(Course description last updated for academic year 2020-21).
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 beneficial. All of the background 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 appreciate 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 as 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 working 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 appreciate the unique role of spin, and its properties. 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 concept of `indistinguishable particles', and its profound consequences. To understand the difference between Fermions and Bosons, and to see how quantum mechanics necessarily leads to remarkable concepts such as entanglement.

Synopsis

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

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

Schrödinger's Equation: Time-independent and time-dependent Schrödinger Equation. Probability current.

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

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

Time-Dependent Quantum Mechanics: Time-dependence, and state propagation: expectation values, Ehrenfest’s theorem, stationary states, the time-evolution operator, conserved quantities. Ehrenfest's theorem and classical trajectories. Time-energy uncertainty relation. Heisenberg's Picture. 

Quantum Mechanics in Three Dimensions, three-dimensional wavefunctions: 3D potential box. Orbital angular momentum: eigenfunctions, ladder operators, and parity.  Wavefunctions in angular coordinates. The rigid rotator and diatomic molecules. Rotational specific heats of gases. Central potentials: conservation of angular momentum, quantum numbers, separation of variables. The hydrogen atom. Non-central potentials and hybridization.

Two-Particle Systems: Two-particle states. Vector basis for two-particle systems. Conservation of total momentum. Internal and external motion of diatomic system.

Spin: The Stern-Gerlach experiment. Spin angular momentum, spin operators, spin eigenstates. Spin raising and lowering. Combining spin and orbital angular momentum, combining spins. Matrix methods.

Matrix Mechanics: Matrix elements. Example of two-level-systems. Bonding and anti-bonding states. Matrix mechanics of orbital angular momentum and spin. Density matrices.

Indistinguishable Particles: Identical and indistinguishable particles. Exchange symmetry. Multi-particle states. Fermions and Bosons. Multi-particle exchange. Exclusion principle. (And, depending on time: Particle exchange operator. Interacting and non-interacting particles. Ortho- and Para-Hydrogen).

BOOKS

There are many text books on quantum mechanics, and it is a matter of ‘taste’ as to which of these are ‘good’. Recent, standard text books include:

Introduction to Quantum Mechanics, Griffiths D. J.  (Pearson Education Limited, 2014)

Quantum Mechanics: Concepts and Applications, Zettili N.  (Wiley, 2009)

Older, but suitable books include:

Quantum Physics, Gasiorowicz S. (Wiley, 2003)

Quantum Mechanics, McMurry S. M. (Addison-Wesley, 1994).

Quantum Mechanics, Rae A I M (Hilger, 1992)

Quantum Mechanics, Mandl F (Wiley, 1992).

It is also interesting to look at Dirac’s classic text:

The Principles of Quantum Mechanics, Dirac P. A. M.  (Oxford Univ. Press, 1999).

Course section:

Other Information

Staff
Prof Claudio CastelnovoLecturer