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Full Year

Principles of Quantum Mechanics (PHYS10094)

Subject

Physics and Astronomy

College

SCE

Credits

20

Normal Year Taken

3

Delivery Session Year

2023/2024

Pre-requisites

Course Summary

It provides a good starting point for SH quantum courses, in particular Symmetries of Quantum Mechanics, Quantum Theory, and QFT. The second half of the course, Quantum Physics, will continue to be available as a level 10 10pt S2 SH course for Physics, Astrophysics etc: the syllabus is essentially unchanged from the current syllabus of this course, and contains, as at present, some introductory lectures to bring together the cohorts.

Course Description

Semester 1: Mathematical foundations of quantum mechanics- Introduction: Particles vs Waves: photoelectric effect, double slit diffraction, linear superposition, the need for a probabilistic interpretation. [1]- States of a quantum system as vectors in a Hilbert space. Dirac notation. Inner product of state vectors, probability. [1]- Operators and observables, properties of Hermitian operators (orthogonality and completeness). Commutators and compatible observables. Complete Sets of Commuting Observables. Degeneracy. [1]- The Hamiltonian as the generator of time translations: the generic Schrodinger equation. Energy and Hamiltonian. Two-state systems, examples of finite-dimensional Hilbert spaces. [1]- Completeness and orthogonality relations for the eigenfunctions of a continuum spectrum. Dirac's delta and distributions. Position and momentum space representations. The commutation relations for x and p: the Heisenberg uncertainty relation. [2]- The Schrodinger equation for a point particle. One-dimensional square wells, tunneling. [2]- Solution of the one- dimensional harmonic oscillator using both Hermite polynomials, and creation and annihilation operators. Spectrum of the Hamiltonian. Explicit construction of the eigenfunctions of the Hamiltonian. [3]- Symmetries of the Hamiltonian and eigenfunctions. Degeneracy and symmetries of the Hamiltonian. Rotational symmetry: hydrogen atom as an example. [1]- Angular momentum and the generators of three-dimensional rotations. Algebra of the generators. Simultaneous eigenstates of L^2 and L_z. [1]- Angular momentum using raising and lowering operators. Derivation of the spectrum, quantization of the eigenvalues. Normalization of the eigenstates. Explicit form of the eigenfunctions in terms of spherical harmonics. [2]- Derivation of the spectrum of the hydrogen atom (both differential and algebraic). [2]- Spin as an intrinsic property of a quantum system. More on representations of the group of rotations. Arbitrary spin and bases for the space of physical states. [1]- Tensor product and addition of angular momentum. [2]- Identical particles, link with the group of permutations. Two-electron wave function, combining spin and spatial wave functions. Pauli exclusion principle. Spin and statistics: bosons and fermions. Atomic structure. [2]Semester 2: Quantum Physics- Revision: state vectors and wavefunctions, observables and operators, repeated and successive measurements, uncertainty, compatible observables, Hilbert space. [1]- Degeneracy and measurement. Commuting sets of operators. Good quantum numbers and maximal measurements. [1]- The time-dependent Schrodinger equation, the Hamiltonian operator, stationary states and the time-independent Schrodinger equation. Constants of motion. [1]- Non-degenerate time-independent perturbation theory: the first-order formulae for energy shifts and wavefunction mixing. Higher-orders. [1]- Perturbing degenerate systems: the first-order calculation of energy shifts. Lifting of degeneracy by perturbations. Special cases. [1]- Hydrogen fine structure. Kinetic energy correction, spin-orbit correction and Darwin correction. [1]- The Helium atom. Two-electron wavefunctions and their symmetries. First-order perturbative treatment of the inter-electron repulsion in the ground state. The first excited states of Helium: singlet-triplet splitting and exchange interaction. [2]- Brief outline of multi-electron atoms and their treatment in the central field approximation. Slater determinants. [1]- The Rayleigh-Ritz variational method. Ground state of Hydrogen as an example. Result for Helium atom. Variational bounds for excited states. [1]-Hidden variables, EPR Paradox, and Bell's inequality. [1]-Characterising entanglement - qubit, no-cloning theorem, quantum communication - teleportation.[1]-Quantum secure communication, superdense coding. [1]-Quantum computing basics, gates and registers.[1]-Quantum computing algorithms - Deutsch, Grover.[1]-Role of information theory in quantum entanglement - Shannon entropy, density Matrix, von Neumann entropy, entanglement entropy.[1]- Time-dependent Hamiltonians, the Heisenberg picture, and Heisenberg equation of motion. Equivalence of Heisenberg and Schrodinger representations. [1]- Time-dependent perturbation theory, the Dirac picture. Transition probabilities. Special case: time-independent perturbations. [1]- Transitions to a group of states induced by a constant perturbation: Fermi's Golden Rule. Harmonic perturbations and transitions to a group of states. [2]- Interaction of radiation with quantum systems. Electromagnetic radiation. Interaction with a 1-electron atom. The dipole approximation. Absorption and stimulated emission of radiation. Laser action. [2]- Spontaneous emission of radiation. Einstein A and B coefficients. Electric dipole selection rules. Parity selection rules. [1]- Quantum scattering theory. Differential and total cross-sections. The Born approximation via Fermi's Golden Rule: density of states, incident flux, scattered flux. Differential cross-section for elastic scattering. [2]- Scattering by central potentials. The screened Coulomb potential. Coulomb scattering and the Rutherford cross-section. Two-body scattering and the CM frame. [1]- The H+2 ion & molecular bonding. Born-Oppenheimer approximation. Variational estimates of the ground-state energy. Brief discussion of rotational and vibrational modes. [1]

Assessment Information

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

Additional Assessment Information

20% Coursework80% Examination

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