Quantum Physics 2014-15

*Prof Stafford Withington*

## Prerequisites

This Course assumes knowledge of the Part IA Physics and Part IA NST Mathematics courses, and certain additional topics taught in the IB Mathematics courses. Topics of primary importance include: Fourier Transforms, discrete and continuous probability theory, complex numbers and functions of complex variables, elementary notions of vector and operator methods including mappings, matrix algebra, inner products, eigenvectors and eigenvalues. A working knowledge of classical linear and angular dynamics is essential, and an appreciation of the mathematical description of elastic and electromagnetic waves in one, two and three dimensions is assumed. A general appreaciation of optics is beneficial. All of these subjects will be covered to the necessary levels in other IB courses.

## Learning Outcomes and assessment

The structure of the course is driven by the following learning outcomes: To appreaciate how classical physics breaks down at microscopic scales. To understand the conceptual foundations of quantum mechanics, and the notion of the state of a system. To gain a working understanding of Schrodinger's equation, and to become practised in using it to analyse the behaviour of a variety of one-dimensional systems having various potential functions: non-classical phenomena will emerge, such quantum-mechanical tunnelling. To appreciate the difference between time-dependent and time-independent quantum mechanics. To understand the meaning of non-commuting observables, and the origin and meaning of Heisenberg's uncertainty principle. To gain fluency in operator algebra, and a wokring knowledge of how it underpins quantum mechanics at a fundamental level. To appreciate the meaning of Dirac's bra-ket notation, to develop the mathematical skills associated with the algebra of linear operators, to understand the conceptual meaning of operator methods, and their use in solving problems. To understand the quantisation of angular momentum, the representation of angular momentum, and to see how it can be used to describe the internal behaviour of the hydrogen atom. To appreaciate the unique role of spin. To start to gain insight into the quantum mechanical description of multiple-particle systems, and to gain an appreciation of the rich physics that inevitably follows. To understand the difference between Fermions and Bosons, and to see how quantum mechanics necessarily leads to remarkable concepts such as entanglment.

## Synopsis

**The Quantum Revolution: **Photoelectric effect. de Broglie Hypothesis. Bohr's atom and atomic structure. Electron diffraction, Davisson and Germer. Compton scattering. Blackbody radiation, u-v catastrophe. The central role of Planck’s constant.

**Wave-Particle Duality and the Uncertainty Principle: **Young’s double-slit experiment. Free particle in one dimension: wavefunctions and wave-packets. The Heisenberg Uncertainty Principle. Time evolution of wave-packets: dispersion and propagation.

**The Schrödinger Equation: **Time-independent and time-dependent Schrödinger Equation.

**Wave Mechanics of Unbound Particles: **Particle flux. One dimensional potentials and boundary conditions. The potential step: reflection and transmission. The potential barrier: tunnelling. Radioactivity.

**Wave Mechanics of Bound Particles: **The infinite square well potential and bound states. Normalization, parity and orthogonality. The Correspondence Principle. The finite square well potential. The harmonic oscillator: vibrational specific heat.

**Operator Methods: **Operators, observables, linear hermitian operators and operator algebra. Dirac notation: eigenstates and eigenvalues. Orthogonality, degeneracy and completeness of eigenstates. Compatible and incompatible observables: commuting operators and simultaneous eigenstates, non-commuting operators, generalized uncertainty relations, minimum-uncertainty states. Ladder operators: the harmonic oscillator, equipartition. Density matrix, pure and mixed states.

**Time-Dependent Quantum Mechanics: **Time-dependence: expectation values, Ehrenfest’s theorem, stationary states, the time-evolution operator, time-energy uncertainty relation, conserved quantities.

**Quantum Mechanics in Three Dimensions: **General formulation. 3D potential box. Orbital angular momentum: eigenfunctions and parity. The 3D harmonic oscillator. The rigid rotator: rotational specific heats of gases. Central potentials: conservation of angular momentum, quantum numbers, separation of variables. The hydrogen atom. Non-central potentials and hybridization.

**Spin: **The Stern-Gerlach experiment. Spin angular momentum, spin operators, spin eigenstates. Combining spin and orbital angular momentum, combining spins. Matrix methods. Pauli spin matrix and spinors.

**Identical Particles: **Identical particle symmetry: multiparticle states, fermions and bosons, exchange operator, exclusion principle, symmetry and interacting particles, counting states. Entanglement. Two-electron system: helium ground and excited states.

**BOOKS**

*Quantum Physics*, Gasiorowicz S (Wiley 2003) A fine exposition of the subject, suitable for Part IB and Part II.

*Quantum Mechanics*, McMurry S M (Addison-Wesley 1994). Quite well suited to the course and includes a disc with interactive illustrative programs.

*Quantum Mechanics*, Rae A I M (Hilger 1992) A good alternative to Gasiorowicz or McMurry, much shorter and consequently less full in its treatment of difficult points.

*Quantum Mechanics * Mandl F (Wiley 1992). A good book, suitable for Part IB and Part II.

*Introduction to Quantum Mechanics*, Bransden and Joachin (Longman, 1989). A thick book, with very full coverage but perhaps less elegance and clarity than Gasiorowicz.

*Problems in Quantum Mechanics*, Squires (CUP, 1995). Worked solutions and summaries, extending beyond this course.

*Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles*, Eisberg R and Resnick R (Wiley 1985). Too elementary to recommend as a main textbook, but very good descriptive coverage of a wide range of quantum phenomena.