The Department of Physics

Course Description

This is a calculus-based advanced level physics course covering the basic theory of classical and statistical mechanics. The course is compulsory for the major in Physics. Its purpose is to describe the behaviour of microscopic systems in physics such as atoms and particles in confined space. The mathematical formalism is developed to describe these microscopic systems. Through in-class discussions, and problem solving sessions, Students would have an opportunity to improve their ability to reason through challenging situations in the physical Universe using basic principles to develop appropriate solutions. Assessment and evaluation is done in the form of in-course tests and a Final examination.

CONTENT

This course will focus on the following:
Classical Mechanics:

• Introduction to Newtonian Mechanics: Newton’s laws, inertial and non-inertial systems. Systems of particles and centre of mass. Conservation laws and collisions.
• Central Force Motion: Central force motion as a one-body problem. Properties of motion under a central force. General force field orbits and effective potential. Kepler’s laws of planetary motion. Perturbed circular orbits.
• Lagrangian and Hamiltonian Dynamics: Mechanics of a particle and a system of particles. Generalized coordinates and constraints. D’Alembert’s principle and Lagrange’s equation. Simple applications of the Lagrangian formulation. Hamilton’s principle. Derivation of Lagrange’s equations from Hamilton’s principle. Hamilton’s equation of motion. Derivation of Hamilton’s equations from a variational principle. Hamiltonian phase space. Liouville’s theorem. The Principle of Least Action.

Statistical Mechanics:

• Thermodynamics: Equilibrium and state quantities. The laws of thermodynamics. Thermodynamic potentials. The statistical basis of thermodynamics.
• Classical Statistics: Phase space of a classical system. The micro-canonical ensemble. The canonical ensemble. The grand canonical ensemble. Derivation of thermodynamics. Equipartition theorem. Kinetic theory of a dilute gas. The Maxwell-Boltzmann distribution. Gibbs paradox.
• Quantum Statistics: Quantum mechanical ensemble theory: the density matrix. Statistics of the various ensembles. Systems composed of indistinguishable particles.
• Ideal systems: Photon gas , phonon gas, electrons in metals, classical ideal gases,
• Phase transitions

GOALS/AIMS

• To enable students to develop a good understanding of classical and statistical mechanics required for more advanced studies of Physics.
• To produce graduates with quantitative and analytical skills enabling them to apply quantum mechanical principles to the study and analysis of phenomena in Physics.

LEARNING OUTCOMES

After completing this course, students should be able to describe and analyze the following:

• Newton’s laws and applications. Reference frames. Single particle dynamics under various types of forces. Systems of particles and their centre-of-mass. Newton’s laws, conservation laws and symmetry principles. Elastic and inelastic collisions. Inverse square repulsive force- Rutherford scattering.
• Relationship between central force and potential energy. Central force motion as a one-body problem. General properties of motion under a central force. Angular momentum and energy conservation. Law of equal areas.
• Central force field orbits and effective potential. Orbits in an inverse square force field. Kepler’s laws of planetary motion. Perturbed circular orbits and orbital transfers.
• Generalized coordinates and generalized forces on a single particle and a system of particles. Conservative systems. Lagrange’s equation of motion for a single particle.
• Lagrange’s equation of motion for a system of particles. Lagrange’s equation of motion with undetermined multipliers and constraints. Generalized momenta and ignorable (cyclic) coordinates.
• The Hamiltonian function-conservation laws and symmetry principles. Hamiltonian dynamics-Hamilton’s equation of motion.
• Systems, phases and state quantities. Equilibrium and temperature-the zeroth law of thermodynamics. The principle of maximum entropy. Entropy and energy as thermodynamics potentials. Macroscopic and microscopic states. Contact between statistics and thermodynamics. The entropy of mixing and Gibb’s paradox. Enumeration of microstates.
• The postulates of classical statistical mechanics. Postulate of equal a priori probability. The meaning of the micro-canonical ensemble. Application of this concept to the ideal gas. Gibb’s paradox for ideal gases. The canonical ensemble for systems in equilibrium. The grand canonical ensemble for systems with variable number of particles. The chemical potential.
• The postulates of quantum statistical mechanics. The density matrix and ensembles. Examples –an electron in a magnetic field, a free particle in a box and a linear harmonic oscillator, simple gases.
• Thermodynamic behaviour of ideal Bose systems such as the ideal Bose gas, blackbody radiation and phonons in solids.
• Thermodynamic behaviour of ideal Fermi systems such as the ideal Fermi gas and the electron gas in metals.
• The use of the Ising model to understand general phase transitions. Use of the Ising model to describe the lattice gas and binary alloy.

Assessment