. Students will be divided into four groups, each of which will attend one one-hour class every three weeks.

*Problem exercises:* See http://www.dur.ac.uk/physics/students/problems/