Almost all the properties of everyday solids --- their chemistry, electrical conductivity, strength, hardness, ductility, melting point, dielectric constant, refractive index, magnetic moment, and so on --- depend on quantum mechanics. One can measure the density of a brick without using quantum theory, of course, but suppose one asks why a brick has the density it does? The density depends on the sizes of the atoms, and the sizes of the atoms depend on the sizes of the electron orbits around those atoms. These orbits can only be described using quantum mechanical wavefunctions satisfying the Schroedinger equation. Most questions about molecules and solids work like this: as soon as you say "why" more than once or twice, you find yourself face-to-face with quantum theory.
The Schroedinger equation can only be solved by hand in the very simplest systems, and so most of my work is highly computational. The technique with which I am most closely associated is the quantum Monte Carlo method, in which massively parallel computers are used to carry out huge numbers of idealised "experiments", averages of the results of which yields the results of real experiments almost exactly. In large systems, in particular, quantum Monte Carlo methods are far more accurate than any other available approach. My group was among the first to apply quantum Monte Carlo techniques to real solids and remains one of the leading groups in the field. The main drawback of quantum Monte Carlo simulations is that they are very hard to do and the range of tractable physical problems is correspondingly small. Since I am interested in physics as well as computational methods, I also use simpler approaches such as density-functional theory when appropriate.
Recent quantum Monte Carlo work has included studies of idealised surfaces, of electron correlation (aimed at improving the approximations on which density-functional calculations rely), of metal-insulator transitions, and of point defects in transition metal oxides. During the last couple of years I have also become interested in extending the quantum mechanical approaches used to study solids beyond the Born-Oppenheimer approximation, taking proper account of the exchange of energy between electrons and ions.
When not researching or answering email, most of my working life is spent teaching. In recognition of the quality of my lecture courses on quantum physics and differential equations, I was recently given an Imperial College teaching award.