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In this thesis the variational optimisation of the density matrix isdiscussed as a method in many-body quantum mechanics. This is a relativelyunknown technique in which one tries to obtain the two-particle reduced densitymatrix directly in a variational approach. In the first Chapter we introducethe subject in the broader context of many-body quantum mechanics, and brieflysketch the history of the field. The second Chapter tries to summarise what isknown about N-representability of reduced density matrices, and derives theseresults in a unified framework. The optimisation problem can be formulated as asemidefinite program. Three different algorithms are discussed in Chapterthree, and their performance is compared. In Chapter four we show how symmetrycan be exploited to reduce the computational cost considerably. Severalapplications of the method are discussed in Chapter five. We show that thestandard two-index constraints fail in the strong-correlation limit of theone-dimensional Hubbard model. In Chapter six we identify the spin-adaptedlifting conditions as the most compact constraints that can correctly describethis limit.

Variational determination of the two-particle density matrix as a quantum many-body technique