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Semester 1

Foundations of Quantum Mechanics (PHYS09051)

Subject

Physics and Astronomy

College

SCE

Credits

10

Normal Year Taken

3

Delivery Session Year

2022/2023

Pre-requisites

Course Summary

This course covers fundamentals of quantum mechanics and this applications to atomic and molecular systems. The course includes non-relativistic quantum mechanics, supplying the basic concepts and tools needed to understand the physics of atoms, molecules, and the solid state. One-dimensional wave mechanics is reviewed. The postulates and calculational rules of quantum mechanics are introduced, including Dirac notation. Angular momentum and spin are shown to be quantized, and the corresponding wave-function symmetries are discussed. The eigenvalue equation for the energy is solved for a number of important cases, including the harmonic oscillator and the Hydrogen atom. Approximate methods of solution are studied, including time-independent perturbation theory, with application to atomic structure.

Course Description

- Wave mechanics in 1D; wavefunctions and probability; eigenvalues and eigenfunctions; the superposition principle. - Notions of completeness and orthogonality, illustrated by the infinite 1D well. - Harmonic oscillator. Parity. Hermitian operators and their properties. - Postulates of quantum mechanics; correspondence between wavefunctions and states; representation of observables by Hermitian operators; prediction of measurement outcomes; the collapse of the wavefunction. - Compatible observables; the generalised Uncertainty Principle; the Correspondence Principle; time evolution of states. - Heisenberg's Equation of Motion. Constants of motion in 3D wave mechanics: separability and degeneracy in 2D and 3D. - Angular momentum. Orbital angular momentum operators in Cartesian and polar coordinates. Commutation relations and compatibility. Simultaneous eigenfunctions and spherical harmonics. The angular momentum quantum number L and the magnetic quantum number mL. - Spectroscopic notation. Central potentials and the separation of Schrodinger's equation in spherical polars. - The hydrogen atom: outline of solution, energy eigenvalues, degeneracy. - Radial distribution functions, energy eigenfunctions and their properties. Dirac notation. - Applications of operator methods; the harmonic oscillator by raising and lowering operator methods. - Angular momentum revisited. Raising and lowering operators and the eigenvalue spectra of J2 and Jz. - Condon-Shortley phase convention. Matrix representations of the angular momentum operators; the Pauli spin matrices and their properties. - Properties of J=1/2 systems; the Stern-Gerlach experiment and successive measurements. Regeneration. - Spin and two-component wavefunctions. The Angular Momentum Addition Theorem stated but not proved. Construction of singlet and triplet spin states for a system of 2 spin-1/2 particles. - Identical particles. Interchange symmetry illustrated by the helium Hamiltonian. The Spin-Statistics Theorem. Two-electron wavefunctions. The helium atom in the central field approximation: ground-state and first excited-state energies and eigenfunctions. The Pauli Principle. Exchange operator. - Quantum computing, entanglement and decoherence. - Introduction to Hilbert spaces. - Basic introduction to linear operators and matrix representation. - Advanced concepts in addition of angular momenta: the uncoupled and coupled representations. Degeneracy and measurement. Commuting sets of operators. Good quantum numbers and maximal measurements.

Assessment Information

Written Exam 80%, Coursework 20%, Practical Exam 0%

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