QUANTUM MECHANICS Dr Hannabuss 16 lectures MT 2008 Tuesday 3.00, Thursday 3.00

An electron in a corral of iron atoms

Aims and Objectives

Quantum theory was born out of the attempt to understand the interactions between matter and radiation. It transpired that light waves can behave like streams of particles, but other particles also have wave-like properties.

Although there remain deep mathematical and physical questions at the frontiers of the subject, the resulting theory encompasses not just the mechanical but also the electrical and chemical properties of matter. Many of the key components of modern technology such as transistors and lasers were developed using quantum theory, and the theory has stimulated important 20th century advances in pure mathematics in, for example, functional analysis, algebra, and differential geometry. In spite of their revolutionary impact and central importance, the basic mathematical ideas are easily accessible and provide fresh and surprising applications of the mathematical techniques encountered in other branches of mathematics.

This introductory course explores some of the consequences of this, including a treatment of the hydrogen atom.

Synopsis

Generalised momenta, the Hamiltonian, Hamilton's equations of motion, Poisson brackets.

De Broglie waves, the Schrödinger equation; stationary states, quantum states of a particle in a box; interpretation of the wave function, probability density and current. Boundary conditions; conservation of current, tunnelling, parity.

Expectation values of observables, eigenvalues and eigenfunctions.

The one-dimensional harmonic oscillator, higher-dimensional oscillators and normal modes.

The rotationally symmetric and general radial states of the hydrogen atom with fixed nucleus.

Heisenberg's uncertainty principle. Creation and annihilation operators for the harmonic oscillator.

Measurements and the interpretation of quantum mechanics. Schrödinger's cat.

Angular momentum, commutation relations, spectrum and matrix representation. Orbital angular momentum, rotational symmetry and spin 1/2 particles. Application to a particle in a central potential and the hydrogen atom.

Reading

Hannabuss, Introduction to Quantum Theory, OUP, (1997). Chapters 1-5.

Further Reading

• A HEY and P WALTERS, The new Quantum Universe, Cambridge (2003), gives a general background picture.
• B J BRANSDEN and C J JOACHAIN, Introduction to quantum mechanics, Longman (1995)
• I P GRANT, Mathematical Institute Notes (1991)
• A I M RAE, Quantum Mechanics, 3rd edition, Institute of Physics (1993)
• L I SCHIFF, Quantum Mechanics, 3rd edition, Mc Graw Hill (1968).