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Department of Mathematical Sciences

# MATH3111 Quantum Mechanics III

Quantum theory has been at the heart of the enormous advances that have been made in our understanding of the physical world over the last century, and it also underlies much of modern technology, e.g., lasers, transistors and superconductors. It gives a description of nature that is very different from that of classical mechanics, in particular being non-deterministic and "non-local", and seeming to give a crucial role to the presence of "observers".

The course begins with a brief historical and conceptual introduction, explaining the reasons for the failure of classical ideas, and the corresponding changes needed to describe the physical world. It then introduces the defining postulates and reviews the necessary mathematical tools - basically linear vector spaces and second order differential equations. Applications to simple systems, which illustrate the power and characteristic predictions of the theory, are then discussed in detail.

## Outline of Course

Aim: To give an understanding of the reasons why quantum theory is required, to explain its basic formalism and how this can be applied to simple situations, to show the power in quantum theory over a range of physical phenomena and to introduce students to some of the deep conceptual issues it raises.

### Term 1

• Formal quantum theory: Dirac bra-ket formalism, Hermitian operators, eigenvalues, complete sets of commuting operators. Schrödinger, Heisenberg and interaction pictures.
• Path integrals: Position operator, Propagatro, and its description as a path integral.
• Three dimensional potentials with spherical symmetry: Angular momentum operators. The three dimensional isotropic harmonic oscillator. The hydrogen atom.

### Term 2

• Angular Momentum in 3D: Commutation relations, eigenvalues, states, spherical harmonics.
Systems with a central potential.
• Hydrogen Atom.
• Approximation Method: WKB approximation for bound states and tunnelling in one dimension.
• Approximation Methods: Time-Independent Degenerate and Non-Degenerate Perturbation Theory and applications to the fine structure of the hydrogen atom and relativistic corrections to thespectrum of hydrogen atom.

### Prerequisites

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