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Outline:

This module builds on the foundations laid in PHAS0022 Quantum Mechanics. It aims to extend the students' knowledge base and to give a deeper understanding of the subject. The module material is essential for many modules offered in the MSci year. The module gives a formal exposition of quantum mechanics in the form of matrix mechanics using Dirac notation. Topics include: Dirac Notation; quantum harmonic oscillator by operator techniques; Theory of orbital, spin and generalised angular momentum, with an introduction to coupling of two angular momenta; time-independent perturbation theory to second-order including the degenerate perturbation theory, the variational principle, systems of identical particles; the CHSH inequality, Pauli principle, bosons and fermions.

Aims:

This module aims to:

• Introduce the basic concepts of Heisenberg’s matrix mechanics. The second year module PHAS0022 dealt primarily with the Schrödinger’s wave dynamics. In PHAS0042, matrix mechanics using Dirac notation is introduced as an alternative approach to quantum dynamics. It is also shown to provide a complementary approach, enabling the treatment of systems (such as spin systems) where solutions of the non-relativistic Schrödinger wave equation in position coordinates is not possible
• Apply matrix mechanics and operator algebra to the Quantum Harmonic Oscillator and its relation to the 2nd year wave dynamics solutions using Hermite polynomials
• Apply matrix mechanics and operator algebra to quantum angular momentum and its relation to the 2nd year wave dynamics using Legendre polynomials and spherical harmonics.
• Demonstrate that matrix mechanics predicts and permits solution of spin-1/2 systems using Pauli matrices
• Develop understanding of fundamental concepts using these new methods. The Heisenberg uncertainty principle is shown to be just one among a family of generalized uncertainty relations, which arise from the basic mathematical structure of quantum theory, complementing arguments (eg Heisenberg microscope) introduced in the second year
• Explore some concepts of two-dimensional and two-particle systems. Understanding matrix mechanics with higher dimensional spaces (illustrated for example with two coupled QHOs as well as coupled angular momenta). The addition of two-spins is analysed including fundamental implications, exemplified by the Einstein-Podolsky-Rosen paradox and Bell inequalities
• Introduce approximate methods (time-independent perturbation theory, variational principle) to extend the PHAS0022 analytical solution of the hydrogen atom to encompass atoms in weak external electric and magnetic fields and two-electron systems like helium atoms
• Introduce symmetry requirements and the Pauli PrincipleÌý

Intended Learning Outcomes:

After completing the module the student should be able to:

• Formulate most quantum expressions using abstract Dirac notation and understand that it is not simply shorthand notation for second-year expressions.
• Understand how to formulate and solve simple quantum problems expressing quantum states as vectors and quantum operators as matrices.
• Use commutator algebra and creation/annihilation operators to solve for the Quantum Harmonic Oscillator.
• Derive generalized uncertainty relations; calculate variances and uncertainties for arbitrary observables. In this context, have a clear understanding of the relation and difference between operators and observables.
• Use commutator algebra and raising/lowering operators to calculate angular momentum observables.
• Calculate the states and observables of spin-1/2 systems using Pauli matrices.
• Understand the motivation for studying local hidden variable models in relation to quantum mechanics, and be able to derive the CHSH inequality, as well as demonstate that quantum mechanics violations this.
• Use time-independent perturbation theory to compute approximate energy expectation values and eigenvalues.
• Apply the the variational principle.
• Understand the Pauli Principle and symmetrisation requirements on quantum states including for combinations of space/spin states.

Teaching and Learning Methodology:

This module is delivered via weekly lectures supplemented by a series of workshops and additional discussion.

In addition to timetabled lecture and supplementary hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

Formal quantum mechanics:

Revision of year 2 concepts using Dirac bracket notation: Introduction to Dirac notation and application to PHAS2222 material including orthonormality of quantum states, scalar products, expansion postulate, Linear Hermitian Operators and derivations of their properties; simple eigenvalue equations for energy, linear and angular momentum. Representations of general operators as matrices and states as vectors using a basis of orthonormal states. Basis set transformations; proof that eigenvalue spectrum is representation independent.

The Quantum Harmonic Oscillator; Generalised Uncertainty Relations:
Introduction of creation and annihilation operators: main commutator relations, the number operator. Solution of the QHO eigenstates and spectrum using this method. Relation to wave dynamics and Hermite polynomial solutions obtained previously. The zero-point energy as a consequence of both commutator algebra as well as wave solutions. Uncertainty relations for general operators. Solutions of simple examples using pairs of incompatible operators, including Heisenberg Uncertainty Principle.

Generalised Angular Momentum:
Commutator algebra and raising/lowering operators. Obtaining angular momentum eigenvalue and eigenstates using raising and lowering operators. Matrix representation and solution of simple problems.

Spin-1/2 systems:
Introduction to spin-1/2 systems. Matrix representations of spin using eigenstates of z-component of spin: spinors and Pauli matrices. Matrix representations of eigenstates of spin operators along arbitrary directions. Basis set and similarity transformations between different basis sets. Addition of two angular momenta, with detailed treatment of addition of two spins.

Systems of many particles:
Entanglement. EPR Paradox and CHSH Inequality. Symmetry, fermions, bosons and the Pauli principle. Two-particle space-spin symmetry. Slater determinants for many body systems.
Approximate methods and many-body systems
Derivation of time independent Perturbation theory. First and second order theory and examples. The variational principle; examples.