  In this thesis the variational optimisation of the density matrix is
discussed as a method in many-body quantum mechanics. This is a relatively
unknown technique in which one tries to obtain the two-particle reduced density
matrix directly in a variational approach. In the first Chapter we introduce
the subject in the broader context of many-body quantum mechanics, and briefly
sketch the history of the field. The second Chapter tries to summarise what is
known about N-representability of reduced density matrices, and derives these
results in a unified framework. The optimisation problem can be formulated as a
semidefinite program. Three different algorithms are discussed in Chapter
three, and their performance is compared. In Chapter four we show how symmetry
can be exploited to reduce the computational cost considerably. Several
applications of the method are discussed in Chapter five. We show that the
standard two-index constraints fail in the strong-correlation limit of the
one-dimensional Hubbard model. In Chapter six we identify the spin-adapted
lifting conditions as the most compact constraints that can correctly describe
this limit.
