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# Department of Mathematical Sciences

# MATH2071 Mathematical Physics II

This module follows the development of mechanics after Newton's laws to the introduction of quantum mechanics. Although no new physical principles were introduced in the nineteenth century, this era saw the beautiful and powerful reformulation of classical dynamics by Lagrange and Hamilton. This was in many ways more powerful than Newton's approach based on forces, and revealed the deep connexion between symmetries and conservation laws which continues to play a central role in the development of physical theories. Furthermore, with the introduction of the concept of the action, and of Poisson brackets, the step towards quantum mechanics is made much more natural.

### Term 1

This term concentrates on the formulation of the dynamics of systems in terms of the Lagrangian, a scalar function of "generalised coordinates" describing the position of the system. Via the introduction of the calculus of variations, It is shown that all of mechanics can be framed in terms of the "the principle of least action" which states that a system will follow the path which makes the action a minimum (or more accurately stationary). Systems with many degrees of freedom are studied in terms of their normal modes, and an example of a system with infinite degrees of freedom, the string, is also introduced as a simple example of a classical field theory. In the Hamilton formulation, which is introduced in the latter part of this term, the democracy between all generalised coordinates is extended to generalised momenta, putting position and velocity on a more equal footing. The concept of variables being tied to transformations via the Poisson bracket is highlighted, as this idea turns out to be crucial in quantum mechanics.

### Term 2

This term begins with a discussion on how by the end of the nineteenth century it was becoming apparent that classical mechanics not account for physics at very small scales, and how this led to the introduction of quantum mechanics. The subject is approached via Schödinger's equation and the wave function, complementing the Dirac and path integral approaches taken in Quantum Mechanics III. The statistical interpretation of the wave function is given in terms of the position probability density and it is applied in simple systems to build the required intuition. As the term progresses, the statistical interpretation is generalised to more general observables, other than position, ultimately leading to the postulates of quantum mechanics and their impact on our understanding of nature.

## Outline of Course

Aim: To appreciate the conceptual framework of classical physics and give and introduction to quantum mechanics.

### Term 1

• Lagrangian and Hamiltonian Dynamics (3 lectures): Degrees of freedom of a system. Holonomic and non-holonomic constraints. D'Alembert's principle and the derivation of Lagrange's equations for conservative forces.
• Conservation Laws (5 lectures): Ignorable coordinates and conservation laws. Conservation of Energy. Calculus of Variations. The Brachistochrone. Principle of Least Action. Noether's theorem.
• Small oscillations of systems of particles (3 lectures): Positions of equilibrium and stability. Normal modes of oscillation and normal coordinates.
• Waves: Review one-dimensional wave equation (4 lectures). Potential and Kinetic energy. Energy flux and conservation. Boundaries and Junctions. Derivation of Euler-Lagrange Equations for fields from principle of least action.
• Hamiltonian Dynamics (5 lectures): First order formalism for Newton's second law. Definition of the Hamiltonian from the Lagrangian. Hamilton's equations. Formulation in terms of Poisson brackets. Conservation of Energy. Dynamic quantities as generators of infinitesimal transformations on phase space. Connection in Hamiltonian mechanics between symmetry and conservation. Liouville's Theorem.

### Term 2

• Canonical Transformations and Hamilton-Jacobi formalism (5 lectures): Canonical transformations from generating functions. Invariance of Poisson Brackets Hamilton-Jacobi equations. Action-angle variables. Adiabatic invariants.
• Introduction to Quantum Mechanics (4 lectures): Physical motivations; particle-wave duality: double slit experiment, photo-electric effect, decay of the Rutherford Atom. Old quantum theory. Canonical quantisation, energy and momentum as operators; the Schrödinger Equation and the wave function. Probability density and conservation of probability.
• Simple Solutions to the Schrödinger Equation (3 lectures): Gaussian solution to free particle. Stationary solutions. Particles in a box, bound state solutions for square well. Harmonic oscillator.
• Operators and Measurement (6 lectures): Expectation of operators. Ehrenfest's Theorem. Heisenberg's uncertainty principle. Reality of eigenvalues and orthogonality of eigenfunctions for Hermitian operators. Measurement and collapse of the wave-function.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.